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ht is a point, the shadow is that which we have called tin; "geometrical" shadow, if wo neglect diffraction phenomena, as We Shall f,,r the time beiO I f, however, the SOUrce is large, object^, representing the on • tureen rG\y »n opaque ,ik|, | tl;iMi(i MI. ;m i]lllmm;lted piece of paper, the shadow phenomena are evidently not quite so simple. 426 LiailT Thus, if the large source of light is represented by AB, and the opaque obstacle by CD, the shadow cast by the waves from A on a screen FG- is limited by EG-, and that cast by the waves from B by FH. Therefore the region EH on the screen receives no light ; and the region outside FCr receives light from all points of the source ; but the intermediate regions, EF and GH, receive light from only portions of the source. Therefore, the intensity of light on the screen fades away gradually toward the central region EH, where there is no light. The space back of the screen into which the waves do not penetrate is called the "umbra," while the partly illuminated space surrounding it is called the "penumbra." An illustration is afforded by solar eclipses, where the sun is the source, the moon is the opaque obstacle, and the earth the screen. In the dia- gram, S represents the sun, and M the moon. The umbra and penumbra are indicated by dark spaces. If the earth enters this region, the eclipse is total for all points on its sur- face which are inside the umbra, and par- tial for points outside this- but lying in the penumbra. If the earth just misses the umbra, it may happen that at certain points of its surface a ring of sunlight may be seen around the FIG. 187. — Diagram showing shadow cast in space by the moon, M, owing to the rays from the sun, S, edge of the moon ; this condition is called an "annular" eclipse. Another interesting case of rectilinear propagation is given by the formation of what are called "pin-hole " images. If a small hole is made in an opaque screen, any luminous object — e.g. a building in sunshine — situated on one side of it will produce on a screen on the other side an inverted image of itself, which is comparatively sharply denned. Thus if there is a small opening at O in the screen, and A is a point of an illuminated figure, there will be a cone of light from A passing through the opening. If this meets a screen, it will make at the point /* a bright './•:\/:/,M/. i'in-:.\n.Mi-:\.\ LK.IIT i-JT spot, which will hsivi- tin- >ha|>«- of tli«- opening. If the opening is small and tin- t\\o .M-m-ns an- close together, the spot at B will be practically a point of light ; and hence, as each point of the illuminated figure pro- duces a point of light on the screen, there will be formed a well-defined image. As tin- images of the various points of the illuminated object overlap, the general appearance of the image is almost independent of the shape of the opening, if it is small. (The round or ellip- tical spotsof light which are v^ m._Pln_holeim*g<*. seen on the floors near cur- tained windows or under trees are images of the sun formed by minute openings in the curtains or leaves.) This is the principle of pin-hole photographic cameras, of the camera obscura, etc. Opacity; Transparency; Translucency. — A distinction is made between material media which are " transparent " and which are "opaque." If an object when introduced between the eye and a source of light stops all the light, e.g. a board, a piece of tin, etc., we say that it is opaque, meaning that it does not transmit those ether waves which affect our <• of sight. An object may be opaque to some waves and not to others: thus, a piece of red glass is opaque to all visi- ble waves except those which produce the sensation of red. Again, a given material body may be opaque to waves which do not afieet our sense of sight, and may transmit all visible and there may be media with just the reverse prop- . Opacity is due to the fact that the waves which are incident upon the bodv do not pass through it: they are either reflected bn-k from it or are absorbed by it. Thus, a polished metallic surface is opaque largely owing to relleetioii, while a blackened surface is opaque owing to absorption. If we can look through an object and see sources of light 428 LIU / IT on the other side sharply defined, we say that it is "trans- parent." Here, again, the body may be transparent to some waves and not to others. Waves, then, when incident upon a transparent body, will under ordinary conditions be trans- mitted by the ether in the body, maintaining a definite wave front. This is true only if the transparent body has surfaces which are " smooth," in the sense that there are no inequal- ities comparable in size with the length of the waves. Thus, a window pane of glass, layers of water or alcohol, etc., are transparent. If the surface is rough, the waves suffer irreg- ular reflection. Again, we will see later that in certain cases the waves incident upon the surface of a smooth transparent body are entirely reflected. (This is called total reflection.') It must not be thought, however, that a bod}' which is not transparent is necessarily opaque ; a piece of opal glass or of oiled paper is not transparent, nor is it opaque. One cannot see objects through them, and yet they allow light to pass to a certain extent. Such bodies are said to be " translucent." What happens is this. When waves from any source fall upon a translucent body, they are broken up and scattered by it in such a manner that each point of the body becomes a new and independent source of waves. So when such a body is held between the eye and a source of light, the waves which reach the eye come directly from the points of the body, not from the source ; and what the eye sees, then, is the surface of the body. A transparent body cannot be seen ; it is only owing to dirt on it that we are able to see a surface of water or a window pane. The explanation of translucency will be given in the following pages. Reflection and Refraction. — It has been shown that, if there are two media separated by a bounding surface, waves in one will be reflected at the boundary in general, if the velocity of the waves in the two media is different. The boundary surface must be large in comparison with the length of the waves, otherwise the waves will pass around the "obstacle." 9XNMRAL /•///•: \</.v/-; .\.i or /,/'./// 429 If the surface is large, but possesses inequalities which are in it small in comparison with the wave length, each minute I M) it ion of the surface acts like a separate reflector, and so the waves are scattered. If, however, the surface is ••smooth," in the sense that any inequalities are extremely small, the reflection is regular, and the surface is called a '• mirror." This may have any shape, but in practice the only forms used are plane, spherical, cylindrical, and para- bolic surfaces. (These curved surfaces may, of course, be either concave or convex.) Similarly, in these cases there will In' waves entering the second medium with a definite wave front. As already stated, these are called " refracted " waves. If the medium is transparent, they continue indefi- nitely. Imt if it is absorbing, they soon die down. Diffuse Reflection. — When waves fall upon a roughened surface they are broken up and reflected irregularly, as just explained: they are said to be "diffused." Each point of the surface now becomes a new source of waves. This dif- fusion of light is taking place from all natural objects, all walls, pieces of furniture, etc.; and it is owing to it that we are able to see any object. If a body reflects regularly, we do not see it, but the source of light reflected in it. Similarly, if a transparent body contains immersed in it a i numherof minute foreign particles, it diffuses the liijlit falling iip<»n it- In one such bodies the minute for- eign particles are opaque; whereas in another class they are transparent. l»ut have irregular figures. In both cases the incident waves are broken up and sutler reflection to and fro from particle to particle ; and finally t lie disturbances emerge in the foil ii of spherical waves proceeding out from each point of the surface. (If the body is thin, these waves may emerge on l»oth of its sides.) Diffusive reflection is thus due to one of three causes: etching '"' roughness of the surface, the presence of opaque particles in a transparent medium. Of that of minute irregular 430 LHHIT transparent particles in such a medium. Ground glass, ordi- nary unglazed paper, etc., are illustrations of the first class of bodies; opal glass, celluloid films, etc., of the second; and foam, thin starch water dried on glass, etc., of the third. All these bodies which reflect light diffusively, diffuse it also when they transmit it ; that is, they are translucent. If these foreign particles in a transparent medium are very minute, it may happen that the shorter waves only are affected, while the longer ones will be transmitted through the body. In this case the body would appear bluish when viewed in reflected light. This is the explanation of the blue color of the sky, of fine smoke, of a hazy atmosphere, of blue eyes, etc. In each of these cases there are minute par- ticles existing as foreign bodies in a transparent medium, which diffuse the short waves, but allow the longer ones to pass. Regular Reflection and Refraction. — We gave in Chapter XX the treatment of reflection of plane waves at a plane sur- face, quoting from Huygens. But, as noted at the time, this method is not rigorous. In order to make it so, we must follow the same plan as did Fresnel in discussing rectilinear propagation ; namely, we must divide the wave front up into zones and deduce the effect B/ N \ of the secondary waves, thus combining Huygens's principle of secondary waves and Young's princi- ple of interference. When this is done, we obtain Huygens's solution, and FIG. 189. -Reflection of plane waves by a plane also leam the effect of |
re- flection upon a ray. Thus, by Huygens's solution of the problem, plane waves, whose wave front at any instant is given in section by AB, inci- dent upon a plane surface MM, are reflected, and form the fXBAL I'llKMtMENA OF LK,11T 431 plane waves whose wa\e front at a later time is given by T)C, where the angles (BAG) and (DC A) are equal. If draw a straight line OjO perpendicular to AB and meeting the reflecting surface at 0, and another OP, per- pendicular to the reflected wave front C/D, it is seen by geometry that the broken line OjOP is the shortest line that can be drawn from P to the wave front AB by way of the surface, and so Ol is the "pole" of P (see page 382) ; and. in drawing the zones around Or which are to be com- pounded in order to deduce the effect at P, it is seen that they are exactly the same as if the reflecting surface were* removed and the wave front had the position AB1 where the angles (B'AC) and (BAC) are equal. From this fact the ordinary laws of reflection follow at once. The disturbance at P is due directly to that at 0, and this in turn to that at Oj ; consequently, the ray Ol 0 is turned into the ray OP by reflection. If a normal ON is drawn to the surface at 0, the angles ( 01 ON) and (PON) are equal ; the former is called the -alible of incidence," the latter that of "reflection." The plane including the lines 010 and ON is called the "plane of incidence"; it evidently includes also the re- flected ray OP. Reflection of li^ht may be demonstrated with ease by caus- ing a beam of sunlight, or a beam from a lantern, to fall upon an ordinary 1 MM king- glass ; for the path of the \ beams may be seen if dust 0> ;iioke is distributed through the air. ,; l.irl y, in the case of refraction of plane waves at a plane surface, FW. i90.-R««hwt«onofPun«wav««by»pton« Hu\ -.rens's solution is that. >' ]J ifl the trace by the paper of the refracting surface, and AB that of the incident plane waves, tin ., ted 432 . LIGHT waves are plane and their trace is CD, where the angles (BAC) and (DC A) are connected by the relation sin ( BA C) _ velocity of waves in the upper medium sin (DC A) velocity of waves in the lower medium This ratio is a constant for the two media and for waves of a definite wave immU'r, and is, therefore, the same for all angles of incidence ; it is called the " index of refraction " of the lower medium, with reference to the upper for this wave number. This proof of Huygens's, however, is not rigorous ; but if Fresnel's method of treatment is followed, it may be made so. If we draw a line 0-fl perpendicular to the wave front of the incident waves at Ov and continue it by the line OP drawn perpendicular to the wave front of the refracted waves, it may be shown by geometry that the time taken for a dis- turbance to pass from 01 to P along this line is less than along any other line. (Huygens proved this.) Therefore 01 is the pole of P on the incident wave front ; and by constructing Fresnel's zones around Ov we are led at once to Huygens's solution. The disturbance at P is due directly to that at 0; and this, in turn, to that at 6^; so the incident ray Of) lias its direction changed at the surface into OP. Drawing a normal NON' to the refracting surface at 0, the angle (01ON^) is the angle of incidence; and the angle (PON'), the angle of refraction. These angles are equal respectively to (BAC) and (DC A). Therefore we can state the law for a ray: the Bine of the angle of incidence = ^ index Jf refraction. the sine of the angle of refraction Calling the angle of incidence i, that of refraction r, and the index of refraction n, this relation is smr QXNSRAL rui-:.\<>Mi,\A OF LIGHT 433 The other law of refraction is evident, viz., the refracted ray is in tin- plane of incidence. Refraction may be studied with ease by allowing a beam •inlight to fall upon the surface of cloudy water in a tank. It will be observed that, if sources of light of different color arc used, the refraction is different; if the waves have the same angle of incidence, the angles of refraction are different. Thus the index of refraction varies with the wave number or color of the light. This is not easily shown in the experiment just described, unless the beam of light is made extremely narrow ; because the differences in the refraction are not great, but by means of a "prism" or a lens this phenomenon is m<>st apparent. If the incident light is white, each of the component trains of waves (see page 423) has its own index of refraction, and so the light is broken up or dispersed, forming a "spectrum." In the case of ordinary transparent media such as glass, water, etc., the waves having the shorter wave lengths are refracted more than those having longer ones; i.e. blue light is refracted more than green, green more than red. This proves that in these media short waV6fl have a less velocity than do long ones. (In the pure ether all waves, so far as we kno\v. have the same velocity.) This kind of medium is said to have ordinary or " normal " disper- sion. In other media, it may happen that some waves are refracted more than others which have a shorter wave length; they are said to have "anomalous dispersion" (see Chapter \\\ ,. When waves are passing in the ether inclosed in anv material medium, such as water, the minute particles of this medium an- in motion also to a greater or less extent, owing to the waves; so, if there is a long train of waves, and if \\e consider any one point in space, the effect produced there by the read [on «»f the matter on each •• wave" as it passes it is different from what it would he if the matter were at rest; as, for instance, if a sudden •• pulse " came up to t his point and AMI *'i i iivsics — 28 434 LIGHT passed. Since the velocity of a disturbance through the ether depends upon this reaction of the matter which incloses it, it is evident that the velocity of a train of waves is different from a pulse. Further, the method of Fresnel for consider- ing wave motion presupposes the existence of a train of waves. Thus the laws of refraction apply only to trains of waves. Geometrical Optics. — Other cases of reflection and refrac- tion will be considered in the following chapters : spherical waves upon a plane surface, plane waves upon a spherical surface, and spherical waves upon a spherical surface. There are two modes of procedure possible : one is to study the changes in the wave front produced by reflection or refraction ; the other is to study the changes in the direction of the rays, making use of the theorems just deduced ; for in the case of incidence upon a curved surface, we may con- sider the reflection or refraction of a ray at any point as due to an infinitesimal portion of the tangent plane of the surface at that point. The application of this latter method makes up what is called the science of " Geometrical Optics." We shall use this in these chapters, but shall also outline in cer- tain cases the demonstrations in terms of waves. Real and Virtual Foci. — If spherical waves diverging from a point source are spherical also after reflection or refraction, we may have either of two conditions : the centre of the reflected or refracted waves may be in the medium in which the waves are advancing, i.e. the waves converge to a "focus"; or the centre of the waves may be in the other medium, i.e. the waves will diverge away from their centre. (Of course, in the former case, the waves, after converging to a point, will diverge again beyond it if no obstacle 'pre- vents.) The centre of the converging waves is called a " real " focus ; it is said to be a " real image " of the source or " object." The centre of the diverging waves is called a "virtual" focus; it is said to be a "virtual image." There GEXKHAL PHENOMENA of LIGHT L86 are cases, however, in which, even though the incident waves are spherical, the reflected <>r retracted waves are not. Homocentric and Astigmatic Pencils. — Similarly, from the standpoint of rays, if we consider any incident pencil of rays proceeding from a point source, it will, after reflection or i-i- fraction, form another pencil with its vertex in the medium into which the rays are advancing, if there is a real focus, or one with its vertex in the other medium, if the focus is vir- tual. But there are cases when, after reflection or refrac- tion, the rays do not form a cone. A pencil of rays which does form a cone is said to be "homocentric"; while one which does not is said to be "astigmatic." In this latter case, as we shall sec, the rays of a homocentric pencil, after reflection or refraction, have as a focus (either real or virtual) not a point, but two short lines perpendicular to each other and a short distance apart; these are called "focal lines." In describing the incidence of a pencil of rays it is simplest to give the direction of its central ray; so by speaking of " normal incidence " of a pencil we mean a case when the central ray of the small pencil is perpendicular to the surface at the point where the pencil meets it; and by "oblique incidence" is meant a case when the central ray of the pen- cil makes an an.^le different from zero with the normal to the surface at the point where this ray meets it. We shall see shortly that in all eases a pencil which is normal to a surface produces by reflection or refraction a homocentric pencil; and in nearly all cases an oblique pencil produces ,in astigmatic one. Properties of a Focus from the Standpoint of Waves. — Since the locus of the points readied by the dis- turbances at any one instant, we may consider the existence of foci from a different standpoint. If \\ a ves di ver^injr from a point BOH 06 '-on verge after reflection or refraction to an- other point, we can draw various rays proceeding out from the former point and all meeting again at the latter. These 436 LIGHT rays have different paths ; but the time taken for the disturb- ances to pass along all of them must be the sa |
me. Thus, if Zj is the length of the portion of a ray in one medium in which the velocity of the waves is vv and if ?2 is the length of its portion in a second medium in which the velocity is v2, the time taken for the propagation of the disturbance is v\ , or /Z, + Sl.0-' Bufc ^ is the index of refraction, n, of the second medium with reference to the first; so this time is (li + nl^)—. The quantity (7X -f- nl2~) is called the " " of the ray. length optical is It dis- tance in the first medium which the waves would advance in the time taken for the actual propagation in the two media. Then we may state that the optical lengths of all rays from the point source to the focus are the same. We shall make use of this principle in discussing lenses. evidently equal the to Another fact in regard to foci should be emphasized. Re- flection and refraction are always produced by pieces of mat- ter of limited size, e.g. looking-glasses, prisms, lenses, etc. ; and so only a portion of the wave front undergoes the change. The effect at any point in the advance of the wave front must then be deduced by following Fresnel's method of combining the principles of Huygens and Young. The point at which the disturbance is greatest is the focus ; but this does not mean that there is no disturbance at other points. The effect at these latter points must in each case be calculated ; for some it is zero, and in no case does it approach in amount that at the focus. This fact is of great importance when we discuss the reflection or refraction of trains of waves from two points that are close together. CHAPTER XXVI PHOTOMETRY Intensity of Sources of Light ; Intensity of Illumination. — One of the most important practical questions hi regard to sources of light deals with their illuminating power. Of course only part of the energy radiated by these sources is in the form of waves of such lengths as to affect our sense of sight; and we speak of the "luminous energy" or the ••quantity of light" that is radiated. The quantity of light emitted by a source in a unit of time measures its "intensity." If the source is small compared with the dis- tance at which the light is received, the waves may be regarded as spherical. In other cases, for instance a white wall or screen, this is far from being true. In any case, however, if by means . of opaque screens with L L suitable openings in tin -in the radiation is limited to a "beam," we can lind the relation between the quantities of li-_rht falling <>n sur- faces inclined at differ- Fio. 191. — Incidence of a beam of light upon »r. nMiijue screen. ent angles to the direction of the beam. Thus, as slio\\ n in the cut, let the beam be that luwsing through t\\«- equal and parallel openings AB and AVB^ of area A; and let the beam fall upon a screen whose normal makes the angle N with the direction «.f the heam. The illuminated surface on this screen, CZ), will have an ar< here 438 LIGHT BcosN=A. (See page 28.) If the screen is a dif- fusing one, this portion of it will appear bright when viewed from any direction. The " intensity of its illumina- tion " is defined to be the quantity of light received per unit area in a unit time. Therefore, if Q is the quantity of light carried by the beam in a unit of time, the intensity of illumination, 7, of a screen perpendicular to the beam is -£• That is, 1=-^- The intensity of illumination of the 0 oblique screen is £« Calling this Iv we have, then, If the source is small, we can, as has been said, consider the waves as spherical; and, if Q is the quantity of light emitted by the source in a unit of time, assuming that it radiates uniformly in all directions, the amount falling on a portion of the spherical surface of area A at a distance r is H tne illuminated surface is small and is oblique to the direction of propagation, let its area be B and let the angle made between a perpendicular to it and the direction of propagation be N\ then the projection of this surface on a plane perpendicular to the direction of propagation has the area B cos N. So in the above formula if A is this projected area, A = B cos JV; and the light received in a unit time by the oblique surface whose area is B is — -- — — • The intensity of illumination of this oblique surface is then _Q_ cosjgV 47T r2 4?r r2 This formula offers at once a method for the comparison of the intensities of two sources of light. The general method is to illuminate a portion of a diffusing screen by one source and contiguous portions of the screen by the other, and then to vary the distances of the sources until the two portions of I'UOTOMETHY Jo'.' tin M-iven appear equally bright. When this is the case, the intensity of illumination must be the same for both portions; and, if the angles of inclination of the two sources are the i i Luminosity of Sources of Light. — If the source of light is not a point but an extended surface, like the surface of a white-hot metal or of a diffusing screen, let us consider the radiation of any small portion of the surface, whose area we may call A. If this were a point source, and it emitted a quantity of light Q in a unit of time, the quantity received in that time on a screen of unit area at right angles to the direction of propagation at a distance r, would be, as we have seen, —^—- Therefore the total amount of light actually 4 received per unit area by a screen, parallel to the luminous surface and at a distance r from it, varies inversely as r2, provided the area of the luminous surface and that of the portion of the illuminated screen considered are both small and face each other. That is, calling P the quantity of light received in a unit of time per unit area of the sen-en, P = — , where c is a constant depending upon the properties of the luminous surface. Evidently c varies directly as the area of this surface ; so we may write c = LA, or P = — , win -iv I, is a factor of proportionality. The quantity L is called tin- •• intrinsic Ininiimsit y " of tbe luminous" surface prrjirmlienlar emission." It is a constant for the given luminous surface. The light received from a luminous surface by a parallel surface of unit area at a distance r is, as just shown, —• Tin- brightness of a In in inoufl object as judged by our eyes <1- 1 M n,l u]>on the light received per unit area of the retina. Thus, if tin- light recei\.--l by the eye, passing tli.- .liaphr;i-m f..nnrd by tin- pupil. i> <v'. ami if tin- an -.1 "f tin- 440 LIGHT image of the luminous object formed on the retina is a, the brightness of the object is proportional to — • If the object is a surface parallel to the a eye and at a distance r, and if B is the area of the pupil, this light Q enter- ing in a unit of time is, from the above definition of L, — — • The image formed has the area a ; and it will be shown later in speaking of lenses that a is proportional inversely to r2; or a = — , where A: is a constant de- pending upon the construction of the eye. Hence the brightness of the 1* luminous surface, - is — — divided by -, or — — • For a given a r2 r2 k luminous surface and a given eye, this is a constant quantity; or, in words, a luminous surface appears equally bright at all distances from the eye. Certain luminous objects, for instance a star, do not produce images on the retina, as do ordinary luminous bodies ; they give rise to a diffraction pattern determined by the size of the pupil. (See page 390.) Therefore the above statement does not hold for them. A telescope will increase the brightness of a star, because it introduces more of its light into the eye, but will not increase that of the sun or the moon. Experience shows that a luminous surface when viewed obliquely appears practically as bright as when viewed per- pendicularly. Thus, a luminous spherical solid appears to our eyes like a luminous disk of uniform brightness. If the area of the oblique surface is A, and if it is inclined at such an angle that a perpendicular to it makes the angle N with a line drawn to the eye, A cos N is the area of the projec- tion of this surface perpendicular to this line ; and a surface of this area placed parallel to the eye appears of the same brightness as the one of area A placed at the angle N. Let L1 be the intrinsic luminosity corresponding to the direction N; i.e. if the area of the surface is unity, the light received Tt per unit area at a distance r in this oblique direction is — • Then the light received by the eye from the oblique surface AU B is — — — ; whereas, if the parallel surface of area A cos N were used, the light received would be A™*N' LB. But PHOTOMETRY 441 •rience proves that these an- practically equal; so L' = L cosiV. This is called " Lamhert's Law." The intrinsic luminosity in any direction of a small luminous surface is, in words, the quantity of light received per unit area by a screen perpendicular to this direction at a unit distance, divided by the area ..«• luminous surface. So, if the screen is placed obliquely to this direction, making an angle NI with it ; and if its area is />'. the light received on it in a unit of time from a lu mi nous source of area A, making the angle N with this line referred to, and at a distance r, is AcosN - L- BcosN, A I B cos N cos Nt -I or - H As noted above, this statement is not absolutely true. We may regard L as a quantity which is not a constant factor but varies slightly with N. A method is thus evident for the comparison of the lumi- nosities of different sources of light. Each is surrounded by an opaque screen provided with a rectangular opening, or slit : the sources are so situated that a suitably placed diffusing screen receives light perpendicularly from these two open- ings; one portion of the screen receiving light from one source only, and contiguous portions receiving light from the other source only. The screen is now illuminated by li'^ht coming from the two rectangular openings as sources. Then 1>\ some means the conditions |
are so altered that the hri'_rhtncss of one portion of the screen is diminished until that of the two portions appears equal. This diminution in intensity may be seemed in various ways: (1) remove one source to a greater distanc alter the widths of the rectangular slits. (:', ) interpose between the more intense sourer ;md the illuminated s.-r.-m ;m opaque disk which has certain i -it out, and cause this to revolve rapidly: by altering the si/e of the sector* the intensity may be diminished at will and in a known ratio, When the two por- tion reen are equally illuminated, they are receh 1 quantities of lj._rht per unit ,n< -\. I 'or simplicity let us assume that there is no interposing revolving disk. J*i 442 LIGHT Al be the area of one opening, L^ its luminosity, and rl its distance from the illuminated screen ; the light received by it per unit area per unit time is then — ^ ' • Similarly, giving A2, Ly and r2 corresponding meanings for the second source, the light received by the screen per unit area per unit time from the second source is — ^ Since, in the arrangement described above, these are equal, 121 = 2"^2 ; and so L1 and Lz may be compared. 1 r% Standards of Light; Photometers. —Various standard sources of light have been defined. One of these is the flame of a "standard candle," which is a sperm candle weighing one sixth of a pound and burning 120 grains per hour. Another is the flame of a " Hefner- Alteneck lamp " (which burns amyl acetate), when the height of the flame is kept at 4 cm. Still another is a surface of platinum when it is raised to such a temperature that it is on the point of melting. An instrument for comparing the intensities or the lumi- nosities of two sources of light is called a "photometer "; and the special science involved in the study is called " photom- etry." In Rumford's photometer, an opaque rod is placed close to the diffusing screen, so that two shadows are cast on it by the two sources ; the shadow cast by one receives light from the other only ; so, when the brightness of the two shadows is the same, the wished-for condition is obtained (care must be taken to have the angles of inclination the same). In Bunsen's photometer, a screen consisting of white unglazed paper, in the centre of which there is a small round or star-shaped grease spot, is placed between the two sources. Looking at this screen from either side, any portion is illu- minated by the transmitted light from the source on the other side and also by the reflected light from the source on that side. The screen is moved until the grease spot and the other portions appear equally bright when viewed from either PHOTOMETRY 443 side: and then tlu- above relation holds. For, let a be the proportion of light reflected by the unglazed paper, and b that reflected by the greased paper, and assume that there is no absorption. Then, if P1 is the quantity of light per unit area incident upon one side of the screen, and P2 that upon the other, the amount of light reflected per unit area by the unglazed portion on the former side is aPv and that received by transmission from the other side is (1 — a)P2 > similarly, the light reflected by the grease spot is bPr and that received by transmission is (1 — b)P2. Hence, when the two portions are equally bright, aPl + (1 - a)P, = bPl + (1 - &)/>„ or (a - 6) Pl = (a - 6) Py And therefore, since a does not equal 5, Pl = P2. The best photometer in use to-day is one designed by Lummer and Brodhun. For full details of these and other instruments reference should be made to some treatise on Photometry, such as Stine, Photometrical Measurements, or Palaz, Indus- trial Photometry. Naturally, the intensities of two sources of different color cannot be compared directly ; and, in general, if any two sources are to be compared, tin ir luminosities corresponding to each wave length should be investi- gated. This can be done by combining with a photometer a dispersing apparatus such as a prism. The complete apparatus is called a " spectro- photometer," the simplest and most accurate form of which is one devised by Professor Brace of the University of Nebraska. CHAPTER XXVII REFLECTION WHAT is meant by regular reflection, and by a mirror, has already been explained ; and the law of reflection for a ray has been deduced (see page 431). We will now consider several special illustrations. /N, Plane Waves Incident upon a Plane Mirror. — This is the case already discussed on page 367, and needs no further treatment here. There is one illustration of it, however, which may be described. It is that of a plane mirror which is being rotated when plane waves are incident upon it. Let the trace of the mirror by the paper at any instant be MM, and let PO be any incident ray ; draw ON perpendicular to the mirror at 0; the re- flected ray OR will make with the normal an angle (RON) equal to the angle (PON). Let the mirror now be ro- tated about an axis through 0 perpendicular to the plane of the paper; that is, about an axis parallel to the inter- section of the plane wave At the end of a certain time front with the plane mirror. the mirror will have turned into the position indicated by MlMl ; so, iHXZVi is the position of the normal, the reflected ray will be ORl where the angles (P&ZVj) and (NflR^ are equal. The angle turned through by the reflected ray is 444 Fio. 192. — Rotating mirror: the incident ray iaPO. REFLECTION 445 . This equals the difference between the angles (POR) and (POR^j that is, twice the difference between the angles (PON) and (PONJ, or twice the angle (NONJ. But this is the angle of rotation of the mirror ; so the reflected ray turns twice as fast as the mirror. This prin- ciple is made use of in many optical instruments : the sextant, which is used to measure the angle subtended at the eye of the observer by two distant points; the mirror attachment to a galvanometer ; etc. Spherical Waves Incident upon a Plane Mirror. — Let the sheet of paper be perpendicular to the plane of the mirror, and let the trace of the latter be MMr If 0 is the source of the waves, we may consider any two rays OP and OQ. l>v reflection they become PPj and QQ^ where the angles Q, (MPO) and (M^PP^ and (MQO) and (Jfi^Ci) are equal. If PPj is prolonged backwards, it will meet in the point (7 a line drawn from 0 perpendicular to the mirror. Let this line meet the mirror in the point A. Since the angles (MPO) and (MP(y) are equal, the triangles ( OAP) and (CTAP) which have the AP in common are equal ; and so Ad = (FA. The point Of is therefore a« tar In -low the surface of the mirror as 0 is above it, and is independent of the position of P\ that is, its position does not depend upon the ray which we u^e to locate it; ami tl ihe prolongation backwards of the reflected mist also pass through (X, as U evident at once I'mm geometry. It follows that all the rays from 0 \vhich are incident upon the mirror pro- 446 LIGHT ceed after reflection as if they came from 0' . This, then, is the virtual image of 0 ; and all pencils, normal or oblique, have their vertex at 0' after reflection. Therefore an ob- server looking into the mirror will apparently see a source of light at 0'. Similarly, any extended luminous object, AB, has a virtual image, A' B' , of the same size as itself and symmetrically placed with reference to the plane of the mirror. This explains the ordinary use of a looking glass. If two plane mirrors are placed close together, several images of a point source are formed by the reflected rays. Thus, if the mir- rors are at right angles, and the point source is at 0, images are formed at 0^ 02, and O3 ; the first two are the ordinary images of 0 in the two mirrors; the third is the image in one mirror formed by those rays which fall upon it after reflection at the other. It may be shown in a similar manner that, if the mirrors make an angle N with each other such that, when FIG. 194. — Image formed by a plane mirror. oorv expressed in degrees, — — is a whole number n, there are n — 1 images. (This is the principle of Brewster's kalei- doscope, in which three mir- rors are placed so that their section is an equilateral tri- angle, and broken pieces of colored glass serve as luminous objects. In this case N= 60°, so there are five images.) This case of reflection of FIG. 196. — Images formed by two plane mirrors which are at right angles. spherical waves by a plane mirror may be stated simply in terms of waves. A point source emits waves which advance in the form of ever- REFLECTION 447 mlin.u spli.-rical surfaces. Let such a spherical wave front proceed out from the point source 0; it will meet the surface at A, where OA is perpendicular to the plane surface. The waves will then be reflected. If there had been no reflecting sur- face, the wave front would have reached the position PQP' after a definite time ; but owing to reflection, the disturbance has the wave front PQ'P', where this spherical surface is simply PQP' inverted. This new wave front will therefore proceed back FlQ 196 _ Reflectlon of 8pherical waye8 by a plmne mlrror from th<> surface exactly as if it came from a point O1 below the surface, where OO' is a line perpendicular to the surface and bisected by it. Spherical Waves Incident upon a Spherical Mirror. — There are two cases to be considered, depending upon whether the concave or the convex surface is turned toward the illuminated object. 1. Concave Mirrors. — Let the centre of the spherical sur- face be (7, and the point source be 0; and let a section of the surface by a plane 0 through these two points be PRM. The line OCM, joining the centre of the spherical surface to the point source, is called the " axis of the mirror bya eoD«Teiph«ried mirror. O aod O' ar» •• conjugate A normal pencil of 197. - Formation of an Image of a point touree NV j { ' l reference *O 0. " from <> liiia then the as its central ray. Let ffF !••• any ray of such a pencil ; 448 LIGHT that is, the angle (MOP |
), and therefore the distance MP, is supposed to be small. The radius CP is normal to the sur- face at P ; and therefore, if the line PS is drawn making the angle (SPC) equal to the angle (OP (7), it is the re- flected ray caused by the incidence upon the mirror of the ray OP. The ray OM will be reflected directly back, be- cause the line is perpendicular to the mirror. Hence #', the intersection of PS and OM, is the point to which the two rays come. Further, all rays from 0, after reflection at other points of the surface, have directions which pass through 0', provided the surface is only slightly curved, and that only a small portion of the mirror around M is used. For, since the line PC bisects the angle PO-.PO' = C or, putting Jlo = u, MO' = v, MC = r, PO'.PO' = u-r:r-v. If, however, the above conditions as to the curvature of the mirror and the closeness of P to M&TG satisfied, the distance PO nearly equals MO, and PO' nearly equals MO'. That is. Hence for definite values of u and r, that is, for waves from a definite point source 0 falling upon a concave mirror with the radius r, the value of v, which determines the position of the image 0', is independent of P. It is, in the case illustrated in the cut, a real focus of 0. Conversely, if Of is a source of rays, they will after reflection converge to 0. The two points are therefore called "conjugate foci." The equation for v may be put in the form or REFLECTION 449 A simple geometrical method for determining 0' is as follows: Draw OOM through the centre of the mirror (7, draw OP to any point P near M, and OR through C parallel to it; draw a line PF so as to bisect OR at F\ where this line intersects the line OM is the image 0' '. For, as has just been shown, O1 lies on the line OM and on PS, where the angles (£P(7) and (OPC) are equal ; and the intersec- tion of PS with CR may be proved to bisect it. CR is drawn parallel to OP, and F is its point of intersection with PAS'. The angles (RFP) and (FPO) are equal, and (RFP) equals the sum of (FPC) and (FCP) ; hence (FOP) equals the difference between (FPO) and (FPC), i.e. (OP0). But (CPO) and (FPC) are equal (angles of incidence and reflection); therefore (FCP) equals (.FP(7); the triangle (CFP) is isosceles ; and the sides JV and FP are equal. P is supposed to be close to M, and therefore to R ; and so FP {Tactically equals FR. Consequently FC equals FR. Q.K.D. A special case is when u is infi- nite; that is, when the incident waves are plane, with their wave normal parallel to OCM\ hence v = -, or (X <:ts the line CM. This point is called the "prill- : cipal focus" on the line OM. Conversely, if a point source is at the middle point of CM, i.e. if u = r?, the reflected \\ aves will be plane and will liavr tln-ir wave normal parallel to CM. It is seen from the formula - + - = -, that, if u < -, v < 0. 112 T u v r 2 That [a, if the point source 0 is between the surf i the AMES'S PHYSICS — 20 450 LIGHT principal focus, v is negative; and therefore 0' is on the opposite side of the surface. This means, then, that it is a virtual focus; and so the rays from 0, after reflection, diverge "o7 away from the sur- face. It is thus shown that a normal pencil of rays gives rise on reflection to another such pencil, having FIG. 199. — Special case: 0 is nearer to the mirror M than either a real Or a virtual focus. If the pencil is oblique, this is not the case. Thus, if OPl and OP2 are any two oblique rays which are close together, after reflection they will cross at a point Fl off the axis ; and we can deduce at once the main phenomena for the whole oblique pencil of rays from 0, by rotating the plane fig- to its centre C. The image O is virtual. ure in the cut through a small angle around OM as an axis. The two rays OPl and OP2 will thus describe a cone having as its base a small rectangular IYM \X c 0 area, of which PjP2 is one of the sides. Fio. 200. — Formation of focal lines at FI and Ft by an oblique pencil incident upon a concave spherical mirror. The point F^ will de- scribe a short straight line perpendicular to the plane of the paper. So all the rays making up this oblique pencil will, after reflection, pass through two small elongated areas which are practically short straight lines: one at F± perpendicular to the plane of the paper; and REFLECTION 151 one at F^ in the plane of the paper at right angles to the reflected rays. These two lines are called "focal lines." Tli us an oblique pencil gives rise to an astigmatic one. The phenomenon is called "sjiheTicalaberration./' meaning that after reflection at the spherical surface the oblique pencil does not have a point focus. If we con- siiU-r all the rays from 0 which fall upon the concave surface, it is seen that after reflec- tion consecutive rays intersect at points which lie on a surface having a cusp at O1, the image in the mirror <>f 0 for a normal pen- cil. This surface, a sec- tion of which is shown in the cut, is called Fio. 201. — Caustic formed by reflection at * concave spherical mirror. the "caustic by reflection at a spherical surface." (Sections of this surface may be seen by looking at the reflection of a light in the mirror formed by a tumbler or cup containing milk or some opaque liquid. The cylindrical surface of tin* nip or glass forms tin- mirror, and the opaque surface is the n mi which ih«' ima^e is formed.) For the treatment : M sties by means of waves, the student should refer to the description of a method de\ised by Professor \V,>nd be .1 nh us Hopkins University, which is given in Edser, /-/•///', page 298. Formation of Images. — If there is an illuminated object " V. its ima'_;e will l>e O'N', as shown in Fig. 202, in which OP is parallel to NCR, and when- F bisects the line < 7/. K.i.-h point of (Wghes rise to an image at a point of (PIP ; and, if ON is perpendicular to the lino NCR^ (FN* will be 452 LUillT also, if ON is small. Two cases are illustrated in the cut. In one, the image is a " real " one, because the waves con- verge toward it O after reflection ; and if a screen is placed at O'N', a sharp inverted image will be formed on it; in the other it is virtual. The ratio of the linear magni- tudes of the ob- ject and image, i.e. ON is evi- dently by geom- etry equal to the OC ratio —^ i.e. to FIG. 202. — Formation of images by a concave spherical mirror. u— r r — v Owing to the relation - + - = -, this ratio equals - . O'N1' The ratio of the areas of the object and image is then ^~- 2. Convex Mirrors. — The same proof maybe carried through for convex mirror. Let O be the source, C the centre of the spherical surface, P any point of the surface near M, the intersection of OC with the surface ; draw OP and PS so as to make equal angles with the radius CPC'. Prolong SP backward and let it meet OC in 0' ; then 0' is the virtual image of 0. Further, by geometry, FIG. 203. — Formation of an image 0' of a point source 0 by a conyex spherical mirror. REFLECTION L53 Call, as before, MC = r, 3/CX = r, 3/0 = u; and the equation becomes tt:v = u — r :r — v, if Pis close to 3f. Hence, - + - = _ M i' r If CR is drawn parallel to OP, and if F is the intersection of 50' and CR, it may be proved with ease that CF equals FR, if the curvature of the surface is slight and P is near 3f. Hence the construction of images Fio. 204. —Formation of an image of an object 0.Y by a convex spherical mirror. is as shown in the cut : an illuminated object ON has a virtual image (yN'. The distances of the object and image from the mirror are con- nected as before, by the relation 1,1 21 2 1 — I- - = -, or - = --- . u v r v r u Hence, so long as u is negative, i.e. so long as O is on the opposite side of the surface from C, or the waves diverge from a point on the convex >i«l«« of the mirror, v is positive and the image lies on the same side of the mirror as the centre C. A special case is when u is infinite, that Is, when plane waves having the wave normal OMC fall upon the mirror; v = * and the virtual is therefore a point bi- secting th«- liii.-r.i/. Thin is called the " principal focus." Thus a normal pencil gives rte to a homocentrio one; and it may be shown, by following the same method as was used for concave mirror.'-, that an oblique pencil produces an astigmatic one by reflection. 454 Plane Waves Incident upon a Parabolic Minor. — This is a mirror whose surface may be imagined described by rotating a parabola around its axis ; it is called a " paraboloid of revo- lution." If the rays are all parallel to the axis, they will after reflection all converge to the focus, F, of the parabola. (To prove this requires a knowledge of the analytical prop- erties of the parabola.) In this case, then, there is no spherical aberration. Conversely, if a point source of light is placed at the focus -F, all the rays which are reflected by the surface proceed out parallel to the axis. This is the reason why such mirrors are used in search lights, the headlights of locomotives, etc. FIG. 206. — Parabolic mirrors. CHAPTER XXVIII REFRACTION Plane Waves Incident upon a Plane Surface. — This is the case already discussed on page 432. The incident plane waves give rise to refracted waves, which are plane and in such a direction that the incident and refracted portions of any ray and the normal to the surface at the point of incidence are all in the same plane, viz., the "plane of incidence"; and, if Ni and JV2 are the angles made with the normal by the two rays, sin is the same for all angles of FIG. 207. - Refraction of a r»y. incidence. As already explained, this ratio, or the index of refraction of the second medium with reference to the Jirxt. as it is called, equals the ratio of the velocities of the waves in the two media. It should be noted that the waves are sup- posed to be homogeneous, i.e. to have a definite wave num- ber. Calling v1 and va these velocities, and writing n^ l for Conversely, the this index of refraction, na> t index of refraction of the first m |
edium with reference to the sn second, n%l equals • Therefore, if v^v^ sin JV^sin NT and so ^ > N^ ; that is, the refracted ray is bent in closer to the normal than is the incident ray. This is the case illustrated in the cut. On the other hand, if V|<VP -ZV1<-ZV3; and tin- ivfrartrd ray is In-lit away t'nmi tin- normal. This 456 456 LIGHT may be illustrated by the cut, if the arrows indicating the directions of the rays are considered reversed. Total Reflection. — It is evident in this second class of refraction that, if the angle of incidence, Nv is sufficiently increased, the angle of refraction, Nv may finally equal 90°, i.e. ^. So the refracted ray just grazes the surface. When this occurs, sin JVj = 1 ; and, therefore, according to the for- mula, sin N2 = — — This angle of incidence is called the " critical angle " for the two media and for waves of a defi- nite wave number. If it can FIG. 2Ut>. — i'otal reflection. be measured, 7i2 j, or the corre- sponding index of refraction, may be at once calculated. If the angle of incidence exceeds this critical angle in value, there is no refracted ray, for the sine of an angle cannot exceed unity, and the ray suffers total reflection. The velocity of ether waves in water is less than in air, as is shown by direct experi- ment, or by the fact that a ray in air incident upon a plane surface of water is bent toward the normal. So this phenomenon of total reflection will be observed if rays are incident obliquely upon a surface of water from below. This condition may be secured if one holds a tumbler of water in such a manner that the eye looks up through the glass at the surface of water and turns so as to face an jllu- mmated object. If the direction in which one looks is suffi- ciently oblique to the surface, nothing is seen through it ; for it acts like a plane mirror. A piece of apparatus that is often used to change the direction of a beam of light, called a " totally reflecting prism," consists of a glass "1,0 ' REFRACTION 457 triangular prism whose cross section is an isosceles right-angle triangle. Light incident normally upon one of the smaller faces suffers total reflection at the hypotenuse face and emerges perpendicular to the third face. Index of Refraction. — As defined above, the value of the index of refraction of a substance depends upon its optical properties with reference to some other medium. The natural medium to which i7g prism.7 ' all indices of refraction should refer is the pure ether; and so, when the words "the index of refrac- tion " are used, this is implied. Thus, if VQ is the velocity of \\iives in the pure ether, and wlf0 the index of refraction of any substance, wx 0 = — ; similarly, w2 0 = — ) for a second substance, and therefore n2> 1 = -^- Experiments show that unity, nearly of air is very of refraction the index about 1.0003; and therefore the index of refraction of any solid <»r litpiid substance, such as glass or water, with reference to air is practically the same as with reference to the pure ether. In actual measurements of indices of refraction, it is customary to determine them with reference to air and then to in ahe the necessary correction. In the illustrations which f(»lh»\\. // will indiejite the index of refraction of a substance witli reference to air. fact that air r - shown by the unsteadiness of objects \V|HMI s»'«-n through airriiini; from a hot stovr or tirl.l : t In- ditY.'r.-nt J-T • »f the air, being at different t<-,, n-frart diffnvntly. and tin- rtT.-.-t i, thuaame as if a pane of glass whirh i> un. \. u and "streaky" is moved up and down in front of t I'hr refraction of n ini|Nirtant. too. in all a^tron«'niic;il ,>\^, 'I'll.- fa.-t that the waves travel with diff.-r.-nt fdoottiM in layers of air at iliffi-r.-nt t.-nn-ratnn-i i-< shown by the fact that li^ht is rtjltcted l.v ^nr-h lavi \|-lanation of mirage and similar phenomena depends upon this. 458 Lid I IT Special Cases. — The refracting matter is generally made into a figure with regular geometrical surfaces. There are three cases of special interest : (1) a " plate," which is a figure bounded in part by two parallel planes ; (2) a " prism," which is a figure bounded in part by two non-parallel planes ; (3) a "spherical lens,'-' which is a figure bounded in part by two spherical surfaces, and which is symmetrical around the straight line joining their centres. We shall discuss briefly the path of a ray in passing through these various figures. 1. Plate. — Let the plate be placed with its parallel faces per- pendicular to the sheet of the paper, and consider a ray incident in this plane. This is illustrated in the cut. If JVj is the angle of incidence upon one plane face and N2 that of refraction, the angle of incidence of the ray upon the FIG. 210. — Refraction of a ray by a plate . other plane face, N2', must by the laws of geometry equal the angle N2 ; and the angle of refraction, or of emergence, out into the original medium, NI, must equal JVj ; for and, since N^ Therefore the emerging ray is parallel to the incident one, but is displaced sidewise an amount depending upon the angle of incidence, the thickness of the plate, and its index of refraction. It follows, then, that plane waves incident upon a plate emerge in the form of plane waves parallel to the incident waves. The case of spherical waves will be considered later. 2. Prism. — The straight line in which the two non-paral- lel surfaces meet (or would meet if prolonged) is called the REFRACTION 459 "edge," and the angle between them is called the "angle" of the prism. Let the prism be placed with its edge perpen- dicular to the plane of the paper, and consider a ray incident in this plum-. This is illustrated in the cut. Call the angle of incidence upon the prism N-^ ; that of refraction, Nz ; that of incidence upon the second face of the prism, N% ; that of refraction out into the original medium, NJ ; that of the angle of the prism, A ; and that between the directions of the entering and the emerging rays, D. Fio. 211. — Refraction of a ray by a prism. It is evident from the geometry of the figure that the angle b« -tween the two normals drawn to the two surfaces equals the angle of the prism ; and that the following relations are trii.- : Further, JVj and NT and NJ and NJ are connected by the ivt'rart i«»n I'nrinula; and so the value of D, the "de- . " as it is called, can be deduced in terms of A, Nv and n. It \\a\rs nf dilTrnmt wave numbers (or colors) are inci- dent upon a pn>in, it is observed that it deviates these waves 460 LIGHT to different degrees, thus showing that these waves have dif- ferent indices of refraction. If n is large, the ray is refracted more than if n is small ; and therefore the deviation is great, as is evident from the cut apart from the formula ; as a result, if white light enters the prism, it is dispersed into a spectrum of colors. (With glass or water the shorter waves, e.g. the " blue ones," are deviated more than the longer ones, e.g. the "red ones.") (See Chapter XXX.) It may be seen by actual experiment, and it may be proved by methods of the infinitesimal calculus, that as the angle of incidence is varied gradually from normal to grazing inci- dence, i.e. from 0° to 90°, the deviation gradually decreases, reaches a definite minimum value, and then increases ; and, further, that this minimum deviation is obtained when the angle of incidence, Nv equals that of emergence, N^ ; in other words, when the ray is symmetrical on the two sides of the prism. Call this angle of minimum deviation D; then, since N^_ = N^ N2 = NJ, and the two formulae above become It follows that, since we may write n = • A and D may both be measured with accuracy; and so n may be obtained. (Reference for details of the method should be made to some laboratory manual.) If homogeneous plane waves parallel to the edge of a prism are incident upon it, they will therefore emerge in the form of plane waves, but will be deviated through a certain angle. The case of spherical waves will be discussed later. ni: FRACTION 461 3. Spherical Lens. — Let the line joining the centres of the two spherical surfaces, the "axis" of the lens, as it is called, lie in the plane of the paper. The section of the lens will be one of the forms shown in the cut, for a plane surface is a Fio. 212. —Different forms of lenses. special case of a spherical one. These are called "double convex," "plano-convex," "concavo-convex," "double con- cave," etc. Consider a ray incident upon the lens. It is refracted into the lens at one point and out at another, exactly as if there were minute plane surfaces tangent to the lens at these points. These minute planes constitute a prism ; and therefore a lens may be treated as a special case of a combination of a great number of pris- matic faces. (Other lenses than spherical ones are often used, especially in the construction of spec- tacles.) When Diane waves are Fio. 818. - Refraction of » r»y by a tons. incident ii] inn ;i lens, its different rays meet the surface of the lens at different angles of incidence and on emerging they are refracted out at different angles; so nothing can be said immediately in regard to the nature of the trans- mitted waves. Plane waves are a special case of spherical ones ; and the theory of the action of a lens upon the latter will be given shortly. The values of tli«- indices «.f refraction of a few substances are given in the following table: LKillT INMCKS OF REFRACTION FOR YELLOW LIGHT OF WAVE LENGTH 01. SlTMTAXOX INDBX o°c. TEMPERATUBB 0° pressure 0° 76 cm Air Oxygen Alcohol Carbon bisulphide .... Water Rock salt Flint glass Crown glass 1.000043 1.000L>9±> 1.000140 1 000297 1.000272 1 360 1 449 .624 .334 5441 .651 .524 5 6 ° ° 2 1 Spherical Waves Incident upon a Plane Surface. — Let the surface be perpendicular to the plane of the paper, and let the point source 0 lie in this plane. Let OB be any ray incident upon the surface at a poin |
t B near the foot of the perpendicular OA dropped from 0 upon the surface. Let BO be the re- fracted ray, and ° 4 2 Fw.314. — Formation of Image* of a point source 0 by a plane B01 its prolonga- tion backward until •nrfkoe: (1) when n>l ; (2) when n<l. ,• •> •, j ,-T it meets the normal OA. There are two cases to be con- sidered, depending upon whether the velocity of waves in the first medium is greater or less than that in the second. In either case let n be the index of refraction of the second medium with reference to the first. If the waves in the ° ° 5 5 1 1 ° 5 2 0° REFRACTION 403 latter have a givati'r velocity than in tlie former, n > 1 ; if thi-ir velocity is greater in the former, w<l. These two cases are illustrated by the two cuts. In each the angle of incidence equals (BOA); and that of refraction j&OA. Therefore, nnce the sine of , and the sine of (^01^L)=' the index of 0 B l refraction, n = — — . If B is extremely close to A, that is, OB if the ray OB is one of a normal pencil, we may replace the ratio Qi by -. So n = or O^A = n OA. It fol- lows, then, that Ol is a point at a fixed distance from the plane surface for all the rays of the normal pencil provided the waves are homogeneous, so that n is a constant. If n > 1, Oj is farther from the surface than 0 ; if n< 1, it is nearer the surface. In other words, a normal pencil from a point source 0 gives rise to a pencil of rays by refraction whose centre is Or the virtual image of 0. Conversely, if we imagine the directions of all the rays reversed, a normal pencil of rays in the second medium converging toward a point Oj in the first will actually meet at a point 0. A luminous object in one medium will thus give rise to a virtual image of itself. This image will not be of the same M/.r a^ the object; but their relative dimensions may be easily calculated. The treatment of this case of refraction of spherical waves at a plane surface l.y the method of waves is as follows: Let the source O be in the medium in which the waves hav. velocity, \\hich may therefor*' !"• railed the "slow medium," and let the waves meet tin- MI, face of the "faster medium" at A, as shown in -!."». lierefore perpendicular to the surface. If the velocity ..f the wav.-., in thr t\\«> media \\--r. • tin- same, the wave front would advance in a definite tim •• t<> the ].<.siti,,n PQP* • but, since the new medium is the di>turl'an.-e t, :i ,tead of Q, and the actual curve of the entering waves i- /'*,' /' . an arc of a circle whose centre n' is on the lin- O.4,but nearer the surface than O. O1 is then the virtual image of O. 464 LIGHT Consider also the case where the source 0 is in the " faster medium " ; the spherical waves meet the surface at A, the foot of the perpendicular from 0 upon the plane surface. The waves entering the other medium would have the form PQP' if the velocity were unchanged; but, since the medium is "slower," the Case when n>l. Fio. 215. — Refraction of spherical waves by a plane surface. REFRACTION 465 actual curve is PQ'P', an arc of a circle whose centre is 0', a point on ()A , but farther away from the surface than O. (/ is, as before, the vir- tual image of O. The distance between the point source and its image may be calculated from the formula already deduced : O^A = nOA. For OOj = OA - O^A = OA (1 - n). Thus, if n > 1 , the object appears to be farther from the surface than it really is ; while if n < 1, the opposite is true. This is illustrated when one looks at a stone lying at the bottom of a pond of water ; in this case 0 is at the bottom of the water, and 01 is nearer to the surface by a distance OA (1 — w), where OA is the actual depth of the water and n is the index of refraction of the air with reference to the water; i.e. it equals — , where n' is the index of refraction of the water n 4 with reference to the air. (n' for water is approximately -; OA \ so 00l is . J A method is thus offered for the measure- ment of w, since both OA and 00l can be measured. If the pencil of rays from 0 is oblique, it forms by refraction an astigmatic pencil. Thus, ^ Fw. 116. — Fnrin.itl..n of focal line* at A\ and A', by the refraction of an oblique pencil at a plane Rurface. if OB and OS are two oblique rays, they form by refraction two rays which if prolonged backward cross at Fv -,\ point off the perpendicular line OA. Therefore, the 217. — Caustic formed by reflection at a plane surface. AMES'S PHTSICS — 30 466 LIGHT whole oblique pencil gives rise to two virtual focal lines: one at Fl perpendicular to the sheet of the paper : the other at JP2 along the line OA. Further, if we consider all the rays from 0 falling upon the surface, they give rise by refraction to a virtual caustic surface with a cusp at Ov the image of 0 for a normal pencil. Special Cases 1. Plate. — We shall consider the plate made of a material in which the waves have a less velocity than in the surround- ing medium, i.e. n>\. Let 0 be the source of the spher- ical waves. The path of any ray is indicated by OB, BB', WC. 0' is the image of 0 in the first surface ; and 0" is the image of 0' in the second one. So, if the pen- cil is a normal one, all the rays leaving 0 will diverge after emerging from the plate as if they came from FIG. 218. — Formation of an image of a point source O by refraction through a plate. 0". Its position may be at once calculated. 2. Prism. — We shall consider the prism made of a material in which the waves have a less velocity than in the surround- ing medium, i.e. n>l. Let 0 be the source of spherical waves. The path of any ray is_indicated by OB, SW, WC. 0' is the image of 0 in the first surface ; and 0" is the image of 0' in the second surface. It should be observed that, if the ray ., Fro. 219. — Formation of an Image of a point IS One Of the normal pencil flource 0 by refraction through a pri8m. REFI! ACTION 467 for one surface it is oblique to the other when it is incident upon it. Therefore, in general, an incident pencil of rays will, after two refractions, emerge in the form of an astig- matic pencil; the rays will apparently come from two focal liiu's. It may be shown, however, that, if the pencil of rays is incident upon the prism at that angle which corresponds to minimum deviation, the virtual image from which the emerging rays apparently diverge is practically a point. So in this case the emerging pencil is homocentric. Owing to the difference in the index of refraction for waves of different wave number, it is evident that, if there is at 0 a source of several different trains of waves, each of them will have a different virtual image O" . The greater the value of n, i.e. the shorter the wave length, so much the more is O1 displaced from 0. The prism then disperses the light. 3. Spherical Lens. — It is evident that any ray which falls upon a lens suf- refraction •«», once on entering the lens ;ind a^ain on leaving it. We must therefore discuss, as a liminary to the treatment of the lens, the re f mo- tion of a ray at a spherical surface. There are two cases, which we shall consider separately: refraction at a concave surface and at a convex one. i. / ' «. Fio. MO. -Formation of Images of a point nourre O by refrmction at • concave spheric*! surfcoe: (1) when n>l ; (2) when n<l. a. Concave Surface. — Let the section made by a plan, tli rough the oentre of the surface O and the point source 0, 468 LIGHT i.e. through the "axis" OA, be that shown in the cuts; let OB be any incident ray, and O'B the prolongation backward of the refracted ray ; and draw (7#, which is therefore normal at B. (The two cuts illustrate respectively the cases when the index of refraction of the second medium with reference to the first is greater and when it is less than one.) The angle of incidence is (OBC) = Nr The angle of refraction is (O'BQ) = N2. Then fay the well-known trigonometrical formula, which states that in any plane triangle the ratio of the sines of any two angles equals that of the lengths of the opposite sides, we have from the triangles (OBC) and (O'BC): sin ^ : sin (BOC) = OC : BC, sin N2 : sin (BO'C) = WC : BC. Therefore, since = n, and si"^^> = ff (as is smBO'C) OB seen by dropping a perpendicular line from B upon OA). OC O'B O ' c O B ' If B is very close to A, that is, if the ray is one of a normal pencil and if the curvature of the surface issmall,we may replace the ratio by ; and so n = x - This OB OA 0 C OA formula shows that, if a normal pencil from 0 is incident upon the surface, it gives rise to a homocentric pencil with its centre at Of ; so 0' is the virtual image of 0. This can be expressed in a simpler mathematical form. Call the dis- Then tances OA, u ; V~A, v\ CA,r. OC = u - r, O'C = v — r; and » = ^H£. v — r u Hence or > PEFR ACTION n- 469 If w < 1, it may happen, therefore, for a suitable value of 0 that v is negative. This means simply that Of is in on the opposite side of the surface from 0 ; for in this formula the positive direction is that defined by r, namely from 0 to A. b. Convex Surface. — The same treatment and the same formulae as those oc (y A above lead to the same result for a convex surface, viz., n = — — • — — . L/C OA And since «= OA and r = CA, PC = OA + AC = OA - CA. Hence, as before, OC = u — r, and O'C = v — r ; and on substitution, we again have = , or - = - + r v u v u r If O is on the opposite side of the surface from C, u has a negative value. For instance, if in the cut the length of the line OA is x, the value of u is — x. If n' is the index of refraction of the first medium with reference to the second, n' = - ; and the general formula becomes n't; u n'r & u l=£, or ! = !' Conversely, if a pencil of rays in the second medium is converging apparently toward a point C? in the first nn-dium, tlu-y will actually meet at the point 0. Fio. 221. — Formation of an Image of a point source 0 by ™*™"™ " • <*>«"«* "P»>w leal nurfa*, ease when n>l the pencil of rays is oblique, it gives rise to an astig- matic pencil |
; and if all the incident rays are considered, the image is a caustic surface. This phenomenon is said to be • In* •, as in other similar cases, to spherical aberration. We shall now return to the problem of the refraction pro- duced by a spherical lens. It is evident from what has just been shown that a normal pencil from any point on the axis 470 LIGHT of the lens will give rise to a homocentric pencil emerging from the lens after the two refractions ; the image of the source, though, may be either virtual or real. We shall deduce the formula for a double convex and for a double concave lens, and then show that, by a suitable agreement as to signs, one formula may be used for all lenses. We shall assume at first that the lens is so thin that a ray incident at any point of one surface emerges from the other surface at a point which is at the same distance from the axis as is the former. a. Double Convex Lens. — Consider a section through the axis. Let 0 be the point source. FIG. 222. — Formation of an image of a point source 0 by refraction through a double convex lens. Let PJ be the radius of the first spherical surface of the lens. Let r2 be the radius of the second spherical surface of the lens. Let n be the index of refraction of the lens with reference to the surrounding air. The formula for refraction at the first surface of the lens *>i ui ri is, then, — = h n ~~ , where u* and v* are positive if 0 and its image lie on the right of the lens, because the centre of the first surface of the lens lies on this side. The refraction REFRACTION 471 at the second surface is from the lens out into the air ; so the formula is — = — h — — , where u2 and v2 are positive if the v2 "2 r2 points to which they refer are to the left of the lens. But, if the lens is thin, v1 = — w2, for the rays incident upon the second surface are those diverging from the virtual image produced at the first ; but a quantity u or v which is positive with reference to one surface is negative with reference to the other, since their centres are on opposite sides of the lens. Therefore we have the two formulae : n _ 1 t n-1 »! tij r, 1 _ n | 1-n V9 t>! r II. -nee, I + l = -(n- ti, t;, In this formula, MX is the distance 0 lies on the axis to the right of the lens (referring to the cut) ; and v2 is the distance the image produced by the second surface of the lens lies to its left. Therefore, if we agree to call the distance the point source lies to the left of the lens w, and the distance the image lies to the right of the lens v, u = — ur v = — va. So the formula becomes Tin; quantity on the right-hand side of the equation is a con- staiit quantity t<u a -ivcn lens, and it is essentially positive if n>l, as it is in all ordinary cases, e.g. glass, quartz, etc. lenses surrounded by air. We write this quantity — ; and the formula then assumes the final form * 472 LIGHT b. Double Concave Lens. — In this case the formulae for refraction at the two surfaces are, as before, where u^ and v1 are positive if 0 and its image in the first surface lie on the left of the lens ; and Fio. 223. — Formation of an image of a point source 0 by refraction through a double concave lens. where uz and v2 are positive if the points to which they refer are on the right of the lens. Hence, or, introducing the same agreement as to signs as in the pre- vious case, u = uv v = v2 ; and hence u v f We can therefore use for both kinds — in fact, for all kinds — of lenses the one formula, 1,1 1 in which u and v have the meanings agreed upon above and / is positive for some lenses and negative for others. We RJCFBAi Y70.Y 473 shall discuss presently the properties of these two types of lenses. The formula for a lens may be deduced in another manner, possibly more instructive, if we assume the fact that a point source on the axis gives rise to a point image. The principle made use of in this second method is that the " optical " lengths of the paths from the point source to its image are the same along all rays. (See page 43G.) Fio. 224. — Refraction by a lens. Consider the case of a thin double convex lens, as shown in the cut on an enormously magnified scale. Let 0 and & be the source and its ", both on the axis ; so 0.4, AB, BO1 is the normal ray from 0. Let OC be any incident ray making a small angle with the axis. Since the I'M i- is assumed to lie thin, the portion of the ray within the lens, namely <'/>. is parallel to the axis; an.l th«- other portion is DO'. Then the " optical " lengths of the two rays are ~OA + nAB + Wf and ~UC + nCD+ D&. Since these are equal, UA + nJB + BV = UC + nCD -I- D&, or U&-OA +D0 -E& = n(AB - CD) = n(AE+ 7-70. to the. *ign* of the. lines, this may be written OC-UE + AE+D&-FO' + FB = n(A$ + FB), or <tE) + (S& - FCO = (n _ \)(AE + FB). 474 LIGHT These three quantities may be expressed in a simple form provided the angles between the rays and the axis are small, as we have assumed to be the case. Thus, describing an arc of a circle RS around P as a centre, and drawing the two radii PR and PS, and the perpendicu- lar line RQ from R upon PS, we have the following formula : FIG. 226. — JiS is portion of a circle whose centre is P. Hence, if PR is large compared with RQ, we have, on expansion by the binomial theorem and neglecting small terms, 2 PR 2 PS 2 PR* PR-PQ = - Therefore, or. In the formula, as given above, for a lens, then, 2 PS 2 P R ' O f f -S 1 CT? Consequently, ± + ^ = („ - 1)(I + I). If the incident ray makes a sufficiently small angle with the axis, and if the curvatures of the two surfaces are small, we may replace OC by OA, i.e. u, and D<y by BO1, i.e. v ; and we have the general formula, as before, rj rzl f REFRACTION 475 We shall now discuss the two types of lenses referred to above : for one/ is positive ; for the other, negative. a. Lenses for which f is positive. — The general formula is A -f _ = _. Let us consider several special cases. u v f If the point source is removed farther and farther from the lens, but kept on the axis, u approaches an infinite value and the waves be- come plane as they reach the lens. When w = oo, it is seen from the for- mula that v =/; and, since / is positive, this means that there is a real imae'e F10-226- — Special case: the source 0 Is at an Infinite distance on at a distance / the axis. from the lens. This point is called a " principal focus " of the lens. We may express this fact in words by saying that plane waves advancing with their normal parallel to the axis are changed by the lens into converging spherical waves whose centre is at a distance / from the lens ; or, rays parallel to the axis are bent by the lens in such a manner as to converge to a point on the axis at a distance/ from the lens. Similarly, if w=/, it is seen from the formula that 0 = 00. In words this states that, if the point source is on the axis of the lens at a distance from it equal to /, the diverging spheri< -al waves are so converged by the lens as to become plane and to advance in a direction parallel to the axis; or, we may say that rays through a point on the axis at a dis- tance / from it are SO detleeted by the lens as to become parallel tn its axis. Conversely, we can have plane waves incident upon the lens from the other side, which will con- verge to a point at a distance/ from it. There are thus two •I7»i ( LIGHT principal foci ; one on each side of the lens, and at the same distance / from it, if the lens is thin. This distance is called the " focal length." Therefore, if we consider the point source as at an infinite distance from the lens, its image is the principal focus on the other side; and, if the source approaches the lens, the image recedes from it, until, when the source reaches the principal focus, the image is at an infinite distance. When the source is between the principal focus and the lens, i.e. when w</, it is seen from the formula that v < 0, so the image is on the same side of the lens as is the source, and is, therefore, virtual. When the source reaches the lens, i.e. when u = 0, v = 0 also ; so object and image coincide. If u has a negative value, the physical meaning is that rays are converging on the lens apparently toward a point on the other side, which may be called a " virtual " source ; and, in this case, as is seen from the formula, v is positive and less numerically than u ; but it should be noted that v does not exceed / in value so long as u is negative. Therefore, the converging rays are converged still more, and form a real image. It is thus seen that a lens for which / is positive always converges waves which fall upon it; for this reason it is called a "converging lens." The ordinary form of a con- verging lens is double convex ; but any thin spherical lens thicker along the axis than elsewhere is a converging one. We may arrange in tabular form the facts proved above in regard to u and v : U = CO . . . . V = f °°>w>/ . . . . *>>v>f U =f . . . . V = 00 />M>0 .... «>>-l>>0 0>w>-cc . . . />y>0 It is a simple matter to find by geometrical methods the position of the image of an object, for we know the effect of UK FRACTION 177 the lens upon three of the rays from any point source : A ray parallel to the axis is deflected so as to pass through the principal focus on the other side of the lens ; a ray passing through the principal focus on the " incident side " of the lens will emerge on the other side parallel to the axis ; a ray meeting the lens at the point where it is intersected by the FIG. 887. — Formation of images by » converging lens. , i.e. the "centre of the lens," keeps its direction unal- tered, because at tliis point the two faces of the lens are par- allel, and it is equivalent to an infinitely thin plate. In showing graphically the formation of images l>y a lens, we shall represent the lens by a straight lino, lot- simplicity. Thus, let Fr F.,. and C !><• the principal foci and the centre <>f the lens, and P any point of the object; its image is atP'. T |
wo cases are xlmwn : if /'is farther from the lens than I he principal fnens. the image is real: if /' is between the lens and the principal focus, the image is virtual. 478 LIGHT If the object is small, close to the axis, and perpendicular to it, as represented by OP, its image O'P' is also perpen- dicular to the axis. So 0' is the image of 0\ and, since these points are on the axis, 00= u and C0' = v. Further, by similar triangles, P 0 : OP1 = u:v. The ratio of the length of the line O'P' to that of OP is called the " linear magnifica- tion " of the lens. It is evident from the geometry of the cut that Therefore, the magnification of the surface of any portion of C O O P O ' P ' the object perpendicular to the axis is ^- ' O C A case of special interest is when the source, P, is placed in a plane perpendicular to the axis at the principal focus, or FIG. 228. — Special case : the point source P is in the focal plane of the converging lens. in the "focal plane," as it is called. It is seen from the geometry of the cut that the two emerging rays which we can draw from known principles are parallel to the line join- ing P to the centre of the lens, (7; for the triangles (P-Fjtf), (PAC), and (ACF^) are all equal. Therefore, all the rays diverging from a point in the focal plane, which meet the lens, emerge parallel to the line joining this point to the centre of the lens ; or, in other words, spherical waves diverg- ing from such a point are made plane by the lens and proceed in the direction of the line referred to. Conversely, if we REFRACTION 479 these ravs reversed, parallel rays incident obliquely upon the lens. Imt. at a small angle, converge to a point in the focal plane where this plane is intersected by a line through the centre of the lens parallel to the beam of rays ; or, plane waves incident obliquely at a small angle upon a lens are brought to a focus at a point in the focal plane, as just defined. P, F,. Fi... 229.— Construction for the refraction of any ray PQ by a converging lens. This fact enables us to construct at once the refracted por- t ion of any incident ray. Thus, let PQ be a ray incident at @ upon the lens, whose principal foci and centre are Fv Fy and (7. Draw the focal plane through Fv and a line BC through 0 parallel to PQ\ this meets the former in a point A. Draw the line QS, joining Q to A\ this is the emerging ray. Kor, if there were two parallel rays PQ and BC, they would converge to A ; and therefore PQ must produce a ray which passes through A. Again, r, f. rring to Fig. 230, let there be two beams of parallel rays incident u]..,n the lens, one having the direction /'r. the other tiny \\ill be brought to a focus at Q and Qv in the focal plant-. (I hi- i- the case when an image of a ilistunt object is formed on a screen or photographic plate by a l<-n-. t'or ••arh point of the object sends out ;ippp. \imal. •!> parallel to each oth«-r \\licn they i-.-ach tlir lens.) F« D "angular separation " of the two beams, i.e. (/'CV,) tli- 'I f the focal length of th«> !-•: increased. If this angle is small. is the focal length, the length of the line Q(^ <«quals the | / l.y this nngl.-, for the an (PC/*,) an- 1 C<"\',) an- t-.jnal. If these two rays com*' from t\\o| 480 LIGHT on the edge of the distant object, its image will be bounded by the two points Q and Ql ; and the linear dimensions of this image will vary directly as the focal length of the lens. Therefore the area of the image will vary as the square of the focal length. If a lens having a long focal length is used, the size of the image is great; but, if a pho- tograph is to be taken, the time of exposure must be prolonged because FIG. 230. — Formation by a converging lens of an image of an object at an infinite distance. the energy is distributed over a large area. b. Lenses for which f is Negative. — The general formula is - -f- - = -^ . If the source is at an infinite distance, u = oo, and therefore v=f; but since f is negative, v is also, and the image is a virtual one on the same side of the lens as the incident waves. This is called a " principal focus " ; and its distance, /, from the lens is called the " focal length." Similarly, there is another F vx Fro. 231. — Special case: the point source is at an Infinite distance on the axis. principal focus on the axis on the opposite side of the lens, and at the same distance from it if the lens is thin. If u=f, i.e. if the rays are converging apparently toward the principal focus on the opposite side of the lens, it is seen from the formula that v = oo, i.e. the emerging rays are all parallel to the axis. We can, moreover, in a similar manner to that used in the previous case, discuss the relation between the positions of the object and image as the point source moves from -f- oo to — oo. It is seen at once that the effect /,'/•;/••/,'.!' T/o.Y 481 of the lens is to make the incident rays or waves diverge; for that reason lenses of this type are called u diverging lenses." All thin lenses which are thinnest at their centres are diverging. We can also deduce graphically the position of the image of any point source, because we know the effect of the lens upon three ravs : a ray parallel to the axis emerges in such a direction that if pro- longed backward it would ^ F Fio. 282. — Special case: the Incident rays are converging toward the principal focus on the farther side of the lens. meet the axis at the principal focus ; a ray pointed toward the principal focus on the other side of the lens emerges parallel to the axis; a ray through the centre of the lens retains its direction unchanged. A few cases will be drawn, the lens being represented as before by a straight line. F , O ' Pia. 988.— Formation of an Image by a diverging lens. A real source P gives rise to a virtual image P' ; and a small object O/' perpendicular to the axis has an ini.r •< 0 /' also perpendicular to the a\ Thelinearmagnification prod viced by the lens, that is the ratio ' > /' r r2 — , equals as before - : and the surface magnification is -a. Vr a u* AMES'S PHYSICS — 31 482 LIGHT A case of special interest is when the virtual source is in the focal plane. See Fig. 235. Let this point be P ; draw FIG. 234. — Formation of an image by a diverging lens. The "virtual source" P has a real image P', provided P is between the lens and the focal plane. two rays pointed toward it, one parallel to the axis, the other through the centre of the lens ; it is seen by geometry that the emerging rays are parallel, for the triangles (^(74), ((TAP), (CF2P) are all equal. Conversely, rays which are parallel to each other and inclined slightly to the axis diverge, Fio. 285. — Special case: the virtual point object P lies in the focal plane. after emerging from the lens, as if they proceeded from that point in the focal plane on the incident side where a line through the centre of the lens parallel to the rays meets the plane. This furnishes us with a method for the construction of the emerging portion of any ray. Let PQ be a ray meeting REFli ACTION 4K3 the lens at Q : draw through (7 a line CB parallel to PQ, ami intersecting the focal plane at Fliu AI draw through A the line AQSi QS is the continuation of the incident ray. For, if there were two parallel rays PQ and BC, they would diverge, on emerging, as if they came from A. F, \S Fio. 286. — Construction for the refraction of any ray PQ by a diverging lens. Spherical Aberration, etc. — It may not be useless to state again the assumptions made in the above treatment of lenses: the lens is supposed to be thin; the object must be small and close to the axis; the pencils must be normal; the lens must have surfaces whose radii are large in comparison with its dimensions. If any of these conditions are violated, the laws cease to hold. Reference should be made to some special treatise such as Lummer, Photographic Optics, for a full discussion of the general subject. We have also assumed ttimughout that the waves were homogeneous, for n has been treated as a constant. Resolving Power of Lenses. — It has already been explained page 436) that when the waves from a point source fall upon a converging lens, the image of this source is the point to which liy far the greater amount of the energy is brought, but that owing to diffraction energy is relieved by other points also. In the case of a lens with a circular edge — such as most lenses have — the diffraction pattern consists of a bright area, whose brightest point is the geometrical image, 484 LIGHT as just explained, and this is surrounded by rings alter- nately dark and bright, if the waves are homogeneous; that is, the light gradually fades, then increases gradually, etc. So, if there are two point sources close together, the line joining which is perpendicular to the axis, there will be two diffraction patterns, which will overlap; and the resultant effect is due to their superposition. It is evident that if the two point sources are so close together that the centres of their diffraction patterns almost coincide, it may be impossi- ble to see these centres as separate bright points; but, if the centre of one pattern coincides with the first dark ring of the other, then the existence of two bright points may be recog- nized. There is thus a limiting value of the nearness of two point sources which can be perceived as such by the use of the lens. This quantity may be deduced by considering a simple case. Let AOB be the cross section of a converging lens by a plane through the axis, and let 0' be the image of 0, a point on the axis. 0' is the image of 0 because the disturb- ances along the 0' different rays from the latter reach the former in the same time, and so are in Fio.287.-Diagramtoillu8tratethere8olvlngpowerofalen8. the game phage Qf vibration; but, if a point near 0' is considered, the different disturbances from 0 reach there in different phases, because they pass over different "optical" paths; and it may happen that |
the difference in path is such that owing to the disturb- ances arriving there in opposite phases they annul each other's action. If this is the case, this point is on a dark ring. If there is another point source at P, where OP is perpendicu- lar to the axis, the disturbance produced by its waves at 0' REFRACTION 485 may be calculated in a similar manner. Draw PA and A&, PB and BO\ these are the two lines from P to (7 whose difference in length is the greatest. If this difference amounts to a whole wave length of the waves, it may be assumed that the resultant action at O1 due to the disturbances from P which pass through one half the lens differs in phase by half a wave length from that due to those which pass through the other half of the lens; and therefore 0' will be on the first dark ring of the diffraction pattern due to P. From what was said above, when this is the case, P is as close to 0 as it is possible for it to be and yet to be seen separate from it. This condition may be expressed in an equation: (PB + B&) - (PA + ZO7) = /, where I is the wave length. But 0' is the image of 0\ and, therefore, drawing 03 and OBy we have, since the optical lengths from 0 to O1 are equal, _ (OB + BO') = (OA+AOf). Subtracting this equation from p the previous one, we have ^ ^ (PB-OB) + (OA - PA)=l. These quantities may be ex- *»•««• -Portion of Fig. pressed in a simpler form. Draw- ing the diagram on a larger scale, and dropping a perpen- dicular 1>(J tmm P upon OA, it is seen by similar triangles _ lint since OP is extremely small, 0Q = &A -PA. It may be shown in a similar manner that 486 LIGHT Hence, substituting in the above formula, it follows that 2dP^ = l,oTdP = l -2fL = / °A. OA 2CA AB This marks the minimum value of OP which allows us to recognize the two independent point sources 0 and P. The 1 2 OA. 1 A.B , reciprocal of this, — , or - — or - — -, is called the " re- I (JA. I (JA solving power " of the lens. There are two cases of special interest : one when we are considering two luminous points of an object near at hand ; the other when the two sources are at a great distance ; one deals with the power of a micro- scope to " separate " fine details of a structure ; the other, with the power of a telescope to separate the two components of a "double star" or to recognize the details of the surface of the moon or sun. The same formula applies to both, but it may be put in more convenient form. 1. Microscope. Call the angle (AOC), N; then ^4 = sin JV; hence (JA 1 = 2sinjy OP~ I Therefore, the larger the angle subtended at the object 0 by the lens, so much the greater is the resolving power, and therefore the lens should have a short focus, so that the point 0 may be close to the lens and yet may have a real image. Further, the shorter the wave length the greater this resolving power. Thus, fine details may be seen with a microscope if the object is illuminated with blue light, which are not distinguishable in red light. 2. Telescope. Since in this case the object is at a great distance, OA OP c/C equals OC, practically ; and — equals the angle subtended at the centre of the lens by the distant object. Call this angle M ; then from the formula OP = I Qj- it is seen that M = J-. This is the smallest angle which two objects can subtend and yet be seen as separate points of light in a telescope. — = - is called the "angular resolving power." AB is the diameter of the lens; and we see the advantage then of using telescopes with large lenses (or mirrors) if one wishes to separate double stars, etc. RKi'n.\< IION 487 (This elementary treatment of the resolving power of a lens is due to Lord Rayleigh. A rigid treatment leads, as he and others have sho\\ n, to the formula OP = 1.22 / QL) . I it In using high-po\u-r microscopes, it is customary to fill the space between the lens and the object with a liquid which has a large index of refraction — oil of cedar wood is generally used. (This oil has nearly the same index of refraction as the glass, so no light is lost by reflection at the surface of the glass; and, further, spherical aberration is avoided to a great extent.) Then the formula as deduced above must be modi- fied, because the optical paths of the rays from P and 0 include portions in a medium whose index of refraction, n, is different from that of air. Thus, (nPB + BO1) - (nPA + A&) = /, (nOB + BO") - (nOA + AV) = 0. Hence, n (PB - OB) + n (OA - PA) = I', so the resolving power of the microsco]*' is increased in the ratio of w : 1. n N is called the " numerical aperture " of the lens.) The greatest value N can have is ?, for which the sine is 1 ; so the least possible value of OP is -— , or, very approximately, one half the wave length of • — the light used. No combination of lenses can therefore enable one to see separately two independent luminous points which are closer than thU together. Combination '/'>;> Thin Lenses. — L<-1 the two lenses be at a (listain. // apart; and let their focal lengths be fr fy The formula' I'm- tin- i\v<> lenses are: 111 111 — +-=-, tod - + — =-i w, ', w, v, /s 488 LIGHT where uz — h — vr To find the focal length of the combina- tion on the right-hand side, with reference to Fig. 239, put MJ = oo. Hence, v1 =fr u2 = h — /x ; and f* or, Therefore, the focal length y2 = FIG. 289.— Formation of an image by a combination of two thin lenses. A special case of this is when the two lenses are placed close together, i.e. when h = 0. Then the focal length equals 0' ; or, calling it /, 1 = 1 + -i- /l-r/2 / /I /2 The reciprocal of the focal length of a lens (or of a combination of lenses) is called its "power." The unit of power adopted by opticians is that of a lens whose focal length is one metre; it is called a " diopter." To find the power of any lens, then, in diopters, its focal length in metres must be measured, and its reciprocal taken. A converging lens is called positive. Thick Lens. — If the lens is of such a thickness that it cannot be assumed to be "thin," in the sense in which this word has been used above, the solution of the problem of re- fraction can be obtained, exactly as in the case of a thin lens, by considering the refraction of a pencil of rays at the two surfaces. But the distance from the first surface to the image produced by it no longer equals its distance REDACTION from the second surface ; that is, v1 does not equal wa numerically (see page 471). If t is the thickness of the lens, M2 = — (Vj + j); and, if this value is substituted in the equations, the iinul formula connecting the positions of the object and image may be deduced. Chromatic Aberration. — Attention has been called re- peatedly to the fact that the index of refraction of a sub- stance is different for trains of waves of different wave number, showing that the velocity of these different waves in the substance is different. (It should be remembered that the wave number of a train of waves does not change as it passes from one medium into another, e.g. air into glass, for the number of "waves" reaching any point in a given interval of time must equal the number that leaves it. But, calling N this wave number, and Fi, L, K, L, the velo- Fio. 240. — Chromatic aborrati-n. cities and wave lengths in the two media, l\ = Nlr V2 = JV72 ; and since Vl is different from I',. /! is different from lv So, as the waves pass from one medium into another, their wave length changes.) In the case of ordinary dispersion the shorter waves are refracted more than the long ones ; or, in other words, the index of refraction varies in an inverse manner from the wave length. Methods for the study of tin- connection between these two quantities will be described in a later chapter. This fact that n varies with the. wave length is of great importance in dealing with the theory of lenses; for this quantity enters into the two fundamental formulae: and Linear Magnification =-. Therefore, if waves of different wave length are used to illu- minate a Ljiveli nl.j.-ct, or if tllC object itself emits SUci 490 LIGHT not alone will the corresponding images be at different dis- tances from the lens, but they will also be of different sizes. Since light waves of different wave length correspond to different colors, this phenomenon is called " chromatic aberra- tion." For ordinary substances n is, as has just been said, greater for blue light than for green, etc. ; and so, if there are two trains of waves corresponding to these colors, / is less for the blue ones than for the green, and consequently the principal focus for the former is nearer the lens, and the magnification is less. If the object emits white light, there will be a series of colored images of different sizes. This fact is, naturally, most detrimental to the proper use of an optical instrument, for sharp clearly defined images are de- sired. As will be shown in the next paragraph, it is possible by a combination of two lenses of different material to remedy this defect to a certain extent by causing any two definite trains of waves of different wave length to be brought to the same focus, but waves of all wave lengths will not be simi- larly affected. So, when white light is used, if two of its components are thus brought to the same focus, the other components will have different foci — they form what is called a "secondary spectrum." Thus chromatic aberration can be corrected only partially. The choice of the two trains of waves which shall be brought to the same focus is arbi- trary; if the instrument is to be used for visual purposes, two trains are chosen which affect the eye most intensely ; while, if it is to be used for photographic work, the two trains of waves chosen are those which act most intensely upon a photographic plate. Achromatism. — Since a lens can be considered as made up of prisms, we shall first show how one prism may be so chosen as to neutralize the dispersive action of another prism for two given trains of waves. In speaking of the effe |
ct of a prism, it was shown that the deviation produced depended upon the material and angle of the prism, the wave number of the REFRACTION 491 waves, and the angle of incidence. The difference in the de- viations for t\v<» di tie rent trains <>f waves is called the "dispersion" of those waves; and by suitably choosing the material and angle of the prisms and the angles of incidence, it is evidently possible to secure the same dispersion for two dftiiiite trains of waves. When this is done, the dispersion of any two other trains is as a rule different ; for owing to the difference in the material of the prisms there is no connection Fio. 241. — Two different prisms of different materials may produce the same dispersion of two rays lt and /,. 1 "-tween their dispersions in different parts of the spectrum. 'Hi us suppose that, when, white light is incident upon two prisms at angles /and /', as shown in the cut, a ray is dis- persed in such a manner that two of its components, of wave length Jj and lv have the same dispersion D. Let the angle hctween the emerging ray ^ and the normal to the second face of the prism be N and N' in the two prisms. Then if abeam of parallel rays of wave length Zj are ineident at an angle N' upon the second face of the second prism, they will emerge, alon<_r the direction of the incident ray in the cut, i.e. making the angle /' with the normal : and. if a beam of parallel rays of wave length l^ is incident upon the same face at an au^le (N1 + D), they will emerge in the same direction as the former rays. This condition may be secured by inverting the first prism and placing it with its edge parallel to that of the other pi-ism, l.iit M inclined that the angle of incidence upon it of the i. iv /, IV.. m the scc,,nd p] N\ f..r. as is seen from geometry, the angle of incidence <»t the ray £s is, 492 LIGHT under these conditions, (N -f D). Therefore a ray incident upon the second prism at the angle I1 is dispersed by it ; and two of its components, ^ and Z2, fall upon the first prism, are refracted by it, and emerge parallel to each other, making an angle I with the normal to the last face of the prism. Therefore a beam of parallel rays of white light incident upon the second prism at an angle I' will be dispersed by the two prisms ; but all the rays of wave length l^ and £2 will emerge parallel to each other in the direction defined by /. In practice it is found _ja that it is possible to choose the prisms so that, when combined as described above, FIG. 242. — One prism may neutralize the disper- their adjacent faC6S may be sion of two rays produced by another prism. n i i • j_i parallel ; but owing to the differences in the material and angles of the prisms the directions of the incident and emerging parallel rays are not necessarily the same. Therefore this double prism still deviates the two rays ^ and Z2, but does not disperse them. Such a prism is called an " achromatic " one ; or, either one of its component prisms is said to be "achromatized." (It is evident also that two prisms may be compounded which will not deviate one particular train of waves, but will deviate the others, thus producing dispersion.) It follows at once that, by combining two lenses of different materials, one diverging, the other converging, a double lens may be secured which will deviate two rays of definite wave length, but not disperse them. Such a lens is shown in the cut. The converging lens is as a rule made of "crown" glass ; the diverging one, of " flint " glass. It is called an achromatic lens for the two definite trains of waves. FIG. 248. — Achromatic lens. FLINT QLA88 CROWN_GUSS REFRACTION 41 »o If the component lenses are thin and have focal lengths /! and /2, the focal length of the combination is given by — = — | Therefore, we can express the facts in regard / /i /2 to the achromatic combination by saying that / has the same value for the different values of n corresponding to l} ami /.,. In the case of this combination of thin lenses, not alone are the focal lengths the same for the two trains of \\a\rs, but also the magnifications. This is not true if the combined lenses are thick. If a system of several lenses — not in contact with each other — is to be completely achro- matic for two definite trains of waves, each of its component lenses must be achromatic also. The first achromatic lens was made by John Dollond, a London optician, in 17.~>s. Previous to this, as early as 17*29, a method for achromatizing lenses had been discovered by Chester More Hall of Essex, England ; and he constructed several such lenses, but never pub- lished any account of his work. Newton and others had maintained that if one prism corrected the dis/n-rsinn of another, it would at the same time annul the deviatitm, and that therefore it would be impossible to make an achromatic lens. But when different kinds of glass were tested, it was found that the generalization drawn by Newton from his experiments on prisms of glass and water was not correct. Achromatic Combinations. — It may be proved that if two thin lenses of focal lengths /J and /2 of the same kind of glass are placed at a distance h apart, where h = •* ! , the combination has the same focal lengths for waves of all lengths. The distance of the principal focus -Jd h -f\ -ft fn.m the end lens of focal length /* is, as we have seen, {* ' ~ ^ an«l the distance of the oth«-i- principal focus from the other end is ft There are two special combinations \\i,i,-h an- <>f interest because of th-ii DM microscopes or telescopes. 1. / <'c?. — In this tli.-n- are two plano-convex lenses of the same kiml oi t!n« same fix-al l.-n-th placed "ith th.-ir convex faces toward each other. The condition for achromatism is that 4U4 LIGHT their distance apart h = ~^=fv since fi=/2- Therefore, any scratches or dust particles on the surface of either lens would be seen if one were to look through the other with the eye accommodated for an infinitely distant object, as is the case when one is using a telescope. Therefore it is better to make the distance of the lenses apart slightly less, although by so doing the chromatic aberra- tion is not exactly corrected. Thus, if k = $/j, the distance of the prin- cipal focus from either end is Fio. 244. — A Ramsden eyepiece: the focal lengths of the two lenses are equal. In this case, any object placed at this distance from one of the lenses will be seen on looking through the other lens at apparently an infinite distance. In other words, this eye- piece has the focal properties of an ordinary converging lens of this focal length, but it is approximately achromatic. 2. Huygens eyepiece. — In this there are two plano-convex lenses, placed as shown in the cut, with focal lengths /, and /2, where /: — %f.,. In order to secure achromatism their distance apart should satis- fy the equation h = The distance of the principal focus from the first lens is given by /l(2/2~/2) _ 3/2 /2 2/,-/,-/« 2 - 2 This means that, if a ray is incident upon the first lens at such an angle as to be pointed toward a point between the lenses at a dis- tance £ from the first lens, it will emerge from the second lens parallel Fio. 245. — A Huygens eyepiece : the focal length of the first lens is three times that of the second. to the axis. Therefore, since the object must be virtual in order to have an image at infinity, this eyepiece cannot be used, like a converging lens, to magnify ordinary objects. CHAPTER XXIX OPTICAL INSTRUMENTS OPTICAL instruments, so called, are pieces of apparatus so designed as to make use of light waves in order to form images of luminous objects. In general they consist of com- binations of mirrors, lenses, and prisms; but some instruments do not have such parts. The number of these instruments is great, but in this chapter only the simplest and most gen- erally important ones will be described. Optical instruments may be divided roughly into three <lasses : " pin-hole," reflecting, and refracting. In the first, no mirror or lens is used ; in the second, some form of mirror is the essential feature ; in the last, some prism or lens, or a combination of prisms or lenses. " Pin-hole " Instruments These have been described in Chapter XXV, and nothing further need be said here. They all depend upon the "recti- ir propagation" of light. The pin-hole camera and the •• e.imera obscura" are the only instruments of this class that of importance. Reflecting Instruments The action ..I plane, spherical, and parabolic mirrors \\ as explained in Chapter \\YII: and their use as looking glasses, in searcl} lights, etc., was described. The great advantage of all forms of reflecting iiM rnmcnis over refract- ing om-s is that they are necessarily five from chromatic 4'.':, 4'.t»i LIGHT aberration. Spherical aberration, however, can be avoids 1 only by using normal pencils of light and mirrors with small curvatures. One important application of a concave spherical mirror was first made by Newton, namely, to form a "telescope." When we look at a distant object, it subtends at the eye a comparatively small angle. Thus, if A and B are two points on opposite edges of the object, and if 0 marks the position of the eye, the linear angle subtended by the object is (AOB). We estimate the distance of an object from us by the angle which it subtends at our eyes ; the nearer it is, the larger is this angle. FIG. 246. -Diagram representing the effect of a The purpose of a telescope telescopejn increasing the angle between two rays jg to brinff a distant ob- AO&nlJBO. . ' , ject apparently nearer, by changing the direction of a ray AO to Af 0, and a ray B 0 to B' 0 ; so that the angle subtended by the rays from A and B is now (A' OB'), an angle greater than (AOB). The "power of the telescope" is defined to be the ratio of the angles (A1 0&) and (A OB). In all telescopes either a concave mirror or a converging lens converges the rays so as to form a real image ; and this i |
s viewed by a lens, or a combination of lenses, called the "eyepiece." The simplest form of eyepiece is a converg- ing lens, but a Ramsden or Huygens eyepiece is ordinarily used. In doing this the eyepiece is so placed that the image to be viewed comes at its principal focus (or just inside it), and it forms a virtual magnified image at apparently an infinite distance. Reflecting Telescope. — In all forms of reflecting telescopes a concave spherical mirror is used to receive the rays from OPTICAL INSTRUMENTS 497 the distant object. The mirror is mounted in some frame- work or tube, so that it may be turned to point in any direction. In the cut, let C be the centre of this spher- ical surface, and I\C and PC be lines drawn .from two opposite points ill the edge Of the F'°- 247- — Dia?rnin representing a reflectinp telescope. P remote object. All rays parallel to Pj(7are brought to a focus at a point Fv on PjC', halfway between C and the mirror; and all rays parallel to PC will be brought to a focus at a point F, on PC, halfway between C and the mirror (see page 449). Therefore, there will be a real image of the object in the plane FFr and I\ are two points at an infinite distance. In order to see the magnified image of this image produced by the eyepiece, some plan must be adopted which will ren- der it unnecessary for the observer to stand in front of the mirror and so obstruct the view. Several have been tried : 1, the mirror is tipped slightly so that its centre lies outside the telescope tube, and the principal focus for parallel rays lies near the edge of this tul>e, and thus the image may be viewed — this arrangement was used by Herschel ; 2, a small plane mirror (or a totally reflecting prism ) is placed close to the principal focus, but between it and the mirror, and is so inclined as to turn the rays off sidewise through an opening in the side of the telescope tube; thus the image is formed outside or near the edge of the telescope — this was Newton V plan; 8, a small convj mirror is used in place of the plan, mirror, and is so turned as to reflect the rays back through a small opening at the centre of the concave mirror and form an image at iinur this arrangement is due to Casse- grain . 4. or. finally, a small concave mirror is placed beyond AMES'S PHYSICS — 32 498 LIGHT the principal focus of the large mirror, and is turned so as to form an image in an opening in the large concave mirror (like Cassegrain's method) — this plan was proposed by Gregory. PIG. 248. —Different forms of reflecting telescopes : (a) Herschel ; (b) Newton ; (c) Cassegrain ; (d) Gregory. The power of the telescope is found directly from simple geometrical considerations. As shown in Fig. 249, parallel rays from the distant point P are brought to a focus at F, OPTK .17. Of STRUM BA T8 halfway l>et \\<-ni (J and tin- mirror: parallel rays from the distant point Pl are brought to a focus at Fv The straight line at .1 represents the eyepiece, which is placed so that F and Fl lie in its focal plane; B is a point so chosen that AB = FA, and therefore F and B are the principal foci of the eyepiece. The rays from Pv after reflection, converge to Fr pass on through the eyepiece, and emerge as if they came from the virtual image of F-^ at an infinite distance in t In- direction^ A. Similarly, the reflected rays passing through F have a virtual image at an infinite distance on the lineal. . I>i:ii:r:im ..f f..nn:iti..M «>f ima-c t.y mirn-r .in.] refore, the angle subtended by tin- distant object itself is ; while that subtended by the virtual inia!_re of this formed by the eyepiece is (!*/! /•',>. lh.se angles may be compared by considering their tangents. The e<,nals (FCF^ and its tangent equals ^J. The FF tangent of (FAF\ ) equals _J. These angles are, however, FA snOieiently small in general t<> allow us to replace their numerieal values 1»\ those of their tangents; so the ratio of the angles (/'.I / , and (P^P) is , that is, it is the ratio i.f the f.u-al IniLTth of the mirror to that of the eyepiece. FA \\*f see. therefnre, the great ad \, intake of having a teles< whose concave minor has a long focal length. 500 LIGHT The limit to the resolving power of a telescope is deter- mined, as we have seen on page 487, by the diameter of the mirror ; the larger it is, so much the greater is this power. A large mirror also gathers more light, and therefore enables fainter objects to be seen. Newton constructed his first reflecting telescope in 1668. It had a diameter of one inch, and magnified thirty or forty times. He later made a larger instrument. Previous to this, such instruments had been designed ; and the invention is attributed to Niccolo Zucchi (1586-1670) of Rome. Refracting Instruments All instruments containing lenses are subject to a certain amount of chromatic aberration, although this may be mini- mized by using only achromatic lenses and combinations. There is also always some spherical aberration ; but this may be avoided largely by using only normal pencils of light, and exposing only the central portions of the lenses. The number of the defects which are possible with a lens is so great that some special treatise on the subject should be consulted. Only a few of the simplest refracting instru- ments will be described here. Photographic Camera. — In this instrument a real image of an object, more or less remote, is formed on a plane photo- graphic plate. This image is produced by a converging lens or system of lenses. If the camera is used for taking photo- graphs of landscapes, the lens is ordinarily a single achroma- tic one, and a diaphragm with a circular opening of variable size is placed in front of it. If it is used for photographing buildings, special pains must be taken to avoid spherical aberration and the consequent distortion; two achromatic lenses symmetrical with reference to a plane halfway between them are used, the diaphragm being in this plane - this is called a " rectilinear " or " orthoscopic " lens. For portrait work, lenses of large diameter must be used in order to secure as much light as possible. OPTICAL IN STRUM E* TS 501 When photographs of distant objects are taken, the images are, in general, small ; but by combining a diverging lens with the converging system, a larger image may be secured. Sueh a compound lens system is called a "teleobjective." Projection Lantern. — This instrument consists of a lamp, or a strong source of light, which by means of lenses illumi- nates a drawing or photograph on glass, or some object which is transparent in parts, and of a converging system of lenses which throws a real image of this illuminated object upon a suitable screen. The lens system between the light and the object — called the "condenser" — consists, in gen- \ \ Fie. SCO. — Projection lantern ; (.-1) arc light; (C) condenser; (5) slide to be Illuminated; (/•') focusing lens. eral, of two plano-convex lenses with their curved surfaces in contact. Its function is to deflect down upon the object to be " projected " as much light as possible, so as to render it strongly luminous. Then by means of the "focusing lens " a real image of it is formed upon the screen. The focusing and condensing lenses must be achromatic, and the former must be corrected for spherical aberration also. Astronomical Telescope. — This instrument consists of a C()nver«rin^ achromatic lens, called the "object gla— ." \\liieh forms a real image of the distant object in its focal plane, and this is viewed by an eyepiece. (In the cut a Ramsden eye- piece is represented.) The eyepiece is so placed that the image formed by the object glass comes at or just inside its 502 LIGHT principal focus, so the object is seen apparently at infinity. The power of this telescope is, therefore, like that of the FIG. 251. — Astronomical telescope: O is source at an infinite distance; O' is image formed by object glass ; 0" is virtual image formed at an infinite distance by the eyepiece. reflecting instrument (see page 499), equal to the ratio of the focal length of the object glass to that of the eyepiece. The resolving power of a telescope is, as we have seen, determined by the size of the object glass, as is also the quantity of light received. (See page 500.) The astronomical telescope forms, as just explained, a virtual image of the distant object which is inverted; that is, if the object is a tree, the image will have the tree pointed down instead of up. This is a disadvantage if the instrument is to be used for purposes that are not purely astronomical. Dutch Telescope (or Galileo's telescope). — This telescope is free from the disadvantage of the astronomical instrument FIG. 252. —Dutch telescope. to which reference has just been made. It consists of a converging lens, forming the object glass, and a diverging <U>TK AL INSTRUMENTS 503 I- us so placed that the principal focus on the side next the observer coincides with the principal focus of the object glass. Thus, parallel rays from a point of the distant object are con- verged by the object glass toward a point behind the diverg- ing lens, which is in the focal plane of both lenses ; these rays are diverged again by the second lens and emerge parallel to a line drawn from the centre of the second lens to the point in the focal plane toward which the rays were converging. The image of the distant object is therefore a virtual one at an inlinite distance; but it is erect; that is, the image formed of a distant tree represents the tree in an upright position. The first telescope was probably constructed by Hans Lippershey of Middleburg in the Netherlands in 1608 ; and Galileo, upon hearing of the invention, but without knowing any details of the construction, made an exactly similar instrument in 1009. Kepler was the first to suggest the use of a convex lens for the eyepiece. Galileo immediately used his telescope for observing the heavenly bodies, and made many most im- portant discoveries. Pr |
evious to the construction of the reflecting telescope by Newton and the invention of the achromatic 1ms by Pollond, the only means of minimi/ing the color effects produced by telescopes was to use lenses of great focal length, which were clumsy and ott* « .1 many disadvantages. 1 1 _cens presented to the Royal Society of London a lens whose focal length was 123 ft. Microscope. — This instrument is one designed to -magnify" a small ohjert, that is, to increase the apparent distance apart <>t any t \vo of its point* wh'u-h are close together. a. Simple microscope. — As was said on page 494, in speak- iii'_r of eyepieces, a single converging lens or .t K'amsden eye- piece can be used as a microscope, the object being placed inside the principal focus. There is thus formed an virtual image of the object (see page 477). The dis- tance at which this image is formed is arbitrary: but it is generally chosen as about 2"> om. (or 1" in. > fn>ni the eye, because this U t: < <• at which j.copl,- with normal eyes hold an object in «>rder to see it most distinctly. 504 LIGHT b. Compound microscope. — The magnification secured by a single lens is not great ; and it is in general combined with another lens system, as shown in Fig. 253. The latter forms a real magnified image of the object, which is viewed and again magnified by the eyepiece. (In this instrument a Huygens eyepiece may be used, as is illustrated in the cut.) The lens system which is nearest the object is called the FIG. 258. — Compound microscope: OP is the object; 0"P", the virtual image formed by the eyepiece. "objective." It consists of several lenses so chosen as to give an image as free as possible from chromatic and spher- ical aberration ; and it is so constructed as to give as much light as possible to the image and at the same time to have a large value of n sin ^V (see page 487), so that the resolving power is large. The magnifying power can be deduced by simple geometrical methods. The use of converging lenses as simple magnifiers was known to the ancients; but the compound microscope was probably invented by Zach- arias Joannides of Middleburg in the Netherlands and his father, some years before 1610. It was invented independently by others also, among OPT1< AL INSTRUMENTS 505 whom was Galileo. All the early microscopes had a concave lens for the eyepiece ; and Franciscus Fontana of Naples was the first to suggest the sul»titution of a convex lens. Spectrometer. — A spectrometer is an instrument primarily <lfsigned to measure the angle of deviation produced in the direction of a beam of light by reflection or refraction. It consists essentially of four parts. There is a substan- tial metal base to which is rigidly connected an upright metal cylinder, whose axis is called the "axis of the instrument." To this is attached a metal plate whose edge is divided into degrees, minutes, etc. Two bent metal arms are also attached to this cylin- der by means of collars, so tliat they can turn around it as an axle ; they carry metal tubes which lie in a plane perpendicular to the axis of the instrument and are thus movable in this plane. One of these tubes is a simple form of astronomical telescope; the other carries at one end an achromatic converging lens whusf focal length is that of the tube, and at the other end it is .-IMN.-.I by a metal cap in which there is a fine slit with straight parallel edges, which can be opened or closed — this is called a 4i enllimator." If the slit is illuminated, it serves as a s<>mvc <>f diverging waves which will emerge from the lens of the collimator in the f,.rm of plane waves. 506 LIGHT The telescope and collimator are turned until their axes intersect at the axis of the instrument ; and the object which is to produce the deviation of the light — a plane mirror or prism — is placed on a platform in the middle of the metal plate referred to above, having the normals to its reflecting or refracting faces in the plane of the axes of the collimator and telescope. If now by means of the former a beam of parallel rays falls upon the mirror or prism, they will be deviated, and their new direction may be found by turning the arm carrying the telescope until there is formed in the centre of the field of view an image of the illuminated slit. In order to determine this condition FIG. 255. -cross hairs exactly, it is customary to insert in the focal plane of the object glass a metal ring across which are stretched two fine silk fibres or spider lines, which are called the "cross hairs." These are made to cross exactly at the centre of the tube. Since they are at the focal plane of the object glass, they will be seen through the eyepiece at the same apparent distance as the object which sends the parallel rays into the object glass. The positions of the collimator and telescope may be noted by means of the divided scale on the edge of the plate, if suitable pointers or verniers are attached to them. used in telescope. * With this instrument the laws of reflection and refraction may be verified ; the angle of a prism and the angle of minimum deviation produced by it for any train of waves may be measured; etc. Therefore, the index of refraction of any substance which can be made in a prism may be deter- . A + D sin— |- mined ; for n = ^ (See Ames and Bliss, Manual of sin- Experiments in Physics, pages 459-475.) OPTICAL M->T/,T.I//-;.\ :,«»; If the laws of reflection are to be studied, ordinary white light may be used to illuminate the slit; but, if the phe- nomena of refraction are to be observed, special precautions must In- taken which will be described in the next chapter. Effect of Diaphragms. — One fact that must be taken into account in the description of all reflecting or refracting instruments is that the wave front of the waves is always limited by certain apertures or diaphragms. Thus, in the case of a telescope, the only portion of the wave front that enters the instrument has the size of the object glass. Again, if there are diaphragms in the tubes, they may limit the cone of rays which proceed from any point of the object and enter the instrument. Further, the only rays that are of practical use in the case of an instrument that is used visually are those which have such a direction as they leave the eyepiece as to enter the pupil of the eye. The effect of the presence of these various circular openings is felt in many ways. The brightness of the image, the resolving power of the instrument, the contrast of the background, the spherical aberration, etc., all depend upon their size and position. For details in regard to the matter, reference should In- made to some advanced text-book, such as Drude, Optics, or Luiuiuer, Photographic Optics. CHAPTER XXX DISPERSION IN speaking of chromatic aberration, achromatic lenses, etc., it was noted that the index of refraction of a given substance varies with the wave length of the light which suffers refraction, and that, in general, the shorter the wave length, the greater is the index of refraction. In this chapter the method of determining the connection between these two quantities will be discussed. Pure Spectrum. — Before any relation between index of refraction and wave length can be established, it is necessary to devise a method for securing homogeneous light of a FIG. 256. — Prism spectroscope. definite wave length. The method ordinarily adopted is to make use of the dispersive action of a prism. If waves of a definite index of refraction, emitted by a small source, fall upon one face of a prism, they suffer refraction and emerge 608 509 on the other side, diverging apparently from a virtual image of the source. They do not, therefore, of themselves come to a focus ; but, if a converging lens is introduced, the rays may be focused upon a screen. This real image is due to the virtual iina^e formed of the small source by the prism ; and it should be noted that the latter image is an astigmatic one, unless the pencil of incident rays meets the prism at the angle corresponding to minimum deviation. (See page 467.) The intensity of the effect can be increased if the point source is replaced by a series of such sources forming a line parallel to the edge of the prism. In general, a fine slit is made in an opaque solid, and the source of light, in the form of a flame, etc., is placed behind it. The image on the screen is in this case a narrow rectangle, practically a line, parallel to the slit. If the source is emitting several trains of waves of different wave length, there will then be formed as many images of tin- slit as there are separate trains of waves. The light is said to be "analyzed"; or a "spectrum" of the lii?ht is said to be formed. If the slit is very fine and the adjustment for minimum deviation as exact as possible the spectrum is said to be "pure," because in this case each train of waves affects only a very narrow rectangle on the screen, and so there is only a small amount of overlapping of the images. (Since t}i«- angle of minimum deviation is different for waves of different wave lengths, it is best to make the adjustment of the prism for the mean wave length of the light which is Further, since the focusing lens — even if aohrom — has a different focus for different waves, the screen must be curved so as to obviate the error, or one must be satis- fied with a slightly i 114)11 re spectrum.) In this manner the nature of the light — or, speaking in a more general manner, of the ether waves — emitted by any source can ivestigated; and we shall discuss this matter in a later chapter. 510 LIGHT Resolving Power. — Owing to the fact that the waves which leave the slit have passed through several "apertures" on their way to the screen (or to the eye), and that their wave front has been limited by this means to the shape of these apertures, the final image is not a line. These apertures are the edges of the lenses, of the prism, and of any diaphragms there may be i |
n the tubes. There is, therefore, a diffraction pattern produced, exactly as described for a lens on page 484. The image of the slit, provided the light is homogeneous, is a broadened line accompanied on each side by a series of alter- nately dark and light fringes, whose intensity is much less than that of the central line. The distance apart of these fringes varies with the FIG. 257. — Diagram to illustrate the resolving power of a prism. f , . , , width of the aper- ture. So, if the source of light is emitting two trains of waves of wave length I and I + AZ, there will be two overlapping diffraction patterns ; and the two trains of waves will produce two distinct images if the central line of one pattern is so far displaced by dispersion as to coincide with the first dark fringe of the other. If #j is the length of the shortest path of any of the rays through the prism, and £2 is the length of the longest one, t2 — t± is practically equal to the length of the base of the prism ; call its value b. If n and n + Aw are the indices of refraction of the two trains of waves, of wave length I and I + AZ, which are just resolved by the prism, it may be shown that An = Y • In other words, if the prism has a thick base 0 An is small, the image formed of the slit is extremely nar- row, and the resulting spectrum is pure ; but, if the prism is thin, the image formed is broad. D/> /•/-:/,•> ION 'A 1 Spectroscope. — The essential parts, then, of an instrument to In- used in order to form a pure speeti urn are a slit, a pi ism (or other means for securing dispersion), and a converging lens. This is called a "spectroscope." In general, the arrange- ment is slightly different, a collimator and telescope heing used as described for a spectrometer on page 505. When the slit is illuminated, plane waves emerge from the lens; these enter the prism and are dispersed by it ; and, finally, as they leave the prism they enter the telescope which is focused for plane waves ; and a series of colored images of the slit is seen 1)\ the observer looking in the eyepiece. Measurement of Dispersion. —If the eyepiece is removed, and the images formed by the object glass are focused sharply on an opaque screen, and, if a fine slit, parallel to that in the collimator, is made in this screen, it will be illuminated by iiract i( -ally homogeneous waves, and will serve as a source K h waves. This entire instrument is called a "mono- chromatic illuminator." By varying the position of this second slit (or by turning the prism) the wave length of the waves transmitted through it is altered. .Methods will be discussed later which enable one to measure tin- wave length of any train of waves; and, granting that the wave lengths of the waves emitted by the source are known, we have thus a method for measuring the refractive index of any substance for waves of definite wave lengths. The substance to be studied is made in the form of a prism, and is placed on a spectrometer table. The entire instru- ment is turned until the slit of its eollimator eoinrides with the slit of the inonnehromatie illuminator. The angle of the prism and the angle of minimum deviation t<u different waves are m- md I mm a knowledge of these quantities the indi -inn may be calculated. Fraunhofer Lines. — It will be sho\\ n later that, when HUH- light , liy a spectroscope, it is found to OO1 trains of waves of varying lengths, forming \\hal a; 512 LIGHT first sight to be a "continuous" spectrum; that is, one in which waves of all wave lengths within certain limits are present. But, if examined with an instrument of fair resolv- ing power, the solar spectrum is seen to be distinguished by the absence of a great many trains of waves, as is shown by the presence of black lines across the bright colored spectrum. (The fact that this absence of the waves is manifested by the presence of lines is due, of course, to the use of a slit illumi- nated with sunlight as the source of the light in the spectro- scope.) These lines are called " Fraunhofer lines," because they were first carefully studied by him. He measured — by methods to be discussed later — the wave lengths of the waves which correspond to these lines ; and to the most prominent of them he gave names, in the form of letters. (Thus the A line is at the limit of vision in the red end of the spectrum ; B and 0 are also in the red ; Dl and D2 are in the yellow, etc. ; K is at the limit of vision in the violet end of the spectrum.) It is customary, therefore, in studying the dispersion of any prism to illuminate the slit of the spectrom- eter with sunlight, and then to note the angle of minimum deviation that corresponds to the waves in the immediate neighborhood of the various strong Fraunhofer lines. Thus, nE means the index of refraction for waves whose wave length is that of the E line, etc. Dispersive Power. — The differences in the refractive index of various substances for different waves is shown in the fol- lowing table, in which the columns give the values for the A, B, C, etc., lines : A B C D E F 0 Water, 16° C. . . Carbon bisulphide, 10° Crown glass Flint glass . Rock salt, 17° . . 1.330 l.«16 1.528 1.678 1.537 1.630 1.681 1.539 1.332 1.626 1.531 1.583 1.640 1.334 1.635 1.534 1.587 1.644 1.338 1.661 1.540 1.597 1.653 1.537 1.502 1.549 1.546 1.606 1.661 // 1.344 1.708 1.561 1.614 1.568 DISPERSION 513 It is evident from this table that not alone is the index of refraction different for different substances, but that also the dispersion of any two rays, e.g. (nD — wc), may be the same for two substances and the dispersion of two other rays may differ ; thus for crown glass and rock salt nB — nA = 0.002, but for the former nn — n?= 0.011, while for the lat- ter it equals 0.015. Further, what is called the "dispersive power " is different. This is defined to be the ratio n»~nA. (In practice, the n in the denominator is taken as n^ or n^ or nE, etc., whichever happens to be best known ; for the error thus introduced is negligible.) Its value for the above substances is therefore : nD—l Water 0.042 Flint glass .... 0.061 Carbon bisulphide . . O.H5 Rock salt .... 0.057 Crown glass . . . 0.043 The dispersive power of a substance is proportional to the length of the spectrum produced by a prism made of it, which has a small angle. For, and if A and therefore D are both small, we may write ...5^, and ».-l >equently, the dispersive power equals "~ •D/i— I>A* the difference in the minimum deviations of waves of wave IfiiLTtlis A ami //. is practically the angle subtended at the prism by the length of the spectrum when the prism is in its position of minimum deviation. Therefore, prisms having ' :le but made of dinVivnt substances pro- duct (,f dillrivnt length and of dillVn-nt dispersion /)/>— 1 PHYSICS — 88 514 LIGHT for the same waves ; and the average deviation for the whole spectrum may be different also. It is owing to these facts in regard to dispersion that it is possible to construct an achromatic lens, or to combine two prisms in such a manner as to produce no deviation of a defi- nite train of waves, but to disperse the other waves. Direct-vision Spectroscope. — The spectroscope that has just been described suffers from the disadvantage that the collimator and telescope are not in the same direction, and therefore in some cases it is difficult to support the instru- ment rigidly. As has been just explained, however, it is a sim- ple matter to choose two prisms of different materials which, when combined with their edges parallel but turned in opposite -A direct vision spectroscope. directions? will not deviate OHO particular train of waves, but will the others. The disper- sion produced in this way by two prisms is not great ; but, if several are thus combined, it may be made as great as desired. If such a series of prisms is combined with a collimator and telescope, the apparatus is all along one straight line. Such an instrument is called a " direct vision spectroscope." Normal and Anomalous Dispersion. — The table given above shows that in all the cases cited the index of refraction in- creases as the wave length decreases ; this is called " normal dispersion." There are, however, many substances with which this is not true. As will be shown later, every sub- stance absorbs more or less completely ether waves of certain wave lengths; and some substances absorb only waves of wave lengths that lie within a narrow limit, e.g. only waves having wave lengths that differ but slightly. In such cases the index of refraction for waves whose wave lengths do not differ much from this wave length of the "absorption DISPERSION 515 band " does not follow the above law connecting it with the wave length. Such dispersion is called " anomalous." There may, of course, be several such absorption bands, introduc- ing further complication. Dispersion Curves and Formulae. — The various facts in regard to dispersion, both normal and anomalous, can be best shown in a diagram whose axes are indices of refraction and wave lengths, or by a formula. Thus, for normal dis- 100 «00 300 tOO WO WO 700 800 800 1000 1100 1200 1300 .1100 UOO 1000 1 'n.. 258. — Dispersion carve for quarts. persion, the connection between index of refraction and wave length is given by a curve like that shown in the cut, which is tin- •• disjiersion curve" for quartz. This maybe also expressed 1>\ a formula, in which n is the index of i< traction and I the wave length, viz.: fin (lie neighborhood of an absorption band this formula ceases to he true. ) If then- is a single absorption kind whose wave length is J0, the dispersion curve has a general form as shown in tin- cut, which is the actual cur\e for cyiinine as determined by 516 LIGHT Professor R. W. Wood. This substance has an absorption band at about wave-lengths 600 MJL. The formula which expresses the facts outside the absorp- tion band is n2 = A + — — • It is seen from this or from T* I — 1Q the curve that for wave |
lengths not far from the absorption band, if 11>1Q>1Z, n1>n2; that is, the refractive index of the longer waves is the greater. So, if a prism is made of this substance, it will deviate the long waves near the ab- sorption band more than the short ones ; and the resulting S8iS3Sg§§2gg§8S!Sg 2.3'5>p'^>« oooSoo pc-t-V-t- WAVE LENGTHS IN fJ.fJL FIG. 2GO. — Dispersion curve of cyanine. spectrum will appear, in general terms, as if it were divided in two parts by the absorption band and as if these two halves were shifted toward and across each other. Whenever there is an absorption band, there is anomalous dispersion, and many substances show these phenomena in the visible spectrum ; such are the aniline dyes, the vapors of sodium and other metals, thin layers or films of metals, etc. It will be seen later that all these substances have other opti- cal phenomena which are closely connected with this fact. Rainbows. — An interesting illustration of dispersion is furnished by the phenomenon in nature called the " rainbow." After a rain shower, DISPERSION 517 if tl.r sun is not far from the horizon, and the rain clouds have passed in such ;i .liivrtion that the observer is between them and the sun, a series of colored arcs or circles may be seen on looking away from the sun. These colored arcs are in the following order : violet on the inside shad- ing off to red on the outside, then a dark space, and another arc colored red on the inside and violet on the outer edge. They are all portions of circles whose centres coincide at a point lying in the prolongation of the line joining the sun with the eye of the observer. There are often also u supernumerary " bows inside the primary one. G — - FIG. 261. — Rainbow : refraction oi a ray by a drop of water, single reflection. A complete explanation of these phenomena requires a consideration of the size of the drops, the nature of sun -light, the size of the sun, etc.; this was first given by Sir George Airy. An elementary, but imperfect, theory was given by Descartes ; an outline of which is as follows. Con- sider a raindrop as a sphere of water, and draw the paths of the rays inci- dent upon it from the sun. Certain rays will enter the drop, suffer reflection once and be refracted out, as shown in the cut, in which 0 is the centre of a drop, OS is the direction of the sun, and AB, BC, CD, DE represent the portions of a ray. The deviation of the ray is shown in the cut by the angle (GFE), where F is the intersection of the prolongations of the incident and emerging portions of the ray, AB and DE. By means of higher mathematics it may be shown that if there in a homo- geneous beam of rays, all parallel to OS, falling upon the drop, the angles of de- viation all exceed a definite value. In ..—Rainbow: JVI» angle of mini- mum deviation. N other words, there is a minimum value of the deviation, and this fixes a certain direction with reference to the line OS. If tins minimum angle is N, draw the line AB making such an angle with SA. Then of all the 518 LIGHT rays parallel to OS falling upon the drop, none are so deviated as to • •iin-rge outside the angle (SAB)] some emerge in the directions ABr A Bp etc. This minimum angle is different for different wave lengths ; for violet it is about 140°, for red about 138°, etc. Therefore, if the observer looks up at the rain cloud in such a direction that his line of sight makes an angle of 180° -140°, or 40° (or less), with a line joining his eye to the sun, he will receive violet light from the rain- l / FIG. 268. — Kainbow. drops. If he looks up at an angle greater than 40°, he receives no violet light at all. Therefore, the observer will see a violet arc, each point of which subtends at his eye an angle of 40° with the line drawn from the sun. Similarly, with the other colors, there is a red arc corresponding to the angle 180°— 138°, or 42°, which is sharply denned on its outer edge, and to see which one must look higher up in the sky than was necessary in order to see the vio- let arc. But some of the rays from the sun suffer two reflections in the rain- drop, as shown in the cut. As before, we may show that there are FIG. 264. — Rainbow: refraction of a ray by a drop of water, double reflection. angles of minimum de- viation for the different colors, which give rise to a red bow at the angle 51° and a violet one at 54°, these bows being sharply defined on their lower edges. CHAPTER XXXI INTERFERENCE OF LIGHT Interference Fringes. — The general phenomena of interfer- ence of waves were described in Chapter XXI, page 374, and the special ones dealing with light were discussed in Chap- ter XXV, page 420. It was shown that the simplest mode of illustrating interference was to place two identical sources of light close together and to allow them to illuminate a screen or to enter the eye directly. If the two sources are parallel slits, the interference pattern is a series of parallel colored fringes ; if homogeneous light is used, these are alternately bright and dark, and at regular intervals apart proportional to the wave length ; if white light is used, the fringes are a superposition of different sets, each of whirl i is due to a different color, thus proving that white light is equivalent to a superposition of waves of different wave length. Unless the two sources are identical, there is no permanent phase relation between the two sets of waves emitted l.y them, and so these cannot interfere. In demonstrating the interference fringes which are pro- duced by the two identical sources, a converging lens is always used, the arrangement being as shown in Fig. 265. Ol and 02 are the two sources ; L is the lens : M \^ a sen-en placed at a distance fmm the lens equal t<> its focal length for parallel rays. The two sources are emitting niys in all directions; let OlAl and O^A^ be two parallel rays. After n-f motion by tin- 1ms, tln-\ will meet at the point B in the screen, \vhn. < I: a line drawn through the centre of tin- lens parallel to the incident rays. (See page IT1.'. > If 619 520 LIGHT the difference in length of the optical paths of these rays is half a wave length (or an odd number of half wave lengths), there will be complete inter- ference at B. This difference in path is found exactly, as on page 377, by drawing a line from 02 perpendicular to 0-^Ar For, if B were a source of waves, two of its rays would be BAfli and BA202i and the position of the wave front at any instant would be a plane perpendicular to A101 and A202. So, when the wave front reached 02, it would be at P on the line Aflv where 02P is perpendic- ular to A^O^ 111 Other words, the optical paths 02A2B and PA^B are equal in length ; and so the difference in length of the optical paths from 01 and 02 to B is O^P. b o M Fio. 265. — Young's Interference expert- By means of the lens, then, the interference fringes are focused onjthe screen. The central image is at 0, and the distance OB can be expressed at once in terms of the differ- ence in path of the rays from 01 and <92. Call this difference But the jingles (O&P) and (BOO) are equal ; and sin (BOO) O\ D = OjP = equals — — . Hence BO = EC sin (ECO) = BC sin (0i02P) = D-^- = -5- O\ 0-2, If the focal length of the lens, 06\ is called /, and if it is large compared with BO, we have the relation IM'Klil'KliENCE OF LIGHT 521 The condition for complete interference is that and for this value BO = (~ Therefore the distance between two dark fringes is There is thus offered a method for the measurement of the wave length of light, as the various quantities in the formula with the exception of I may all be measured directly. We shall now describe several methods of securing two identical sources. Young's Method. — This consists, as has been already de- scribed, in illuminating two parallel slits in an opaque screen by light from another slit parallel to them and at the same distance from each. The slits must be sufficiently narrow to dilVract the waves in all directions. (See page 392.) FresnePs Biprism. — This consists of a combination of two identical thin prisms which are placed base to base, as shown by the cross section in the cut. An opaque screen, with a slit in it, is so placed that the slit is parallel to the two edges of E Fra. 2«6. — Fresnern blprinm : two virtual Images 0, and Ot of the source 0 are produced. tin- biprism and at e<|iial distances fmm 1 linn. This slit, if illuminated, will emit waves, which will suffer refraction and deviation by the two hal\«-s of the prism. If the slit is at 0, as sho\\ n in the cut, one half will form a virtual image at 0,, 522 LIGHT the other at 02. So the waves as they emerge from the biprism will come apparently from the two sources Ol and 02 ; and there are then two identical trains of waves, which will inter- fere, and may be focused on a screen by a lens, as described above. Therefore, if the distance 0^0^ is known (and it may easily be determined by experiment), the wave length of the light may be measured. Fresnel's Mirrors. — These are two plane mirrors which are carefully adjusted until their faces are slightly inclined to each other, but are in actual contact along a line. FIG. 267. — Fresnel's mirrors: two virtual images 0j and 02 of the source 0 WV are produced. A slit 0 is placed parallel to this line ; and therefore two virtual images of it, 01 and 02, are formed by the two mirrors. Let B be the line of contact of the two mirrors. Then a pencil of rays P^OB falling upon the first mirror will be reflected into the pencil Q^ OlBl ; and the pencil B OP2 falling upon the second mirror will be reflected into B202Q2. Therefore there will be a region included in the angle (^B^B,^) which is traversed by two trains of waves coming from identical sources. Lloyd's Mirror. - In this arrangement a slit is placed paral- lel to a plane mirror, at some distance from it, but only a slight distance above its plane. There will be a virtual image formed by the reflected rays ; a |
nd so any Pie. 268.— Lloyd's mirror: a virtual image 0, of the source 0, is formed by the mirror. INTERFI-:I;I-:\<-E OF LIGUT 523 point above the mirror will receive waves directly from the slit, and also by reflection, apparently coming from the vir- tual linage of the slit. There will therefore be interference. (These two sources of waves are only approximately iden- tical, for one is the inverted image of the other.) Colors of Thin Plates. — The first interference phenomenon which was recognized as such and so explained is the pro- duction of the brilliant color effects by such thin films of transparent matter as soap bubbles, films of oil on water, layers of air between two pieces of nearly parallel glass, etc. These colors are due to the interference of the trains of waves which suffer reflection directly at one surface of the lil in with the waves which are refracted out from the film after one or more internal reflections. If we consider any point on the surface of a film which is receiving homoge- neous light from any point source, one ray from the latter is reflected at the point, and other rays emerge there which have entered the film at other points and have suffered reflection at the surfaces of the film. It is evident that these rays have had paths of different lengths ; and that also the ray directly reflected has suffered reflection when incident upon the surface of the film from the surround \\\g medium, while the emerging rays have suffered reflection when inci- dent upon the surface of the Him fmm its interior. Owing to this last cause there is a difference in phase introduced, in addition to that caused by the difference in path, because the reflection in the one case is from a "fast" to a "slow*' medium, and is the opposite in the other. (See page 834.) This additional difference in phase is equivalent, as was shown, to half a period of the vibration. If the total effect of all the rays at the point on the surface of the film is null owing to interference, this point will appear dark : while, if th. ravs do not destroy each other's action, the point will be bright. It is evident also that a film of such a thickness as to cause intc for waves of a definite wave length will 524 LIGHT not, in general, cause interference for other waves; so, if white light is used, a point where there is complete inter- ference for a definite train of waves will appear colored, owing to the fact that the other waves are not cut off ; and the effect is as if one color were completely removed from the constituent colors of white light. The trains of waves which interfere at the top surface of the film are not destroyed, for energy cannot be annihilated ; they are trans- mitted through the film and emerge on the lower side. Thus, when white light is incident upon a transparent film, some is reflected at the upper surface, the rest enters the film ; of this a certain amount is reflected once or more times, and is finally refracted out through the upper surface, while the rest is either directly, or after two or more reflections, refracted out through the lower surface. By far the greater amount of the light is transmitted, owing to the poor reflect- ing power of the film. There will then be colors visible if one looks at either of the two surfaces of the film ; but those seen by looking back at the second surface are much weaker than those at the other, owing to the presence of so much white light. If the film does not absorb the waves, the com- bined effects on its two sides are exactly equivalent to the incident waves; that is, they are "complementary." We shall now consider in detail the case of a thin film with parallel faces, and we shall suppose that the film has a greater index of refraction than the surrounding medium, e.g. a film of water in air. Let the rays come from a homogeneous point source at such a distance compared with the area of the film that they may be regarded as parallel. Let 00 in the cut be an incident ray ; it undergoes reflection at O and gives rise to a ray OD in general. Other rays emerge by refraction at C\ one of these is due to the incident ray PA. This is refracted into the ray AB, then reflected into BO, and finally refracted out. We can calculate the difference in optical path of these rays from the source to O. Draw UtTEBFEREN( /•; LIGHT 525 AE perpendicular to the incident rays; then the phase at A .tiid K is the same because they are on the same wave front; draw FC perpendic- ular to AB, it repre- sents the refracted wave front; and there- fore the phase at C and F is the same. The difference in op- tical path of the two rays meeting at C is, then, n(FB + T*C), where n is the index of refraction of the film with reference to the surrounding medium. Calling the thickness of the film h, and the angle of refraction into the film r, it is seen from geometry that FB + BC= 2 h cos r. Therefore the difference in optical path is 2 nh cos r ; and FIG. 269.— Colors of thin plates. (The tram are omitted.) >itt*d rays ' _ the corresponding time lag is , where v is the velocity There is also the of the waves in the outer medium. v . . 2 >///<•«>•>/• . additional difference in phase of half a period, — , due to the difference in the character of the reflection at B and at (7. So the total difference in phase, expressed in terms of time, between the two rays OCD and PABCD is 2 nh cos r , T This maybe expressed differently, by substituting for v its value — , where I is the wave length in the outside medium : V1/" r/2wAeosr 1\ \ / I'/ Another ra\ which will have a compoip • Lfing at (7 is O 'A \ incident at A' ^ etc. ; and it is n.»t dillicult to prove that 526 LIGHT the resultant emerging ray at C has the same amplitude as the reflected ray, but differs from it in phase by the amount given above. If this difference equals an odd number of half periods, the two rays interfere completely. This condi- tion is, then, that 2 nh cos r + 1 _ (2 m + 1) where m is any whole number 0, 1, 2, 3, etc. This reduces to 2 nh cos r = ml. Therefore if h = 0, or — - — or — - — , etc., 2 n cos r n cos r the interference is complete. If, then, the film is wedge shaped, there will be inter- ference at points on the surface of the film corresponding to these critical thicknesses; and alternately dark and light bands across the film will be seen. The light used to illuminate films generally comes from an extended source, like a flame, which is not far from the film. In this case any point C on the surface of the film receives two rays from a point 0 of the source, viz., OCD and OABOD' ; and in general the directions of these rays after leaving 0 are not the same. Similarly, Q receives rays from other points of the source, which also have different directions after leaving it. wmwrn 'Mb So this point may be regarded as FIG. 270.- colors of thin plates: itself a point source ; and, if the eye localization of colors m the sur- fo focused on the surface of the film, the question as to whether C appears dark or not depends upon how exactly these pairs of rays from each point of the source neutralize each other. If the incidence is practically normal, the value of r in the formula is so small that cos r = 1 ; and the condition for complete interference is that 2 nh = ml, which is the same for all these face. INTERFERENCE OF LIGHT 527 pairs of rays. If the thickness of the film at the point C satisfies this condition, and if the eye is focused on the film, C will be a dark point. So, in general, when an extended source of light is used and the incidence is practically normal, the dark or light bands are to be seen by looking at the surface of the film. Another illustration of these formulae is given when the convex surface of a plano-convex glass lens is pressed closely against a glass plate. The film of air between the two pieces of glass may be regarded as made up of a great number of concentric rings, each ring having the same thickness at ^ — -~ — -"" all points. The film is then like that of a FIG. 271.— Apparatus for New. wedge one of whose surfaces is curved; and ton's rings. there is, of course, symmetry around the centre, or point of contact. Therefore if homogeneous light is incident nor- mally upon this film, there will be a central dark spot surrounded by alternately bright and dark rings. The dark rings are given by h = 0, — -, — , etc., where h is the thickness of the film. These are known L' // 'J n as " Newton's rings." h can be expressed in terms of the radius of the spherical surface of the lens and of the radius of the dark ring ; and a method is thus offered for the measurement of the wave length of light; or, if this is known, for the measurement of the radius of the spherical surface of the lens. If in any of the above experiments white light is used, there are no dark bands or rings — all are colored ; and the details can in each case be deduced from the general prin- ciples given on page 524. Interference over Long Paths.— In the cases so far treated, the inter- l rays have been considered to differ in path only by a small amount, 1'iit there are many interesting and important phenomena in which HUH condition is not fulfilled. Tims let 0 be a homogeneous point source of liLjht : /' \te a plate of tran-par.-ut material; C be a converging lens; It be a screen placed in tin- />rincipal focus of the lens. Draw from C, the centre of the lens, a line CE at random. Then, all rays falling upon the lens parallel to this line \\ill be brought to a focus at E. The point source 0 is emitting rays in all directions; one of them, OP,, is parallel to / '/•;. ThN ray aft.-r incidence upon the plate gives rise after successive reflections and refractions to a series of rays Q,/?P Q9K.f etc., 528 all parallel to each other and to the original ray. Therefore they all uuite at E. The difference in path of two consecutive rays is 2 h cosr; and, if this equals an odd number of half wave lengths, there is complete inter |
ference at E, and also, by symmetry, for all points in a circle drawn around D with ED as a radius. If this differ- ence equals a whole number of wave lengths, the ring is bright. Thus, corresponding to the rays from 0 in all directions there will be a series of circular rings, alter- nately dark and light, around D. Similarly, if instead of having a point source, an extended one is used, each point will give rise to identically the same series of cir- cular rings around D ; and so the effect is more intense. PIG. 272. — Interference over long paths. If the source emits two trains of waves of different wave length, there will be two sets of concen- tric rings around Z); at certain points a ring of one set may coin- cide with one of the other, and at others it may fall between two rings of the other set. A connection may be established between the radius of any one ring, say the tenth, the focal length of the lens, the thickness of the transparent plate, and the wave length of the light ; and it is not difficult to see how a method can be devised for measuring the relation in wave length of the two trains of waves emitted by the source. The transparent plate is in general a layer of air included between two plane parallel glass plates. If one of these is kept fixed and the other is moved, h may be varied at will. Professor Michelson of Chicago has obtained interference fringes in this manner, using radiations from mer- cury vapor, when h was so great that the difference in path between two interfering rays amounted to 540,000 wave lengths; Fabry and Perot of Marseille?" -F-avo more recently obtained interference over a distance equiv- alent to 790,000 wave lengths ; and Lummer has succeeded in using even greater distances. This proves that this light must be extremely homo- geneous, otherwise there would be overlapping series of rings. i\ //:/,' 7-v-; /,' /; vr/,- or LK;IH Stationary Waves. — Since light is a \\a\v phenomenon, it must In- possible to prod nee stationary light waves exactly as was done with waves in cords and in air. (Set- page 349.) This was shown experimentally by Wiener, in 1890, by all ii wing homogeneous light to fall perpendicularly upon a polished metal mirror. - It M is the mirror, and plain- waves are incident normally upon it, there will be nodal planes at the mirror and along A^ AY Ay etc., at intervals of half a wave length ; and there will be loops along planes halfway between these. Wiener placed a thin photographic plate inclined to the mirror, and showed on development of it that light had been received along lines through Bv By By etc., where the loops were, but that there had been no light at all along lines through Av AT A# etc., where the nodes were. FIG. 278. — Wiener's experiment with stationary M CHAPTER XXXII DIFFRACTION THE general phenomena of diffraction have been illustrated in previous chapters in giving the explanation of the recti- linear propagation of light and the fringes observed near the shadows of opaque obstacles, in the description of the pas- sage of waves through small openings, and in the discussion of the resolving power of lenses and prisms. Many other illustrations may be found in larger text-books; but only one additional phenomenon will be discussed here. IJJI The Diffraction Grating. — If a great number of fine scratches are made on the surface of a thin glass plate by means of a diamond point, the lines so drawn being parallel and at equal intervals apart, the effect, so far as transmission is concerned, is exactly as if a great number of fine rectangu- lar openings, identically alike and evenly spaced, were made in an opaque screen. A piece of glass so "ruled" is called a " plane transmission grating." Let us consider homogeneous plane waves incident normally upon such a grating. Each point of a slit, or opening be- tween two scratches, serves as a source of waves, and sends out rays in all directions. Let 630 PIG. 274,— Transmission diffraction grating. DIFFRACTION 531 a converging lens be placed with its axis perpendicular to the plane of the grating, and let a screen be placed in its focal plane. In the cut, draw an arbitrary line OP from the centre of tin.- leii> to the screen; all the rays from the various open- ings of the grating which are parallel to this line will be brought to a focus at P. The difference in path of any two surh rays from the corresponding edges of two consecutive openings, or from two corresponding points in two consecu- tive openings, may be deduced at once. Call the distance from the edge of one opening to the corresponding edge of the next, i.e. the "grating space," a; and the angle (P&A), jy. Then, the difference in path referred to is a sin N-, and, it this is a whole number of wave lengths, parallel rays from corresponding points in all the openings will coincide in phase at P\ and it will be a bright point. (There is an exception to this, when rays from some of the points in any one opening interfere with those from other points in the same opening ; but this case need not be discussed here.) There will there- fore be a line of light through P parallel to the openings in the grating. The condition, then, that P should be bright is : where m = 0, 1, 2, 3, etc., and I is the wave length. Conse- quently, there is a series of bright lines determined by a sin jv"0 = 0, a sin N{ = /, a sin Nt = 2 /, a sin N3 = 3 /, etc. ; that is, by riii JV9 = Ob «nJ\r, = 8inN9 = ? sin JV8 = if, etc. The light along any line through a point P defined by these relations is bright ; thru for neighboring portions of the screen as one takes points farther ami farther away from P the light fades gradually away and vanishes. It rises to a maximum in another line detined l»y the next value of N, etc. Then- Si .1 maximum !'«»r tin- dnvetion sin ^ = 0, or ^V0=0; that is along a line through the point A in the cut where 532 LIGHT the axis of the lens meets the screen; this is called the u central image " ; the next maximum, at the angle Nv is called the "first spectrum," etc. It is evident that there are maxima also on the other side of A, corresponding to negative values of N. The number m is said to give the "order" of the spectrum. If white light is used, instead of homogeneous light, each constituent train of waves has its own series of spectra : a central image, and spectra of different orders on its two sides. The spectra of the different colors overlap ; and the spectrum of any one order is not pure unless the individual FIG. 275. — Photograph of spectra produced by a grating, showing the different orders. spectra formed for any one train of waves are extremely narrow. We can easily determine how wide any one spec- trum " line " is by calculating the position of the point next it on either side where the intensity is zero. This condition involves complete interference at that point of all the rays reaching it from the grating. Let P be a point where there is a maximum, and let P1 be the nearest minimum on the side toward A. Draw the line OPV and call the angle (PjO^.), iVj ; then all rays parallel to OP1 are brought to a focus at .Pj. Let us suppose that there is an even number of openings in the grating ; if there is an odd number, we DIFFRACTION 583 may consider the last one by itself, and its effect in com- parison with that of the others may be neglected. If the number of openings is 2n, the condition that P should be the position of the mth spectrum may be expressed by saying that the difference in path of the two rays reach- ing it from corresponding points in the first opening and the middle one, equals //////. For this difference in path is na sin N* and it has been shown thai the con- dition for a maximum is that P F> A FIG. 276. — Diagram Illustrating the resolving power of a grating. a sin N= ml. If P1 is to be the nearest minimum, the difference in path between two rays from these same points must differ from this value for P by half a wave length; for, if this is true, the rays from the first and //th. the second and the (n+ l)th, etc., will interfere completely. The con- < N dition for a minimum at Pl is, then. //'/ sin NI = nml — -• illaily, for a minimum point on the other side of P, the vain.- \\onhl be nml + ^- ) The condition for a maxi- mum at P is, Ilrlier, :Jnn wi Mil .V - mill. / The entire number of openings in the grating is 2n, and each has the grating space a ; so 2 na is the width of the grating; and the formula shows that in order for N to be nearly e.pial to \}. that is. toi />f to be very close to P, the 534 LIGHT width of the grating must be large. Under these conditions the spectrum "lines" are narrow. The distance from one spectrum line produced by a homo- geneous train of waves to the next one is determined by giv- ing m two consecutive values in the formula a sin N= ml. Thus, the fourth spectrum is at an angle -ZV4 whose sine satisfies the equation 4 / sin N. = — ; a and the corresponding formula for the fifth spectrum is Thus, sin N6 — sin N4 = -. This relation is general ; and it is seen, therefore, that in order to have consecutive spectra far apart the grating space a must be small. If white light is used, or waves from some complex source, the central image will receive light of all wave lengths ; and, in addition, a series of spectra will be produced on both sides of this. In order to have these spectra long, i.e. the disper- sion great, the grating space must be small ; and to have the spectra pure, i.e. the "lines" narrow, the grating must be wide. This condition for " purity " may be expressed differently. If there are two trains of waves of wave lengths I and I + AZ, which differ only slightly, their spectral images will, as a rule, overlap; but, if A£ is so large that the maximum of the waves of length / + AZ coincides with the minimum of the other waves, the two images or " lines " may be seen distinct from each other. The condition for a maximum of the forn id- waves is that N should be |
such that a sin N=m(l + A/); and we have just seen that the condition for the first minimum of waves of length / is that the angle NI should be such that na sin NI = nml + - > or that a sin Ni = ml+ — » 2n -, \\ lu-re 2 n is the number of grating spaces, cide, that is, if N=Nt If these two positions coin- I or This is the spectroscopic "resolving power" of the grating; and it is seen that, in order for A/ to be small, the number of grating spaces must be large and a high order of spectrum should be used. It should be noted that, even if the spectra formed of the components of white light are pure, the ones of different order are superimposed on each other. Thus, waves of wave length I have a spectrum line in the first order, at the same point where waves of wave length - have a line in the sec- ^ I ond order, and where waves of wave length ^ have a line in the third order, etc. It should be noted further that the above formulae are all independent of the material of the grating. If the plane waves are incident upon the grating at an angle /, different from 0°, the necessary modification in the formulae is deduced easily. Let O\ and O-> be the edges of two consecutive grating openings; let PI 01 and PtOi be two ..!' th«- im-idi-nt rays; and let <V7i and n..(i, !>«• th<> two rays from Oi and O2 which are diffracted in the direction defined by the angle N. v OtA perpendicular to PiO\, and 09B perpendicular to 0\Q\. At the points A and Ot the rays are in the same phase ; therefore the difference^ in path <>f these rays when they are unit. -.1 l,\- the converging lens is AO\ + &J5. If the grating space O\0t is called n, as before, this difference in path 1* writfr'i .V ) : an. I i!,.- condition for a spectrum line Pio. 877. — Oblique incidence upon a grating. K then, that a (sin /+ sin#) = ml. All the other formula* an- i lifi.-.l in a similar manner. If w«« mii*iili-r tli- s|.iTtniui lii,,-, fmin. -,| on th<> *T.-.-M in tlu- imme- diate neighborhood of the axis of the lens, which is supposed to be 536 LIGHT uila perpendicular to the surface of the grating, we may replace sin N by N, and the formula becomes fl (gin / + ^ j = mL If white light is incident, there will be a continuous series of colored spectra. Let NI correspond to the waves of wave length /i, and 7V2 to those of wave length 12 ; that is, hfi or, a(Ni-Ni)=m(lt-li). a (sin / This means that lengths along the screen, N2 — NI, are proportional to differences in the wave lengths of the corresponding waves. A spectrum where there is this simple relation between the wave lengths of the waves and their distribution is called a " normal " spectrum. It is obvi- ous that such spectra offer great advantages when one is interested in comparing the wave lengths of different waves. Gratings are also made by ruling lines with a diamond point upon a plane polished metallic surface; these are called "plane reflecting gratings." Those portions of the surface which are not scratched, if sufficiently narrow, dif- fract the light in all directions, as do also the grooves made by the diamond point. Two parallel rays in any direction coming from two different grooves have a difference in path which depends upon the grating space, the angle of incidence, and the angle of diffraction. The general formula for a bright line is, as before, a(sin / + sin N) = ml. (The intensity of the light diffracted in different directions will depend obviously on the shape of the groove and on the reflecting power of the material of which the grating is made.) The method ordinarily used for observing diffraction spectra is not to focus them upon a screen, but to place the grating on the table of a spectrometer, illuminate it by means of a collimator, and turn the telescope until the spec- trum lines in turn come upon the cross hairs of the eye- 587 piece. I»y means of the scale divisions upon tin- apparatus, tin- angles of incidence and diffraction may he read most accurately. Tin- ^ratiiiLC space may also he measured by ordinary laboratory methods, using a comparator. There- fore, in this manner the wave length of the waves may be E, determined with great accuracy. The values of the wave lengths corresponding to the prominent Fraunhofer lines are given in the following table: D\ B C *> 687.0186 1064 589.0186 527.0495 F G II K S25 186.1627 184.0684 (The unit ordinarily adopted for expressing wave lengths is one tenth of a thousandth of a micron, that is, 100oioooo «m- This is called an " Angstrom unit " because it was adopted by the great S\\.-.li-li j h\ for the expression of his results. Thus, the wave length <>t />, is ."VM- Angstrom uni- Concave Gratings. — The use of a plane grating, that is, one ruled <>n a plane Surface, involves the use also ol a eollimator and tdcscopt; for ordi- nary pu: hut. if the lilies are ruled on a concave sph.-i-i.-al metallic surface, this is not so. If \veimagineaplanethrougb the centre of the sphere and 1 »i •*,-.• tini: the rulings on the surface, and if a point >«'ii: -lit is in this plant}, the £ratiii'_r will ••!' its, -If produce spectrum lines \\ it limit the need of lenses. These lines are sh«rt and are parallel tc» the grating nil The\ lilacnim lllu*lr»tlnir |.rtn.t|,i-- .if • connive fnitinf. th« wn: k-r.-itiiik.- >» .-Hrrumlbr- •ooeofUMdrcIc, «>.(/.•( /' the pi. in- d t... whieh and the be spheric. il . in points \\ hich in throu-.rli the point 8€ 538 LIGHT certain cases satisfy a simple geometrical condition. If C is the centre of the grating, that is, the middle point of its ruled surface, and if 0 is the centre of curvature of this surface, draw a circle with 0(7 as a diameter which is tangent to the grating surface at C. Then, if the point source S is at any point of this circle, the spectral images will be formed at points P on this same circle. If the source is emitting white light, there will be a central bright image formed by ordinary reflection, and on each side of this there will be series of spectra, all on this circle. Those spectra formed in the immediate neighborhood of 0, the centre of curvature, are normal (see page 536), because CO is perpendicular to the grating, and so N in the formula is small. There is a simple kinematic method of maintaining this normal con- dition for a grating, and yet varying the waves which are brought to a focus at 0. If O is joined_to 5 by a straight line, the triangle (OSC) is a right-angled one^having OCas a hypotenuse. Therefore, if two rigid beams, SB and SA, are set up at right angles to each other, and are g furnished with tracks along which small car- riages may run, and if a beam of fixed length equal to CO is pivoted at each end to such car- riages, then the points at the ends of this movable beam and the one at the intersection of the two fixed beams are always on the circumference of a circle whose diameter equals OC however the cross beam is moved, its ends always being on the two fixed ones. In practice, then, the slit or source of light is put at 5, the intersection of the fixed beams ; the concave grating is placed at one of the ends, f ', of the movable beam and so turned that its centre of curvature comes at O, the other end of this beam ; the observations of the spectra are FIG. 279. — Rowland's arrangement of the concave grating. Dirn;A< TION 539 nia'l'\ cither visually or by photographic means, at O. For am definite po>ition of the crossbeam, certain waves in overlapping spectra an- in locus at O; but, as the beam is moved, these change, owing to the change in the angle of incidence upon the tt; rating, (SCO). (WhendiffrmctioD takes place through two rows of rectangular openings, which are su}H'rim- posed at right angles to each other, spectra are formed, as shown in the cut, in two direc- tions. This is illustrated by looking through an umbrella at an arc light.) Historical. — Diffraction gratings were invented and first Constructed in **'«• %*#>• — Diffraction pattern through rectangular 1821 by Fraunhofer, after whom were named the dark lines of the solar spectrum. He used both transmission and reflecting plane gratings, ami measured the wave lengths emitted by many sources of litfht. The concave grating and its peculiar mounting were invented by the late Professor Rowland ; and all the gratings in use at the present time for scientific pur- poses throughout the world, whether plane or concave, have been made by ruling machines which he constructed I ui tin- purpose. In each of these there is a long cylin- drical screw, to one end of which is attached a large toothed wheel, and which carries a long nut; the screw is supported in fixrd collars near its ends, so, as it is turned by means of the toothed wheel, the nut can be made to advance; this nut pushes forward a platform, which movM on horizontal "ways"; a diamond is made automatically to move ;it iio;ht angles to these ways, \\ith a to-aml-IYo motion. The metallic or glass surface to he ruled is attached firmly to this platform: and, as the diamond point isdra\\n ftCTOtt it, a line is ruled; then the diamond is raised and pushed hack t o t he ot her side ; in the meantime, by a partial 540 LIGHT turn of the screw, the platform is carried forward a short distance and the diamond point is lowered; then another line is ruled ; etc. The screw is turned by means of pawls, which are worked by levers, and which push the toothed wheel around through the distance of one or more teeth at a time. Thus, if the pitch of the screw is ^ in., and if there are 1000 teeth in the wheel, an advance of one tooth produces a motion forward of the nut of 20}00 in. ; this, then, is the grating space. Nearly all of the small gratings made by Rowland's machines have 14,438 rulings to the inch, while the larger ones have 15,020. A full description of Rowland's machines is given in his Physical Papers, page 691. Other forms of gratings have been made in recent years which give greater resolving power ; but none are so gen- erally useful as the ordinary c |
oncave grating. CHAPTER XXXIII DOUBLE REFRACTION General Phenomena. — It was observed by Bartholinus as early as ItJti'J that, when a luminous point was viewed through Iceland spar, two images were seen. With any ordinary transparent substance, such as glass or water, only one image is observed; so the phenomenon is called N " double refraction." j \ It was later discovered p! \ A that many substances H ri;*°\ ) had this property. If a doubly refracting (— substance is made in the form of a plate, any ray OA incident upon it from a point source 0 will there- fore give rise to two refracted rays AB1 and AB% ; and these will emerge as two parallel rays coming apparently from two virtual images Ol and Oy which are not in general in the line OPQ, drawn tVom O perpendicular to the sur- face; nor do. in general, the retracted rays lie in the plane of incidence. Since refraction in all eases is due to the fact that the Velocity of waves 18 not the same in all media, the explana- tion of double refract ii. ii in any i lium is that a centre of disturbance in such a medium produces waves whose wave M] front at an\ instant OOnUStfl <>| t\\»> parts or •• sheets"; SO 542 LIGHT that, if a straight line is drawn out from this centre, there will be two points on it, at different distances from the centre, which mark the advance of the disturbance in that direction at any instant. In other words, waves spread out from the centre in such a manner that there are two distinct disturbances advancing with different velocities along any direction. In some doubly refracting substances there is, however, one direction in which these two disturbances have the same velocity, while in all others there are two such directions. This direction in a given body is not a, fixed line in the substance ; for the above statements are true for all lines in the body which have this direction ; it is called the direction of the "optic axis." Those doubly refracting sub- stances which have only one such axis are called "uniaxal" ; the others, which have two axes, are called " biaxal." All crystals which belong to the cubical system, so called, are single refracting; those that belong to the pyramidal or second system are doubly refracting and uniaxal ; the other crystals are doubly refracting and biaxal. Any ordinary iso- tropic transparent substance, such as glass, becomes doubly refracting if it is strained in one direction by pressure, by unequal annealing, etc. Uniaxal Substances. — In the case of uniaxal doubly re- fracting bodies, it is found that one of the rays obeys both of the ordinary laws of refraction, while the other in general obeys neither of them. The former is called the " ordinary ray " ; the latter, the "extraordinary." In such a substance, then, a centre of disturbance gives rise to a spherical wave front, which accounts for the ordinary ray, and also to another wave front which advances with a different velocity, and which cannot be a sphere ; otherwise the extraordinary ray would obey the laws of ordinary refraction, but would have a different index of refraction from the ordinary ray. If a plate of Iceland spar is held between the eye and a bright object, two images of it are seen ; and if the plate is turned DOUBLE REFRACTION 543 around an axis perpendicular to its faces, one image will Ive ;ir<>mi<l the other. Huygens suggested that this ml wave front was the surface of an ellipsoid; and by his work and that of later investigators, this idea has been confirmed. Since the velocity of both disturbances is the same along the op- tic axis, these two surfaces, which make up at any instant the wave surface produced by a point source, the sphere and the ellipsoid, must be tangent to eacli other at the extremities of a diameter having the direction of the axis; and, since the phenomena are symmetrical around this axis, the ellip- soid must be one of revolution around this diameter. Tin- ellipsoid may lie inside the sphere or outside. Plane sec- tions of the two types of wave surface through the axis are ;i uniaxal doubly re- fracting substance. shown in the cut. It is a simple matter of geometry to draw the refracted rays, provided the wave surface is known. Thus, let a nniaxal substance be cut with a plane face making the angle N with the axis, and let plane waves be incident upon this surface in such a direction that the plain- «>t incidence is one that includes the normal to the surface at any point and the direction of the axis at that point sueh a plane is d a •• principal section " of the crystal. If the ineident wave front at any instant is represented in Fig. 283 by 00r waves will spread out from 0 in the douhly refracting sub- stance and will advance, as sj,,.un. f,,r a definite distance, while the disturbance from Ol reaches A. Following the method MI llnvgens, as explained in Chapter XX, the two wave fronts in the lower medium produced by the in. ident plme waves are obtained by drawing through a line per- 544 LIGHT pi'iidii-ular to the sheet of paper at .1 t\vo planes which are tangent to the sphere and the ellipsoid of the wave surface around 0. The tan- gent plane to the sphere touches it at C, and OC, the radius, is perpen- dicular to it. The tangent plane to the ellipsoid touches it at D, but the per- pendicular from 0 upon the plane is OE, not OD. (If the plane of inci- dence were not a _Fm. 283. — Infraction of Iceland-spar. OB is the axis OC and OE are the normals to the refracted wave fronts OC and OD are the ordinary and extraordinary rays. principal section of the crystal, the point of tangency of the ellipsoid would not have been necessarily in the plane of the paper.) The point D marks that point on the wave front AED which the dis- turbance from 0 has reached at the instant of time considered. Therefore the line OD is the ray ; while the line OE is the direction of advance of the plane wave front AD, it is called the "wave normal." It is seen that the "ordinary ray," OC, obeys both the laws of refraction; while the "extraordinary" one, OD, does not, as a rule, obey either. The fact that the extraordinary ray has a direction different from the wave normal is an illustration of what was noted on page 386, that the ordinary case of the ray and the wave-normal being coincident was due to the velocity of the waves in the medium being the same in all directions. Separation of the Ordinary and Extraordinary Rays. - It is often important to separate the ordinary and extraor- dinary rays so that each may be used by itself. There are several methods for doing this : DOUBLM U1.FHACT10N 645 1. Prism. — If the uniaxal substance is made in the form of a prism and placed on a spectrometer table, the two beams of light produced by refraction of the beam from the col- limator will have different directions, and may be ob- served separately. If a glass prism of suitable angle and material is combined with a ORDIHARY RAY RAY EXTRAORDINARY prism of Iceland spar, the ordinary ray may be made to emerge parallel to its original direction, while the extraordinary ray will be deviated and the dispersion of white light produced by one prism may be largely neutralized. Such a compound prism is called "achromatic." Fio. 284. — An achromatic Iceland spar prism. 2. Absorption. — In some uniaxal substances one of the 1 is absorbed much more easily than the other. Thus, a t/tiit plate of tourmaline transmits both rays; but a thicker / one absorbs the ordinary and transmits only the extraoriUnnrii. 3. Total reflection. — Since the two rays have different velocities in the crystal, it is evidently possible to select some isotropic sub- ice, the velocity of waves in which is inter- im-diate between these, and to have the ray incident upon a surface separating the latter substance from the doubly refracting one at such an alible that one <>f the rays u ill suffer total reflection and the other will not. Thus, in Iceland spar the extraordinary ray is refracted less than tin ordinary ; and Canada balsam, a transparent cement, has an intermediate veloc- | so if a piece of Iceland .spar 18 CUt into ••cos by an oblique ne. ti.m and these are cemented to by Canada balsam, we have a means of AMES'S PHYSICS — 36 546 LIGHT separating the two rays. If light from any ordinary source, such as a flame, is incident upon such a compound plate, it will, on entering it, be broken up into two beams, ordinary and extraordinary, which will have different directions ; the former will suffer total reflection at the section of balsam, the latter will be transmitted and will emerge at the end of the plate. The sides of the plate are generally painted black ; so that the ordinary rays are all absorbed. This piece of apparatus was invented by William Nicol of Edinburgh, and is called a "Nicol's prism," or a "nicol." Biaxal Crystals. — In the case of biaxal crystals, neither of the rays, as a rule, obeys the ordinary laws of refraction. The form of the wave surface which was proposed by Fresnel is the true one so far as is known ; but it has not been veri- fied in all particulars. Its properties will be found discussed in any advanced book, such as Preston, Theory of Light, or Drude, Optics. CHAPTER XXXIV POLARIZATION Huygens's Experiment. — In his investigation of the doubly refracting properties of Iceland spar, Huygens no- ticed a remarkable fact concerning the two rays transmitted by a plate made of it. They both appeared to be like ordi- nary beams of light; they could be reflected and refracted; they affected the sense of sight; etc.; yet they were differ- ent in one respect, as was shown by an ingenious experiment. The simplest form of this is a slight modification of Huy- If8 original one. Two identical prisms are cut out of a piece of Iceland spar, so that the optic axis in each makes the same angle with the normal t<> the surface. These are made "achro- matic" (see page 545), and are mounted in tubes so that each can be turned around a line perpendicular to it |
s faces. (This line may be called the "axis of figure.") If light from a small source falls upon one of these prisms, two pen- cils emerge : one, the ordinary : the other, the extraordinary. It now the second prism is placed parallel to the first, these two pencils emerging from the latin- \\ill fall upon the for- ii !••!•; and each will give rise to two pencils, one ordinary, th« other extraordinary; thus, four pencils in all will emerge, two ordinary and two extraordinary. This is true n. ral ; l.ut llnvgens observed that as the tube contain- ing the second |>rism Was turned around its axis of figure, there were f t mns, 90° apart, dm inir a complete revo- lution «.f the tube, in which only two pencils emerged ; and in one of tlieqp positions the two emerging pencils coincide 647 548 LIGHT in direction. For an intermediate position between any two of these there are, as said, four pencils. They all appear equally bright for a position halfway between any two con- secutive ones of these four positions ; but, as the tube is turned, two of these grow feeble and vanish, while the other two grow more intense ; then, as the tube is turned farther, these two grow feebler and finally vanish, while two others appear and grow more intense. A rotation of 90° is required to turn from one of these positions into the other. It is thus evident that the two pencils which emerge from the first prism are not like the incident light. Further, they are not like each other. For, calling the two pencils emerging from the first prism 0 and E ; and the two pencils produced in the second prism by the former pencil, 00 and OE; and those produced by the latter, EO and EE, we may state the above facts as follows: in general, 00, OE, EO, and EE are present ; as the second prism is turned, a position is reached for which only 00 and EE appear; as the rotation is con- tinued for 90°, these disappear, and OE and EO only are present; etc. Thus for one position of the second prism, one of the two pencils incident upon it gives rise to an ordi- nary pencil, while the other produces an extraordinary one ; and after a rotation of 90° this condition is reversed. Huy- gens noticed that when the principal sections of the two prisms were parallel, 00 and EE were transmitted; but when these planes were perpendicular to each other, OE and EO were transmitted. The explanation of all these phenomena is simple if we consider the two rays 0 and E as plane polarized with their planes of vibration at right angles to each other. (See page 313.) The waves are then to be thought of as transverse, and the vibrations of any one beam are all in parallel straight lines in a plane at right angles to the direction of propaga- tion of the waves. The first prism breaks up the incident beam of light into two, whose vibrations are in directions POL A lilt. \TION 549 perpendicular to each other. The experiment shows that e two directions are fixed in the prism ; so that, as it is turned on its axis of figure, the directions of the vibrations of the emerging beams turn also. In other words, at any point in the surface of the prism there are only two direc- tions in which vibrations are possible; and these are fixed in it with reference to the principal section at that point. Vibrations along one line make up the ordinary ray ; those along the other the extraordinary. Thus, let C be any point of the prism of Iceland spar; PCP be the position of the principal section through it ; and CA and CB be the two directions of /p possible vibrations, CA that of the I ordinary, CB that of the extraordi- t nary ray. When the two prisms are in whldi *™ tbc two i \a AM. L AM. • • • i ^ possible directions of vibration. so placed that their principal sections are parallel, the possible directions of vibrations in the two are parallel ; so that, calling the direction of the vibrations of th. ordinary ray in the second_one C^Ar and^that of the vibrations of the extraordinary C^B^ CA and ClAl areparal- B lei, and also fifl-and C\Blt It is thus apparent why 00 and EE are transmitted. Again, when the second prism is rotated through 90°, CA and (^Bj are parallel, and and C*A.\ so OE and EO are c,, "^~" * transmitted. If the second is turned so that the principal planes ''^A, are in. -lined to each other, let the ri«.f8T.-DtofTMnfliastrmtia«iiay. line ' ', . ' i make the angle N with lien a vihrat.o,, along CA having an amplitude CP will be resolved on entering the second prism into two Mluations; one along C\Al having 550 LIGHT an amplitude OP cos N, the other along C^Bl with an amplitude OP sin N. That is, if the amplitude of 0, OP, is called A, we may write 0 = A, 00 = A cos N, OE = A sin N. Similarly, if there is a vibration along CB of ampli- tude A, we have E = A, EO = A sin N, EE=A cos JV. We must consider the light received from ordinary sources, such as the sun, flames, etc., as being made up of vibrations in all directions in the wave front, because it is not polarized in any way, and when these vibrations are analyzed by the first prism into two sets, along CA and CB, their intensities are equal; that is, their amplitudes are the same. Therefore the four pencils of light transmitted by the second prism have amplitudes given as above : 00 = A cos N; OE = AsmN; E0 = A sin N; EE = A cos N. For N= 0, 00 = A, OE = 0, EO = 0,EE = A. As N increases, 00 decreases, OE and EO increase, EE decreases. For N = 90°, 00 = Q,OE = A, EO = A, EE = 0. In this manner the phenomena observed by Huygens are all explained; but the hypothesis on which it is based, viz., that ether waves are transverse, was not advanced until the early part of the nineteenth century, when it was proposed independently by Young and Fresnel. The credit of explain- ing the various phenomena of polarization and of defending this hypothesis must be given the latter. Huygens recog- nized that the only way possible to account for his observa- tions was to assume a two-sided character for the ether waves ; but the only waves known to him were the longitudinal air waves which produce sounds, and the idea of transverse Waves does not seem to have occurred to him. POLARIZATION 551 Phase Differences. — These emerging rays have different phases, partly because 0 and E are not necessarily in the same phase at any instant, and also because they take dif- ferent times to traverse the plates, owing to their different velocities. Thus, if the incident waves are homogeneous, and if Vl is the velocity of the ordinary waves and V2 that of the extraordinary ones, and if h is the thickness of a plate, the difference in phase introduced by the plate, expressed in V>i V terms of time, between the two emerging rays is h ( Y The period of the rays is the same, since the light is assumed to be homogeneous ; and since an angle 2 TT corresponds to a time equal to a period T, the difference of phase of the two rays expressed as an angle is —h(- - -\ But calling the * V* I I ays /j and 1T Vl = •£ and v2 = ^; wave lengths of the two rays u „.,, ^ — ,a — so this quantity may be written 27rhf- — — j. (The ve- locity va depends, of course, upon the direction of the waves in the plate, and so is not a constant.) These formula; will be applied later to explain the colors observed in certain polarization phenomena. Polarization by Reflection. — From these experiments of lluygens, and from the explanation of them just given, mulling can be said in regard to the connection between the lion of the principal section at a point and that of the • liivctions of possible vibrations; but a discovery of Mains, which will bo described immediately, led to the proof that one • t these directions of vibration lies in the principal section. toning to the cut_on page 649, this means that POP coincides with either CA or OB. More recent experiments have shown that it coincides with CB.) Malus discovered by chance that the light reflected from a •*j surface was plane pol. or less completely, ami lie observed that, if sunlight (or light from any onlin.uv 552 LIGHT source) is incident upon such a mirror at a definite angle, called the "polarizing angle," practically all the reflected light is plane polarized, while the transmitted light contains waves that are also plane polarized, but with their vibrations at right angles to the former. Malus made his discovery in 1808 when looking through a plate of Iceland spar at the image of the sun reflected from the window panes of the Luxembourg Palace in Paris ; he noticed that the two trans- mitted pencils were of unequal intensity and that, for certain positions of the crystal, one image vanished while on rotation through 90° the other one disappeared. This showed that the incident light on the crystal was plane polarized. If one looks through such a plate of Iceland spar at light reflected from a plane glass mirror at the polarizing angle, and slowly rotates the plate, it is observed that, when its principal section coincides with the plane of incidence, only the ordinary rays are transmitted ; while, if the principal section is at right angles to the plane of incidence, only the extraordinary rays are transmitted. For positions of the plate between these two, both rays are transmitted, but with different intensities except for the position halfway between. The direction of the vibration of the rays reflected from the plane mirror must by symmetry be either in the plane of incidence or at right angles to it, i.e. parallel to the plane of the mirror; and so the fact just described proves the statement made above in regard to the connection between the principal section of the Iceland spar plate and the possible directions of vibrations. (For many reasons it is believed that the vibrations in the rays polarized by reflection from a glass plate are parallel to its plane. This is in accord with the statement that the vibrations of the extraordinary rays are in the principal sec- tion. Therefore the vibrations transmitted by a Nicol's prism a |
re in the principal section.) If the light is not reflected at the polarizing angle, only a portion of the reflected light is plane polarized ; the rest is POLARIZATION 563 like the incident light, made up of rays whose vibrations are in all directions in tin- wave front. At the polarizing angle more of the transmitted light is plane polarized than for any other angle ; but, as said above, these vibrations are at right angles to those of the reflected light, as may be shown by viewing it through an Iceland spar plate. Brewster's Law. — It is found by experiment that light reflected from a plane mirror of any transparent isotropic substance which shows ordinary normal dispersion, such as all kinds of glass, water, etc., may be plane polarized for certain definite polarizing angles. (This is not true of re- flection from metallic mirrors or from substances showing anom- alous dispersion.) A connection between the polarizing angle of any substance and its index of refraction was established by .vster. He found that at the polarizing angle the reflected and retracted rays were perpen- dicular to each other. If, in the Fio.m-Bwwst«r'8Lawiiir<*»Tdto cut, MM is the plane surface of msparent substance, whose index of refraction with ref< T- ence to the surrounding nu-d in m is n, and if BA* AD, and AC are the incident, tin- refracted, and the reflected rays respec- tively, experiments sliow, u just said, that if the angle (BA /') is the polari/mg angle, {CAD) is a right angle. Therefore the angle of refraction {DAP') equals (CAM) ; so, calling tin- angle of incidence N, (DAP') = *-N. The index of refraction n satisfies the emiat inn n s= 8in \ _'; and hence, ut the polarizing angle, n = - -—= tan N\ or, the angle of cos A polari/.atinn -nUunrr ii such that its tangent equals the index of refraction. This is known as " Brews ter's jy *\r\(DAP') 554 LIGHT Law." (More careful experiments have shown that there is no angle of incidence for which all the reflected light is plane polarized ; in reality there is always a small amount not so polarized, but at the polarizing angle this is small.). The value of the polarizing angle of pure water is 53° 11' ; of crown glass, about 57° ; etc. Plane of Polarization. — The light which is plane polarized by reflection from a plane transparent surface is by definition said to be " polarized in the plane of incidence " ; or its "plane of polarization" is said to be that of the plane of incidence. (This definition is entirely independent of any conception of the directions of the vibrations of the rays in this reflected beam; but, accepting the statements made above in regard to these directions, it is seen that in a plane polarized beam the direction of vibration is in the wave front and at right angles to the plane of polarization.) Pile of Plates. — A means is obviously offered of securing plane polarized waves by reflecting ordinary sunlight or light from a flame, etc., from a glass or water surface at the polarizing angle. In general, only a small quantity of light is reflected owing to the poor reflecting power of glass or water; but the effect can be increased greatly in the case of glass by using several thin plates placed one on top of the other, thus forming a "pile of plates." When light falls upon such a pile at the polarizing angle, the reflected light is plane polarized, but part of the refracted light is not ; this falls at the polarizing angle upon the surface where the top plate meets the next one, is partially reflected, and is refracted out so as to coincide in direction with the beam reflected from the plate, being polarized also like it. The light which enters the second plate is still not completely polarized, and when this meets the third plate, reflection again occurs ; etc. Thus both the light which is reflected and that which is transmitted by a pile of plates is practically completely plane polarized, but in planes at right angles to eacli other. POLARIZATION fi5f, If plant1 waves plane polari/.ed in the plane of incidence incident upon a i^lass surface at tlie polarizing angle, they will be entirely reflected; while, if they are polarized in a plane at right angles to this, they will be entirely transmitted. These and all the facts in repaid to polarization by reflection may be shown best by the use of a piece el apparatus invented by Norrenberg, which i own in the cut. It consists essentially, is seen, of two plane mirrors which may be turned about axes lying in their planes and also in such a manner that their nor- mals may lie in different planes. Plane Polarization. — We have, therefore, t\\o general methods for the production of plane polarized light; one is to use a crystal of Iceland spar, or other uniaxal crystal, and get rid of one of the rays by the means described on page 544 : the other is to use a pile of plates at the polarizing angle. Fi».«a.-N6rrenb«rg'« apparatus. In practice a Nicol's prism is almost invariably used. Similarly, either one of these methods may be used to test whether a certain beam of light is plane polarized ; thus, if such a beam is incident upon a Nicol's prism, it will be en- tirely extinguished for some position of the latter, as it is slouly turned on its axis of liLnire; and when it is extin- guished, it is known that its vibrations are at right angles to principal section of the prism. In this manner it may be shown that, if one looks at the blue >k rection at right angles to a line joining tin- eye to the sun, the reflected sunlight — that which is scattered by the tine put icles (see page 480)— is plane polarized in the ie including the two directions just mentioned. We have thus another means of securing plane polari/.ed liijht. (It U predict from theory what the direction of the Nibi 111 tins case; and, by comparing this direction 556 LIGHT with the observed position of the plane of polarization, it is proved that the vibrations are at right angles to it. (See page 554.) Interference of Plane Polarized Waves. — Fresnel and Arago performed by means of two piles of plates a most ingenious experiment to determine whether the vibrations in the waves transmitted by them were exactly in planes at right angles to the direction of propagation of the light. If such is the case, by using independently the two piles with their planes of incidence at right angles to each other, two beams of light may be secured in which the vibrations are in the wave fronts but are perpendicular to each other. Two such trains of waves as this cannot " interfere " ; because, in order to have one train interfere with another, the vibrations of both must be in the same straight line. The experiment re- ferred to consisted in modifying Young's original interference one by introducing a pile of plates in front of each of the two slits. We shall quote from their own description, follow- ing Crew's translation in his Memoirs on the Wave Theory of Light : " It has been known for a long time that if one cuts two very narrow slits close together in a thin screen and illuminates them by a single luminous point, there will be produced be- hind the screen a series of bright bands resulting from the meeting of the rays passing through the right-hand slit with those passing through the left. In order to polarize at right angles the rays passing through these two apertures, ... we selected fifteen plates as clear as possible and superposed them. This pile was next cut in two by use of a sharp tool. So that now we had two piles of plates of almost exactly the same thickness, at least in those parts bordering on the line of bisection ; and this would be true even if the component plates had been perceptibly wedge shaped. The light trans- mitted by these plates was almost completely polarized when the angle of incidence was about thirty degrees. And it was I'ULAHI/.ATION 557 : ly at this angle of incidence that the plates were inclined when they were placed in front of the slits in the copper screen. •• When the two planes of incidence were parallel, i.e. when the plates were inclined in the same direction; — up and down. for instance, — one could very distinctly see the interference hands produced by the two polarized pencils. In fact, they behave exactly as two rays of ordinary light. But if one of the piles be rotated about the incident ray until the two planes of incidence are at right angles to each other, the first pile, say, remaining inclined up and down while the second is inclined from right to left, then the two emergent pencils will be polarized at right angles to each other and will not, on meeting, produce any interference bands. . . . We must hide that rays of light polarized at right angles do not affect one another." Fresnel and Arago performed another most interesting experiment which is also described in the memoir from which we have just quoted. Its object was to determine the con- ditions under which interference may occur when two plane polarized waves with their vibrations parallel are brought together. It B incidentally explains the production pf «.f the beautiful colors which are seen when a thin plate made of a crystal or of any doubly refracting substance is interposed between two principal sections are \ / X line so that one may look through Ixith at a sourer of white light. In any such plate, whether uninxnl or biaxal, there are only two possible direc- i whirli rations of the transmitted waves may have. Let these two directions at a point 0 of the plate be OA and OB\ and let the vibrations of the waves 558 LIGHT upon itjrom the first nicol have the direction and the ampli- tude OP. This is resolved into two, a vibration along OA of amplitude OPV and one along OB of amplitude OPV where these lines are the components of OP in these directions; and these two trains of waves are transmitted by the plate. Let us consider the waves as homogeneous, so that they all have the same period. As they enter the plate they are in the same phase |
because they are components of the same vibration ; but they will emerge with a difference in phase, expressed in time by hi --- ), where h is the thickness of \vl v2J the plate and v1 and v% are the velocities of the two waves. They will then fall upon the second nicol whose principal section may have the direction ON. The waves whose vibra- tions have the direction and amplitude OPl will be resolved into two, but the only one transmitted is that whose vibra- tions have the direction and amplitude OQV where this is the projection of OPl on ON. Similarly, the other train of waves whose vibrations are represented by OP2 will be re- solved into two, but the only one transmitted is that whose vibrations are represented by OQ^ the projection of OP2 on ON. Therefore, two trains of waves emerge from the second nicol, which are plane polarized with their vibrations in the same direction, and which have a difference in phase of h(—— — ]. This expression may be simplified if the values v*i V of v1 and v2 in terms of the period and wave length are sub- stituted, viz., v1 = ^J, v2=-%i> hence tne difference in phase may be written Th(- - -). Expressed in terms of an angle , -\h y If N! is the angle between OP and OA, 6Pl = OP cos Nv OP2 = OP sin NL. I POLARIZATION 559 And if NI is the angle between (Wand OA, OQl = OP, cos Nt = OP cos AT, cos N*, = 0Pa sin N* = OP sin ^ sin JV2. So if the original vibration as it leaves the first nicol is A cos nt, and if this difference of phase introduced by the plate is d, the resulting vibrations as they leave the second nicol may be written A cos JVj cos Nt cos nf, A sin AT, sin #2 cos (nt — d). If ON is perpendicular to OP, the nicols are said to be "crossed," and no light is transmitted if there is no plate interposed between them. If this is the case, in the above formula ^=—(90° — ^); so cos JV2 = sin^Vr and sin N2 = — cos -ZVj ; and the two vibrations are A cos ^ sin N! cos nt and — A cos JVj sin N2 cos (nt — d ). This last vibration may be written A cos JVj sin Nt cos (n/ — d — TT). FIG. 291.— Special case of crossed nlcola. So the difference in phase is rf and the amplitudes are equal. If this quantity d -f- TT is equivalent to an odd numlMT <>f half periods, i.e. if d + TT = (2 m + 1) TT, or rf = 2 WITT, the interference will be complete: and thrst; homOgeneOQfl waves will bo extin- guished. (They suffer total reflection at the surface win i. the two halves of the last nicol are cemented together.) Other cases, in which the two nicols are not crossed, m ax- be found discussed in advanced text-books. So, if white light is used, certain waves will be absent in the transmitted light, ami it will he colored. Which par- ti< Milar waves are absent depends upon the thickness of the double refracting plate, as is evident from the formula. 560 LIGHT The first nicol, which polarizes the incident light, is called the "polarizer"; the second one is called the " analyser." If there is no such plate between the crossed nicols, no light at all is transmitted ; but, if a plate is introduced, cer- tain waves appear, as just explained. The phenomenon is called " depolarization " ; and the experiment serves as an extremely delicate test of the double refraction of a substance. It at first sight appears as if, in the above experiment, the first nicol might be removed so that the light would fall directly upon the double refracting plate and then upon the second nicol, and there might still be interference ; for the plate would break up the light into two beams and introduce a dif- ference of phase between them before they were combined again by the second nicol. But the relations between the ampli- tudes and the phases in this case are not definite, because the light incident upon the plate is not polarized, but consists of vibrations in all directions ; and so there is no permanent interference. This fact is the fundamental one established by the experiments of Fresnel and Arago. Circular and Elliptical Polarization. — The fact that a plate of a doubly refracting substance breaks up an incident beam of plane polarized light into two such beams, polarized at right angles to each other and with a differ- ence in phase between them which varies directly with the thickness of the plate, renders it possible to secure a circular or an elliptical vibration. A beam of light whose vibrations are A of this character is said to be "circu- FIG. 292. — Formation of cir- J J F larly" or "elliptically" polarized. cularly and elliptically polarized Thus let OA and OB DG the tions of possible vibrations in the plate, and let OP be the direction of the principal section of the nicol through which the light is incident upon the plate. POL A UIZA T1ON If the amplitude of this plane polarized light is OP, those of the two beams transmitted by the plate are OP1 and OP2. Whatever the difference of phase between them introduced by the plate, these two vibrations will combine to form an elliptical one. (See page 324.) If this differ- ence in phase is equivalent to a quarter of a period, or to any odd number of quarter periods, the vibration is an ellipse whose axes coincide in direction with the lines OA and OB. If, in addition to this condition for the thickness of the plate being satisfied, the incident vibration OP bisects the angle between OA and OB, the amplitudes of the two trans- mitted beams will be the same ; and they will combine to form a circular vibration. ' Such a plate is called a "quarter wave plate" ; and obviously plates of different thicknesses must be used for waves of different wave length or color. Fresnel's Rhomb. — If plane polarized light is totally re- flected from the surface of a transparent substance such as glass or water, it becomes, in general, elliptically polarized ; for, if the light is plane polarized in such a manner that the direction of the vibration is neither parallel to the surface nor in the plane of incidence, it is resolved by reflection into two plane polarized beams, one with its vibrations parallel to the surface, the other with its vibration in the plane of incidence. Their amplitudes are different, unless the direction of vibration in the incident beam bisects the an^le between two lines in the wave front, one parallel to the surface, the other in the plane of incidence. A change of phase is introduced by the reflection, which is not the same for the two beams; and the difference for the two depends upon the angle of incidence and the material at whose surface the reflection takes place. Fresnel m.ide a rhomb of such a particular kind of glass AMES'S PHYSICS — 86 562 LIGHT and with such angles that when light was incident perpen- dicularly upon one of its end faces, it would suffer total reflection twice and emerge perpendicular to the opposite face with a difference of phase equivalent to a quarter of a period between the two plane polarized beams. By this means it is possible to obtain light circularly polarized, or elliptically polarized with one axis in the plane of incidence and the other at right angles to it. Detection of circularly or elliptically Polarized Light. - If circularly polarized light is incident upon a Fresnel's rhomb or upon a quarter wave plate, it will emerge plane polarized, for the effect of these pieces of apparatus is to in- troduce a difference of phase of a quartet* of a period between the two component plane polarized waves into which the incident waves are resolved. The existence of this plane polarized light may be detected by a nicol. If elliptically polarized light is passed through a Fresnel's rhomb or a quarter wave plate, it will, in general, emerge elliptically polarized ; but, if the plate is turned in its own plane, or the rhomb is turned around an axis perpendicular to the planes of its end faces, there will be four positions in one complete revolution for which this light will be plane polarized. This may be detected by a nicol. Other methods are described in advanced text-books. Ordinary Light. — The light which we receive from ordi- nary sources, such as diffused sunlight, etc., is not polarized in any manner ; yet it can be transformed into plane, circu- larly, or elliptically polarized light by methods which have been discussed. When ordinary light is passed through a doubly refracting substance, both the transmitted plane polar- ized beams are of equal intensity, and there is no permanent phase relation between them. This shows that we must consider ordinary light as due to transverse waves in which the vibration at any instant may be rectilinear or circular, etc.. but in which the vibration is continually changing its POLARIZATION 563 form. We may regard it, then, as due to an ever changing mixture of transverse rectilinear vibrations. Rotation of the Plane of Polarization. — There are certain substances which have a most remarkable property in regard to plane polarized light. If one of them is made in the form of a plate and a beam of homogeneous light plane polarized in a } (articular direction is transmitted through it, the emerg- ing light is plane polarized, but its plane of polarization has been rotated through a certain angle. Such substances are said to be "optically active." Thus, if the incident light is produced by the use of a nicol, and a second nicol is " crossed " with it, no light passes before the " active " substance is intro- duced between them ; but after this is done, the second nicol must be turned on its axis of figure through a definite angle before the light is again extinguished. This angle varies directly with the thickness of the substance, and is different for waves of different wave lengths, being much _ri' ater for the short waves than for the long ones. This last phenomenon is called " rotatory dispersion." There are two classes of these substances; one is made up of bodies which are naturally active, while the other contains bodies which are active only when they are under the influ- ence o |
f a magnetic force. The former phenomenon was discovered by Hint ; the latter, by Faraday. A kinematic explanation of this rotation was given by Fresnel; but it is not necessary to state it here. It may be found in any advanced treatise. a. Naturally active bodies. — Examples of these bodies are quart/, when cut at ri^ht angles to its optic axis, an aqueniis solution of certain tartaric acids, of many of the irs, etc. In all these cases, if the plane polar i/ed li^ht is made to pass through a plate and then by means of a m is reflected back ,i.;,nn, the plane of polarization is rotated in one direction and then in the opposite, so it emerges the second time polari/ed exactly as it was on incidence; 564 LIGHT it is as if one screwed a screw into a board and then unscrewed it. In certain bodies the plane of polarization is rotated in a right-handed direction, while in others it is turned in the opposite sense. Thus, if the light is emerging in a direction_perpendicularly up from the X paper, and if AB is the direction of the prin- cipal section of the second nicol, in the experi- ment described above, when it is so placed as to extinguish the light before the active FIG 294 - Rota- substance is introduced ; and, if after this tion of the plane of takes place, the nicol must be turned in the direction shown by the arrow in order to extinguish the light again, the rotation is said to be "right- handed." If, on the other hand, the rotation of the nicol must be in the opposite direction, it is called "left-handed." There are two varieties of quartz, left-handed and right- handed ; two varieties of active tartaric acids, etc. It was discovered by Pasteur that all optically active sub- stances were made up entirely or in part of certain crystals which had a "hemihedral" form ; of which there are for any substance two possible states. These two are symmetrical with reference to a plane, like the two hands of any individ- ual. Thus right-handed quartz has imbedded in it minute hemihedral crystals of one form ; while left-handed quartz has hemihedral crystals of the symmetrical form, etc. Crys- tals of tartaric acid are hemihedral; and when dissolved in water the molecules retain their asymmetric character. These facts are the basis of what is called "stereochemistry," which is a branch of chemistry dealing with conceptions of the atomic arrangement in certain organic molecules. (See Richardson, Foundations of Stereochemistry, New York.) b. Magnetically active substances. — When any transpar- ent substance, such as glass, is placed in an intense magnetic field, it acquires the power of rotating the plane of polariza- POLARIZATION 565 tion ; but this rotation is different from that just described, because, if the rotated light is reflected back on its path, the plane is rotated still farther, in the same direction as before ; it is not turned back into its previous position. When the subject of electricity is discussed, it will be shown that if an electric current is passed through a wire wound in a helix, there is a strong magnetic field inside it, and that the direction of this field is reversed if that of the current is reversed. It is found by experiment that, if a piece of transparent matter is introduced in this helix, the direction of the rotation of the plane of polarization is that of the electric current in the helix. Metallic Reflection. — When plane polarized light falls upon a polished metal surface, it is reflected according to the ordinary laws; but the light is elliptically polarized, unless the incidence is normal. This is owing to the fact that the incident light is broken up into two plane polarized beams which have a difference in phase. If the metal surface is magnetic, that is, if it is made of iron, steel, nickel, etc., the character of the reflected light — the shape of the resulting ellipse — depends upon whether the metal is in its natural condition or is magnetized; if the latter is the case, the effect of the reflection also varies with the direction and the intensity of the magnetization. This IIMWII as the " Kerr effect," and will l>e found fully described in any advanced text-book. CHAPTER XXXV VELOCITY OF LIGHT THE first experiments to determine whether, like sound, light traveled with a measurable velocity were performed by Galileo. They consisted in having one observer flash a light which was seen by a second one at a considerable dis- tance, who then flashed another light as quickly as possible, and this was seen and noted by the first observer. In any case there would necessarily be an interval between the instant when the first observer flashed his light and when he saw that flashed by the second one, owing to the time required to perform the manipulation, but, if time were re- quired for the passage of the light across the space between the two observers, this interval would vary directly with the distance between the observers. No such effect was observed. The interval of time between the events referred to was ap- parently independent of the distance apart of the observers, and was conditioned only by their quickness of motion and perception. It was therefore concluded that light traveled with an infinite velocity. Method of Roemer. — This opinion was maintained by every one until the year 1676, when Roemer, a Danish as- tronomer, then living in Paris, made certain observations on the eclipses of one of the satellites of Jupiter by the planet, which he interpreted as proving that the velocity of light was finite, but very great. As a satellite revolves around its planet — e.g. the moon around the earth — its motion is periodic, or may be assumed to be so to a very high degree of accuracy. So the interval of time which elapses between 566 7MLOCITT oi' LK.nr two consecutive instants of disappearance of a satellite of Jupiter behind the edge of the planet, when viewed from a point in space fixe<l with reference to the planet, must be a constant quantity. Roemer observed that this interval of time, as noted here on the earth, was not, however, the same when the earth in its motion around the sun was moving away from Jupiter, as when it was approaching it; it was longer in the former case than in the latter. Further, if this period of revolution of the satellite was noted when the earth was nearest Jupiter, and if a calculation was made, based on this, of the instant at which at the end of half a year, when the earth was farthest from Jupiter, an eelipse should take place, there was found to be a differ- ence of 996 sec. between this calculated instant and the observed one. Roemer saw that these facts could all be explained if the assumption were made, that it takes time for the propagation of light across space. As the earth is receding from Jupiter, light has to travel a greater distance at the instant of the second eclipse than at that of the first : and so the apparent period ,,f the satellite is greater than it would l>e if the earth were at rest with reference to J uj liter. Just the reverse is true when the earth is approach- ing the planet. So, when the earth is farthest from Jupiter, light has to travel an additional dUtanee equal to the diameter lie earth's orbit ; and the interval of 996 SCC. observed, as stated above, between the calculated instant of an eclipse and the icc,,rd.-d one is the time required for light to pass ii stance; and so the velocity of light may be rmined. More recent observations have given 1002 sec. as the interval of tin). .'oenier; and the diameter of the earth*! orl.it may be taken as 2998 x 106 Km., as will be shown immediately; so the velocity of Iiurht in space given l.\ this method is this quantity divided by 1003, or 2.984 x 1010 cm. per second. 568 LIGHT The angle subtended at the sun by the radius of the earth, N in the diagram, is called the " solar parallax " ; and it is a quantity whose value may be determined by astronomical observations with a high degree of accuracy. The accepted value of this constant at the present time is 8.79 sec. of arc. The radius of the earth is known, its value being 6378 Km. So the distance from the earth to the sun may be calculated. Referring to the cut, AE = 6.378 x 108 cm., But 27r in angular measure = 360 degrees of arc FIG. 295. — Solar parallax : S is the sun ; E is the earth. and N= 8.79 sec. of arc. = 360 x 60 x 60 sec. ; so 1 sec. = -, and 8.79 sec. =- 360x60x60" 360x60x60 27TX8.79 Therefore, since N is so small, we may write AE = ES x N, or t™ AE 6.378 x 108 x 360 x 60 x 60 27TX8.79 = 1.4966 x 1018cm. The diameter of the earth's orbit is, then, twice this, or 2.993 x 1018 cm. Method of Bradley. — Another method by which the veloc- ity of light could be determined was discovered by Bradley, the great English astronomer, in 1727. He had observed that if a fixed star was observed through a telescope, this instrument had to be pointed in slightly different directions at different times of the year ; so that, if the image of the star were kept on the cross hairs of the telescope during the whole year, the instrument had to be kept in continual motion in such a manner that its end described a small curve. The explana- tion given by Bradley was extremely simple. Consider a long tube closed at its two ends by caps in which there are two openings directly opposite each other. A particle entering at one open- ing, with a motion parallel to the axis Fio. 296. — Diagram repre- senting stellar aberration. 1 C B of the tube, will escape through the opening at the other end, if the tube is at rest. But, if the tube is moving at right VELOCITY OF LIGHT 569 angles to its length, the opening in the opposite end from tin- one at which the particle enters must be displaced in a direction opposite to the motion of the tube if the particle is to escape. The line of motion of the particle, with refer- ence to the tube, makes an angle with the axis of the tube whose tangent equals the ratio of the velocity of the tube to th |
at of the particle; for, referring to the cut, if Vl is the velocity of the tube and V% that of the particle, and if t is the time taken for the particle to pass through the tube, 03= Vj and AB= F3*; so tan (BAG) = = . Simi- larly, if we consider light from a fixed star entering a tele- scope, if the instrument is at rest, the light will emerge directly from its farther end ; but, if the telescope is mov- ing at right angles to the direction from which the light is coming, it must be inclined forward in order to see the star, and the reason for this, according to Bradley, is because the "path of the light" is along the line AC. The angle through which the telescope has to be turned when it is pointed approximately perpendicular to the path of the earth in its orbit, is called the "constant of aberration." (Its value according to recent astronomical observations is slightly less than 20.5 sec. of arc.) If this angle is determined by observations on the fixed stars, the velocity of liijht may be calculated, assuming that in the above formula Vl is the velocity of the earth in its orbit and \\ that of liurht, because the velocity of tin- earth in its orbit is known. The value thus deduced is 2.982 x 1010 cm. per second. This explanation of stellar aberration is insufficient to account I'm- all the facts. In the above formula T, w«>ul«l be the velocity of light inside the telescope; therefore, if the telescope tube is filled with water, V^ is diminished in the ratio of the index of refraction of water to air, and so the aberration angle should l»e increased. This experiment was actually performed by Sir George Airy; 570 LIGHT and no change in the angle was observed. When we con- sider light as due to waves in the ether, and if we assume that the ether as a medium does not move as the earth travels through it in its orbital motion, the same formula, as given above, may be deduced for the aberration angle, only in it V% is the velocity of light in the pure ether, not when it is inside matter. Method of Fizeau. — The first method for measuring the velocity of light directly here on the earth, without making use of any astronomical data, was devised and applied by Fizeau in 1849. The principle is extremely simple. A source of light is placed in such a manner that it shines FIG. 297. — Fizeau's toothed wheel apparatus for measuring the velocity of light. through the space between two teeth of a cogwheel, which may be driven at a high speed. At some distance on the other side of this wheel is a mirror which reflects the light it receives back toward the source of light. If then the cogwheel is rotating so rapidly that the light which passes between any two teeth travels to the mirror and is reflected back at such an interval of time that the wheel has turned through a distance which brings a tooth to where an opening was, no light passes back through the wheel. If the speed of the wheel is increased, an opening will come where the previous one was, and the return light will pass through ; if the speed of the wheel is increased still more, the light will be again shut off; etc. If the speed of the wheel, and the VELOCITY OF LIGHT 571 distance over which the light travels are measured, the veloc- ity of light can be calculated. The actual arrangement of the apparatus is shown in the cut. W is the revolving toothed wheel ; P is the source of light ; .p is a plate of glass which partially reflects the light from P into the direction of the wheel ; L^ is a converging h-ii s which focuses the light from P at /, a point at the edge of the wheel ; Za is u converging lens so placed that / is its principal focus; Lz is another lens placed at a considerable distance from Lv but parallel to it, with its principal focus at the middle point of the concave mirror *S> whose centre of curvature is at the centre of the lens Ly The waves from P are thus converged at /, whence they diverge and are made plane by Z2 ; they are converged again by l/8, and are reflected back on their path, until they reach the glass plate p by which tliry an- in part transmitted; so an observer with an eye- piece 0 focused on / can determine when the reflected light passes through an opening and when it is shut off. 1 1 the number of revolutions of the wheel per second is n, its angular velocity is 2 TTH. If there are N teeth in the wlirrl, the angular distance from one edge of one tooth to the corresponding edge of the next one is -=•; and so the 2 7T -Zv angular distance from the middle point of one opening to the ii ii< Idle point of a tooth is ~ The time required to turn tlm.uirj, this angle is ^ • - — =»- — — • If the distance from N 2 irn 2 nN the wheel to the distant minor is D (or what is practically thr sun,-, t he distance from Za to Z8), the path of the light is2Z>. So, if the angular speed STTTI is that \\hich corre- sponds to the first obscuration of ili«- litflit. it has traversed a distance 2 7) in the time - — — ; and the velocity of light '_' nN is, then, 2 D divided by - J—, or * M fwAw 572 LIGHT In one of Fizeau's original experiments, the distance from La to L3 was 8.633 Kin. or 8.633 x 108 cm. ; the toothed wheel had 720 teeth ; and it was found that the first obscuration of the reflected light occurred when the wheel was making 12.6 revolutions per second. Therefore, in the above formula n = 12.6, N = 720, D = 8.633 x 1010, and so the velocity of light is determined to be 3.13 x 1010 cm. per second. The great experimental difficulty is to maintain a uniform speed of the wheel and to measure it accurately. This method has been used in more recent years by Cornu and Perrotin. In the work of the latter, the distance apart of the lenses L2 and L3 was about 12 Km. The results of these experiments is to give 2.99820 x 1010 cm. per second as the velocity of light in air. The object of using a concave mirror at S with its centre of curvature at the centre of Z-3, instead of a plane mirror, is apparent if it is remem- bered that the source of light at P is not a point, but is extended. So the waves from a point near P are converged at /, and are made plane by the lens L2 ; but their line of propagation is inclined slightly to that of the waves from P, and they are converged by the lens Ls upon a point of the mirror S, a short distance away from its middle point. If the mirror were plane, these waves would then be reflected off one side, and so would not return through the lens L3 ; but, since the mirror is concave, with its centre of curvature at the centre of L3, these incident waves are reflected back through the lens and finally reach the wheel. Therefore the reflected light is brighter than it would otherwise be. Method of Foucault. — Another method was suggested by Arago, but was first put in practical use by Foucault in 1850, and is always called by his name. (Fizeau also made some valuable suggestions in regard to it.) It consists of making use of a rotating plane mirror in the following manner : Referring to Fig. 298, there is a source of light at P ; a plate of glass at p ; a revolving mirror at m, whose axis of rotation is perpendicular to the plane of the paper ; a converging lens .L, which focuses upon a concave mirror S the light from P reflected at m ; the centre of curvature of this mirror S is at m. For a suitable position of the revolving mirror the light from the source P will be focused at the middle point of the concave mirror, and will be reflected back on its path until it reaches the glass plate, when it will be in part reflected and VELOCITY OF LIGHT 573 will form an image at a point P'. The revolving mirror is, of necessity, small, almost linear ; and so, in order to col- lect more light from the source P, the I;P" mirror S is concave. (Another method, adopted later, was to place the lens L at a distance from m equal to its focal length ; and in this case the mirror S may be plane.) If, then, a reflected image Of P FIG. 298. — Foucault's revolving mirror apparatus for i is formed at P' when uring the veloclty of "*ht- the revolving mirror is at rest, a displaced image P" will be formed when the mirror is turning rapidly, because in the time taken for the waves to pass from m to S and back again, the mirror m will have turned a short distance, and so a reflected ray coming from S will have an angle of inci- dence upon m different from that which it would have if the minor had not moved. This ray, after reflection from m, \vill have a different patli from the incident ray. If the mirror is making ti revolutions per second, its angu- lar velocity is 2 Trn, and that of the reflected ray is 4 Trn. (See page 445.) If the distance from m to S is D, if that from m to P is r, and if v is the velocity of light, - - is the time 2 D required for the light to pass from m to A^and back again; in this tim«. the mirror will have tnrne<l through an angle 27) o /> - • 2 Trn, and the reflected ray through - -- 4 Trn ; therefore v the image of P will be displaced by a distance r times this. So, calling P*P". :>74 LIGHT Foucault increased the effective distance D by having the light reflected several times back and forth between five mirrors before it was finally returned to the revolving mirror ; but in no case did he obtain a very large displacement P'P". Michelson, however, by changing the arrangement of the apparatus was able to increase D to 600 m.; and even when the mirror was turning at the moderate speed of 200 revolutions per second, he obtained a displacement of 13 cm. This method was improved still more by Newcomb, who FIG. 298 a. — Michelson's modification of Foucault's apparatus. operated over a distance of 3721 m. The final result ob- tained for the velocity of light in air by this method was 2.999778 x 1010 cm. per second. The mean of all the best values for the velocity of light in the ether is 2.999880 x 1010 cm. per second, with a probable error of about 20 Km. It should be noted that the values of this velocity obtained dir |
ectly by the methods of Fizeau and Foucault are for the velocity in air ; and, since the index of refraction of air is 1.00029, the velocity in the pure ether is greater in this ratio. The figure given above for the final value is corrected in this manner so as to apply to the pure ether. Velocity of Waves of Different Periods. — In the pure ether of interstellar space all ether waves, of whatever period, travel with the same velocity so far as is known, as is shown by the fact that the color of any one of Jupiter's satellites is VELOCITY OF LIGHT 575 the same to our eyes when we observe it as it goes into eclipse and as it emerges. If the short waves traveled faster than the longer ones, the satellite would appear red as it disappeared and blue as it reappeared ; and the converse would be true if the long waves traveled more rapidly. In ordinary transparent matter, however, not alone do all waves travel more slowly than in the pure ether, but the waves of different periods have different velocities. It is this which explains refraction and dispersion. Foucault showed by direct experiment that the velocity of light was less in water than in air, by placing a long tube of water in his apparatus immediately in front of the concave mirror. (Actually he used two concave mirrors, one for the light passing in air, the other for light passing in water, and thus olitaim-d two displaced images.) Michelson showed the same for water and for carbon bisulphide; and he also proved directly that in these substances red light travels more rapidly than blue. CHAPTER XXXVI RADIATION AND ABSORPTION SPECTRA Discovery by Newton of Nature of White Light. In the year 1672 Newton made the interesting discovery that when sunlight was admitted into a darkened room through a small opening and was allowed to traverse a glass prism, the trans- mitted light was no longer white, but consisted of beams of different colors, each color having a different refrangibility and therefore a direction differing slightly from that of its neighbors. He recognized as distinct colors violet, indigo, blue, green, yellow, orange, and red; but all other intermedi- ate shades were present also. He performed further the reverse experiment of combining these colors by means of a second prism, and produced white light again. He also showed that it was impossible by means of a second prism to break up any of these spectrum colors into parts. These observations prove that white light is due to a combination of simple elementary causes; and we know from Young's experiments that these are trains of waves of definite wave lengths, each train being characteristic of a definite color if it is perceived by the eye. These experiments of Newton form the basis of our expla- nation of the color of natural objects and of the science of spectrum analysis. Various bodies in the universe are emit- ting light (or, more generally, all bodies are emitting ether waves) ; all bodies reflect light (or ether waves) to a greater or less extent ; so, if we look at any object, the light (or ether waves) which we receive is due to various causes. We can analyze this radiation into its component parts by means 676 RADIATION AND ABSORPTION SPECTRA 577 of suitable dispersive apparatus, and can then detect these separate trains of waves by proper means. We shall con- sider in this chapter (1) methods of producing ether waves, especially those which appeal to our sense of sight ; (2) dif- ferent forms of dispersive apparatus and different modes of recognizing trains of waves of different wave length ; (3) the results of the examination of the radiations from different sources. Sources and Cause of Radiation Radiation owing to Temperature. — All substances in the universe are, so far as kumvn to us, emitting ether waves, owing to the vibrations of certain parts inside their mole- cules. If the body when placed in a darkened room can be seen by the eye, it is said to give off light. The ordinary method of making a body luminous is to raise its tempera- ture ; thus a body may be exposed to a hot flame, such as one from a Bun sen burner, or it may be placed in the poles of an electric arc light. (See page 665.) At such high tem- peratures, many bodies are vaporized, and their vapors are then at this temperature. The laws of radiation due to this cause have been discussed in Chapter XIV. Electro-luminescence. — Again, if an electric spark is made to pass between two metal points, they are vaporized ami the vapors are luminous ; not, however, owing entirely to the temperature being raised. (The same statement is true of the luminosity of the vapors in the electric arc ; it is only in part due to the temperature of the vapors.) Similarly, if a gas or vapor is inclosed in a hollow vessel, such as a glass bulb, and an electric discharge1 through it is produced 1>\ any means, it becomes luminous. These cases of luminosity are said to be due to "electro-luminescence." Chemical Luminescence. — In certain chemical reactions light is emitted ; for instance, when a piece of decayed wood slowly oxidizes, or when phosphorus is oxidized. These are illustrations of "chemical luminescence." AMES'S PHYSICS — 578 LIGHT Fluorescence and Phosphorescence. — There are many bodies which emit waves as a result of their having absorbed other ether waves, quite apart from any radiation due to tempera- ture alone. Some bodies emit these waves only while they are absorbing the other waves; while others continue to emit them even after the absorption ceases. All bodies of this kind are called " fluorescent," and the phenomenon itself is called " fluorescence " ; while the second division of these bodies, as just described, are called "phosphorescent," and the phenomenon is called "phosphorescence." If a beam of light is passed into a fluorescent substance, certain trains of waves are absorbed and others are transmitted ; the energy of these absorbed waves is not spent in producing heat effects, but in emitting other ether waves, which proceed out in all directions. So this fluorescent light may best be seen by looking at the substance from one side. This phenomenon was first observed by Herschel and Brewster, but was first thoroughly investigated by the late Sir George Stokes. He showed that in all cases observed by him, the fluorescent light was of a wave length longer than that of the waves whose absorption caused the fluor- escence. This relation is not, however, true in all cases. Some common illustrations of fluorescence are the colors seen in certain forms of fluor spar (whence the name of the phe- nomenon); the color of canary glass — which is ordinary glass containing traces of certain salts of uranium ; the color of a decoction of the bark of chestnut trees ; the color of the surface layers of kerosene oil ; etc. Phosphorescence is exhibited by the sulphides of barium, calcium, strontium, etc., and by a great many ordinary sub- stances to a certain extent. Sometimes the light is emitted for only a minute fraction of a second ; but in other cases it continues for hours. Conclusion. — In many cases it is impossible to say exactly what is the cause of the luminosity ; and in nearly all there RADIATION AXD ABSORPTION -1'ECTRA 579 are several phenomena involved. We can, however, divide all cases of radiation into two classes : in one, the substance that is radiating does not change so long as its temperature is maintained constant; in the other, the substance does change even if its temperature is kept unchanged. In the tirst class of bodies, the radiation is a purely temperature effect; and to them Balfour Stewart's or KirchhofFs law — as it is more often called (see page 301) — and the other laws of radiation may be applied. This is not true of the bodies of the second class, in which molecular changes are going on. Spectroscopes Different Forms. — In order to study the radiation of any l><»dy, some method of dispersing it into a pure spectrum, an-1 some instrument which is sensitive to the various radia- tions, must be used. As we have seen, there are three ways in which dispersion may be secured: by the use of a prism. a grating, or some interference apparatus. Further, a slit (or small source of light) and a converging lens must be used. Thus we have prism, grating, and interference spectroscopes. The conditions as to the purity of the spectra formed by i nst n i in. -nts have been discussed in previous chapters. Precautions. — There is one obvious precaution which must be taken with these instruments; allowance must be made for aKsnrptimi <»f the ether waves produced in the apparatus itself. Thus, a glass prism can be used to study the spe.-tra of visible sources of liglit. luit not of those which emit only very short or very long waves, because glass absorbs them. A Ojiiart/ prism is ordinarily used for examining the spectra in.ed by very short waves; a prism of rock salt of tluorite, or of .silvite, those produced I iv lon«r waves. AL: a reflecting grating does not reflect all waves equally; and air absorbs certain ves, eertan visible ones, and all the extremely >lmrt one* These absorption phenomena must be observed by preliminary investigations. 580 LIGHT Receiving Instruments. — In order to detect the radiations various means must be adopted, as has been already explained. For a limited range of wave lengths the eye may be used; for these and for shorter ones, photographic methods may be applied ; for longer waves some thermometric device is ordi- narily made use of, such as a bolometer, a thermopyle, a radi- ometer, etc. These instruments do not necessarily measure the intensity of the radiation, but by suitable calibration methods many of them may be used for this purpose. There are, of course, many other methods of observation. Scale PIG. 299. — Prism spectroscope. Scale Attachments. — Many spectroscopes are provided with divided scales so placed as to coincide with the spectra formed ; and in this manner t |
he position of any definite train of waves may be recorded and thus described. Since by means of a grating the wave lengths of any radiation may be measured, it is a simple matter by using, in a preliminary nAblATI<>\ A.\D ABSORPTION SPECTRA f>81 experiment, certain radiations whose wave lengths are known, to calibrate this scale ; and then the instrument may be used to measure the wave lengths of other radiations. Different Kinds of Spectra Continuous and Discontinuous Spectra. — A distinction has been made already between continuous and discontinuous spectra. In the former all waves within certain limits are present; so there are no gaps in them. In the latter only certain isolated trains of waves are emitted, thus forming separate "lines." Investigations show that all solids and liquids — with possibly a few exceptions — emit continuous spectra ; while all gases and vapors emit discontinuous ones. (This is obviously what one would expect to be the case from the kinetic theory of different forms of matter.) Gaseous Spectra. — The spectrum of a gas depends, natu- rally, upon the manner in which it is rendered luminous. So we have "flame spectra," "arc spectra," "spark spectra." " fluorescent spectra," etc. If, however, a gas is made lumi- nous in any definite manner, the waves it emits are definite and characteristic of the gas. Thus, different gases may be identified by their spectra ; and in many cases the discovery of new lines in the spectrum of a gas that was supposed to be pure has led t<> the identification of new elements. Absorption Spectra. — If the radiation from a solid or liquid falls upon any body, certain waves are absorbed ; and so only a portion of the incident waves are transmitted. The spectrum of this transmitted radiation is called the -orption spectrum " of the body which produces the al KOI -j.t ion. This absorption takes place in many ways, as has l>cen already stated. In all, the absorption is due, in the main, to the resonance of the minute parts of the mole- cule or of the molecules themselves; and in the greater number of bodies the energy absorbed is distributed among the molecules of the body, and is manifest by heat effects. 582 LIGHT This is called "body absorption." In other substances the energy absorbed in the interior is spent in emitting other waves of longer wave length, thus producing fluorescence. In certain bodies, the absorption takes place in a very thin surface layer ; but the larger portion of the energy incident upon the surface is reflected directly. This is the case with the metals and a few other bodies, and is therefore called " metallic absorption." The law of Kirchhoff in regard to the equality of radiating and absorbing powers may be applied to a substance which exhibits body absorption only. Thus, if a substance absorbs certain trains of waves of definite wave lengths, it has the power of emitting them if rendered luminous by means of temperature alone (and, also, often if other means are used), and the intensities of absorption and of radiation are the same if the temperature of the substance is the same in the two cases. While if the temperature of either condition is decreased, so is the intensity of the effect. Thus, if a white- hot solid is placed behind a quantity of cooler gas or vapor, the absorption spectrum is a continuous one from which cer- tain isolated waves are absent ; and these are identical with those which the luminous gas would emit. The gas absorbs certain waves and transmits the others; it also radiates waves of the same wave length as those which it absorbs ; but the intensity of these radiations is so much less than that of those which are transmitted, that the spectrum is practically as if the gas did not radiate. If the gas or vapor is at a higher temperature than the white-hot solid, the spectrum will be that of the luminous gas with a continuous weak background ; i.e. it is a bright- line spectrum. If the gas or vapor and the solid are at the same temperature, the spectrum will be continuous. Solar and Stellar Spectra. — These facts are illustrated in the spectra of the sun and of the stars. The solar spectrum and certain stellar spectra are observed to be absorption 584 LIGHT ones ; while other stars produce bright-line emission spectra. The explanation of the latter is obvious : the stars producing them are surrounded by a luminous gaseous atmosphere, which is hotter than the interior portions. Similarly, in the case of the solar spectrum and other absorption stellar spectra, the explanation is that the interior portions are solid or liquid, and are at a higher temperature than the atmosphere of gases and vapors outside. These vapors are naturally those formed by the evaporation of the interior substances. The absorption spectrum in the case of the sun consists of the Fraunhofer lines ; and they can be identified almost completely with the emission spectra of the vapors of certain substances here on the earth ; and thus the con- stitution of the sun is known. The following are a few of the substances which are in this manner known to be in the sun: calcium, iron, hydrogen, oxygen, sodium, nickel, magnesium, cobalt, silicon, aluminium, carbon, copper, zinc, cadmium, silver, tin, lead, etc. Some of the absorption lines in the solar spectrum are due to* absorption by the atmosphere around the earth. Thus, certain groups of lines known as the " A," " B," " a " and " B " " bands " are due to absorption by the oxygen in the air, while numerous other lines are due to the presence of water vapor. A method for distinguishing between solar and terrestrial lines will be described presently. Similarly, by a study of the spectra of the stars, either emission or absorption, a great deal may be learned in regard to their constitution, and also motion, as will be shown immediately. The study of these and similar phenomena forms the science of Astrophysics. An excellent book to consult on this subject is Miss Clerke's Problems in Astro- physics, New York, 1903. In speaking of wave motion a certain general property, known as Doppler's principle, was described (see page 345). RADIATION AXD A I: >o /,•/•'/ /o.v sl'WTRA 585 It states that when a source of waves is approaching a point in space, the wave number at this point is increased, while the converse is true if the source is receding. In the case of ether waves that are being dispersed by a glass or quart/, prism or grating, this would be shown by a change in their refrangibility — an increase if the source is approach- ing the earth, a decrease if it is receding from it. There- fore, if a star is emitting certain trains of waves, their corresponding spectrum lines will all be shifted side wise by an amount depending upon the velocity of the star in the line of sight. If these lines, then, all apparently agree exactly with lines observed here on the earth in the labora- tory for any known vapor, except that they are all slightly displaced, the obvious explanation is that the star is moving in the line of sight; and its velocity in this direction may be deduced from the amount of the observed shift. (This statement is not absolutely correct, for shifts of the lines may sometimes be due to anomalous dispersion or to abnor- mal pressures in the atmospheres of the stars.) Similarly, if the image of the sun is focused by a lens upon the slit of a spectroscope, and it is so arranged that first one edge and then the other of the sun's ima^e is on the slit, the lines in the solar spectrum that are due to solar absorption will be shifted slightly, owing to the fact that one edge of tin- sun is receding from the earth while the other is approach- in LT it, because of the rotation of the sun. Hut those lines in the spectrum due to absorption in the earth's atmosphere will not be so displaced. student should consult Ames, Prismatic and Diffraction Spectra, for Fraunhofer's original memoirs, and Brace, The * of Radiation and Absorption, for the memoirs of Kin-h- hoff :.nd lJunseii. CHAPTER XXXVII EXPLANATION OF COLOR General Discussion. — The color of an object that is self- luminous depends upon the character of the light that it emits. If it radiates all the visible waves with suitable intensities, it will produce in a normal eye the sensation that we call "white." (The case of defective eyes will be considered in the next chapter.) If the intensity of certain waves is abnormally great, the light appears colored, as is shown when "red fire," a "sodium flame," etc., are used. The color of most objects, however, is due to the fact that they are illuminated and either reflect or transmit light to the eye of an observer. It is obvious that the color of the object will depend fundamentally upon that of the illuminat- ing light ; but we are so accustomed to viewing objects in the white light produced by diffused sunlight, that in describ- ing the color of any object it is always assumed that white light is used with which to illuminate it. When we consider the color of an illuminated body, it is evident that it may be due to any one of several causes. It has been explained in the previous chapter that absorption of light may take place in many different ways, and corre- sponding to each of these there will be certain color phe- nomena. Again, we have seen how colors may be produced by any dispersive action, such as that of a prism, a grating, or an interference mechanism. These various cases will now be discussed briefly. 686 EXPLANATION OF COLOR Absorption Colors Body Absorption. — The most familiar kind of absorption is that shown when the incident light is absorbed in the interior of the body and the energy of the absorbed waves is spent in producing heat effects. The light that is trans- mitted appears colored, therefore, owing to the disappear- ance of certain trains of waves. If a single train of waves of wave length I is absorbed, which corresponds, therefore, to a definite color, the transmitted ligh |
t will include all the other trains of waves, which will combine in the eye to pro- duce a definite color, called the " complementary color " of that of the waves which were absorbed. If the body absorbs two trains of waves, it may happen that the intensity of the absorption is not the same for both trains ; that is, it may n-.juire a greater thickness of the body to extinguish one color than is required for the other ; and it is thus apparent how such a body may appear of a different color as its thick- ness is varied. It is evident that if the absorbing substance is transparent for those waves which it does not absorb, it cannot itself be seen when viewed from the same end as the incident light, or from one side; but if owing to any cause the body diffuses tin- liirht which it does not absorb, then it will appear of the same color when viewed from any direction. Thus a tank ( -ontaininir colored water will appear practically black, except when the transmitted light is viewed, it there are no minute solid particles in suspension : luit if these are intro- duced, it appears colored from all points of view. two portions of matter having body absorption are so placed that the incident liurht tails upon one, and the trans- mit t«l li'jlit is then incident upon the other, the color of the emerging li^ht is that due to the waves which are left after two absorpti The color of all leaves and flowers, of •us clotlis, of paints, of bricks, etc., is due to body 588 LIGHT absorption.' If two paints are mixed, their color is, as just explained, that due to the absorption by both the paints; there is a double subtraction, as it were, from the incident light. The nature of the light that gives an object its color may be determined in two ways : one is to illuminate the object with white light and analyze by a spectroscope that light which is diffused ; the other is to form a continuous spec- trum on a white wall and move the object along this ; if it appears black for any position, it means that the color cor- responding to this position is absorbed, but if it transmits and so diffuses any particular color of the spectrum, it will in the corresponding position appear of this color. Fluorescence and Phosphorescence. — If the energy of the waves absorbed in the interior of a body is spent in produc- ing other waves, which are therefore radiated in all direc- tions, the phenomenon is called, as has been said, fluorescence. The fluorescent light consists in general of waves whose wave length is longer than that of the waves whose absorp- tion produces the fluorescence. In this case the color of the transmitted and the diffused light is not the same. It is evident that if the fluorescent body is thick, the waves which cause the fluorescence may be entirely absorbed in that portion of the body which is first traversed by the light; so that the fluorescence will occur in this portion only. In some cases this color is confined to almost the surface layers. If the emission of light continues after the incident light is intercepted, the phenomenon is called, as has been said, phosphorescence. Evidently there is some molecular trans- formation involved in this. Surface Color. — When polished metals and many of the aniline dyes in the solid form (e.g. a dried drop of red ink on paper) are viewed in white light, they have a peculiar appear- ance which is called "metallic lustre." This is due to the EXPLANATION OF COLOR 589 fact that they reflect certain waves much more intensely than others, or, in other words, they have " selective " reflection. This process does not take place in the interior of the sub- stance, as in the case of a colored liquid, but at the surface. If a substance showing this metallic lustre, or surface color, is made in a film which is sufficiently thin, it will transmit certain waves. But the color by reflected light is not the same as by transmitted ; in some cases they are approximately complementary. These substances which exhibit surface color have anoma- lous dispersion and change plane polarized light into ellipti- cally polarized light by reflection. Scattering by Fine Particles. — If the light traverses a region where there are numerous minute particles, it may happen that they are of such a size as to scatter certain trains of waves, and to let pass unaffected all trains of longer wave length. The light so scattered is plane polarized if it is viewed at right angles to the incident beam. This scatter- ing is the explanation of the blue color of the sky, as has i already said, and of the color of sunset clouds, at least in part. The phenomenon also plays a most important part in (h'termining how much radiation (visible and invisible) reaches the earth from the sun. Dispersion It is not necessary to say anything here in regard to the dispersive action of prisms, gratings, etc., but a few illustra- tions may be given of colors due to it. Prismatic <lis|>«TM(»n is slmun by rainlmws, halos around the sun and moon, dew- drops, diamonds when mutably cut, etc. Diffraction colors are seen when looking ;l( in,,ili,.r-«.f-p«-arl, at certain line feathers, at corome (the colored rings around tin- mmm ». tli rough fine-ni' ->bcd cl.,tb at a bright light, etc. Int< colors are shown by soap bubbles and other thin films of transparent matt< CHAPTER XXXVIII THE EYE AND COLOR SENSATION A TEXT-BOOK of Physics is not the proper place for a de- tailed description of the structure of the human eye or of the various theories which have been advanced to account for the sensation of color. Some treatise on Physiology or on Physi- ological Optics should be consulted. It is simply necessary to state here a few facts which are of physical importance. The Eye. — From an optical standpoint the eye consists of a converging lens which is provided in front with a diaphragm of adjustable diameter, the " iris," and whose focal length can be changed at will to a cer- tain degree. (This power of accommodation is greatly decreased as one grows old.) This lens exhibits both spheri- cal and chromatic aberration, but not to a noticeable degree in general. The medium on one side the lens is the air, but on the other is a liquid filling the cavity of the eye. At the rear of this is the " retina," upon which a normal eye forms an image of the object viewed. A "near-sighted" eye has its focus in front of the retina; while a "far- sighted" one has its back of it. In the former case, the image may be formed on the retina if a diverging lens is used in front of the eye ; in the latter, if a converging one is substituted. Fio. 801. — The human eye. 590 THE EYE AND COLOR SENSATION 591 Perception of Color. — The retina consists of a structure of minute parts which are intimately connected with the endings of the optic nerve. The exact mode of excitation of these nerve endings by the incident ether waves is not known. Certain portions of the retina, viz., those remote from its centre, play no part in color sensation ; for, when waves of all wave lengths lying within the limits of the visible spectrum are incident upon them, one is conscious of a sensation of gray only. This is true of all portions of the retina if the light is faint, with the exception of a small area, called the " yellow spot," which gives color sensations only. This spot is slightly off the axis of the eye considered as a lens. There is also a minute area — called the "blind point "-—near the cen- tre of the retina, where the optic nerve enters, at which no sen- sation of light is produced. Over the other central portions of tin: retina, light of all different colors may be perceived. Addition of Colors. — It has been known since the experi- ments of Newton that, in order to produce the sensation kk white," it was not necessary to have all the trains of waves in a continuous spectrum from violet to red. Corresponding to any color there is another such that if these two sensations are produced simultaneously in the eye, white is perceived. These two colors are called, as has been said, complementary. One way of producing these simultaneous sensations is to paint different sectors of a circular piece of cardboard with the two colors, and then to rotate it rapidly while it is illuminated with white light. Thus at consecutive minute intervals of time if one looks at the rotating disk, the eye receives first one impression and then the other; but since, if impressions reach the eye at intervals faster than about thirty or forty a second, a continuous effect is produced, the e\ this case receives two simultaneous impressions. This is what may be called the addition of colors ; and it is evident that the mixing of paints, or the suht rael i"ii of color, has HO connection with it. 592 LIGHT Similarly, a great variety of choices of three colors may be uuule which when added in suitable intensities will produce white light. Taking any three such colors and adding them in different intensities, any other color which is desired may be produced. This proves, then, that in order to account for the perception of colors of all kinds, it is simply necessary to assume that in the eye there are three sets of nerves corresponding respectively to these three colors. (Some writers claim that there are four such sets of nerves, while Hering has an entirely different theory. The statements made here are in accord with all ordinary facts, and embody a portion of the theory of color sensation that was advanced by Thomas Young and supported and extended by Helm- holtz.) The Young-Helmholtz Theory of Color Sensation. — The choice of these three colors is to a certain extent arbitrary ; but it is limited by an investigation of different cases of color blindness. Many people are afflicted with an inability to recognize certain objects as colored which appear so to the normal eye ; to them they appear gray. Other colored objects appear to them to have colors different from those which would be ascribed to them by an |
observer whose eyes are normal. The simplest explanation is that in these cases one or more of the sets of nerves referred to above do not respond to stimuli. By an examination of a great many cases of color blindness the conclusion has been reached that the three fundamental color sensations are red (about wave length 671/Lt/^), green (505^), and blue (470 /^). The general explanation of color sensation may then be explained by assuming that there are these three sets of nerves which when excited produce these sensations respectively, red, green, and blue, and that when any train of waves of a definite wave length reaches the retina it stimulates all three sets of nerves, but to different degrees. Curves can be drawn as shown in the cut which give the TIIK EYE AND COLOR SENSATION 593 iv>ults of rxprrinu'iits on adding these three colors with varying intcnsi; Thus, considering the vertical line through F, the three curves are so dnnvn that if ml light of \va\v length 671 /JL/JL and of an intensity proportional to the length of the line F\^ • Ided to green light of wave length 505/iyLt and of inten- sity proportional to ™LET ~ BLUE GREEN ¥ELLOW" "^ the length F2, and to • t i I TTO j j '-'• <' ii rvos of color sensation. blue light of wave length 41 Op/* and of intensity proportional to the length F3, the color perceived is blue corresponding to a wave length 486ft/i. Consequently, if we can assume that when waves of this wave length fall upon the retina they stimulate the "red," "green," and "blue" sets of nerves to degrees which are proportional to F 1, F 2, and F 3, the phenomena of color sensation have been explained. BOOKS OF REFERENCE ;:. Light for Adranoed Students. London. 1902. An excellent text-hook, contain in.: <!• •-••n|'tioiis of all the fundamental phenomena and references to all the recent investigations. PftBftTOW. Tin- Th.Miry of Liuht. Loii-lon. iM edition. !*!»:>. Tin- recognized book of reference for all the elementary phenomena of Light OptiOS. (Translation.) \,-w York. 1902. I h. l....st mo. I. M -ii text-book ing the accepted theories and explanations of all optical phenoi LUMMKR. Photographic Optics. (Transit i- . ) LoDdOD. 1000. A text-book on geometrical Optics, with special reference to lenses. PUT8IC8 — 38 MAGNETISM CHAPTER XXXIX PERMANENT AND INDUCED MAGNETIZATION Magnets. — A body which has the property of attracting pieces of iron is called a "magnet" ; that is, if a magnet is brought near a piece of iron there is a force between them which is shown by their approaching each other if either (or both) is free to move. Such bodies occur in nature, for one of the forms of iron ore which is not uncommon, a mix- ture of FeO and Fe2O3, is magnetic. It is, moreover, a simple matter to make any piece of iron or of ordinary steel into a magnet. There are two general methods for doing this : one depends upon a property of an electric current, the other, upon what is called magnetic induction. If an electric current is made to traverse a wire which is wound in the form of a spiral spring, or helix, the apparatus is called a " solenoid " ; and experiments show that, if a piece of iron or steel is placed inside this solenoid, it becomes a magnet. Or, if a piece of iron or steel is brought near, not necessarily in contact with, a magnet, it is made a magnet also. If a piece of iron is magnetized in this manner, ami the magnetizing agency is removed, the iron will lose its magnetism very easily, if it is jarred or subjected to an increase in temperature; but this is not true of the piece of steel — it remains a magnet under all ordinary conditions. All magnets in ordinary use are made therefore of steel, some kinds of which are much better than others. Much progress in this respect has been made in recent years. ri-:i;MANENT AND IMK' MAC \ l-:i ItATWN 595 Magnets are usually made in the form of bars, rods, or elongated lozenge-shaped "needles." Sometimes the bars or rods are bent into the shape of a U or of a horseshoe; and in this form they are called horseshoe magnets. Experiments show that long magnets are more permanent than short ones Fio. 808. —Horseshoe magnet. and that they remain magnet- ized longer if their ends are joined by a piece of soft iron. Thus an iron bar, called the "arma- ture," is always placed across the ends of a horseshoe magnet when it is not in use. Horseshoe mag- net with soft Iron Bar magnets with armatures. armature. FI.J. Magnetic and Diamagnetic Bodies. — It is found by ex- periment that a magnet can attract other kinds of matter than iron ; such as many forms of steel, nickel, cobalt, man- ganese, etc. These bodies are called " magnetic," and any one of them can be made a magnet by the methods described above for iron or steel. Again, there are many other sub- stam-es which are repelled by a magnet; such as bismuth, antii i\. ami /inc. These bodies are called "diamagnetic." ulay made the most important observation that the • jiit-^tiuii as to whether a body is attracted or repelled by a magnet depends fundamentally upon tin- man-rial medium in winch the magnet an<l tin- body arc innm-rx «1. In the above dcti nit ions of magnetic ami diamagnetic bodies this medium is assumed to be the ordinary atmosphere. Faraday •bowed that, while in OH€ im-dium a l»od\ mi^ht be attracted 1>\ i m.i'jiirt. iii another it might be repelled. Thus tin- 596 MAG* I.TISM importance of the medium in the consideration of magnetic phenomena is shown. Poles. — If an iron or steel rod or " needle " is magnetized by means of a long solenoid, and if it is then removed and suspended by a fine thread or 011 a vertical pivot, so that it is free to rotate in a horizontal plane, it will turn and after a number of vibrations gradually come to rest in a direction which is approximately (or exactly) north and south. (This fact in regard to a mag- FIO. sos. -pivoted mag- netized bar or needle has been known for many centuries, and has been made use of by mariners and travelers.) The end which points toward the north is called the " north pole " of the magnet ; the other, the "south pole." The direction in which it points is called "magnetic north and south." If a magnet is suspended as just described, and another is brought near it, it may be shown that there is a force of attraction between a north and a south pole, but one of repul- sion between two north poles or between two south poles. "Unlike poles attract, like ones repel." It is easily proved, further, that the greater the distance apart of the magnets, the less is the force. In order, then, to explain the reason why a magnet when freely suspended points in a north and south direction, all that it is necessary to assume is that the earth itself has the properties of a magnet. The particular magnetic properties that experiments show it to have will be described later in Chapter XLI. Magnetism a Molecular Property. — If a magnet is broken up into smaller pieces, each fragment, however minute, is found to be a magnet, with a north and a south pole. This leads one to believe that magnetism is a molecular property of all magnetic substances ; and all observations are in sup- port of this idea. Every property of a body except its mass and weight is changed when it is magnetized ; and con- \ /:v/ AND L\i>r<h'i> M.\<; \ ETIZATIOX 59 ely, any ehan^e that is known to afi'ect the moleeiii. a body will att'eet tin- magnetism of a magnet. Thus, when an iron rod is magnetized, its length, its volume, its elasticity, etc., are all changed; and when a magnet is hammered or twisted or heated, its magnetism is altered. as is shown by a change in the force which it exerts upon another magnet or upon a piece of iron at a fixed distance from it. Induction. — We make the assumption, then, that each molecule of a magnetic substance, e.g. of a piece of iron, or of nickel, etc., is a magnet; in other words, that each mole- cule of any one magnetic substance has a certain mass and other mechanical properties and is at the same time a magnet. When the substance is in its natural condition, we can assume that these molecular magnets are not arranged in any order, hut are distributed at random; so that, as far as external actions are concerned, each tiny magnet is neutral- ized by those around it. But if a magnet is brought near such a piece of magnetic sub- stance, each of the r*7vT " becomes a sooth pole. la tter'8 molecular FIG. 806.-Magm-tlo induction: the end A of the Iron bar magnets is acted upon by a force due to the magnet; and the molecules are all turned, more or less completely, in an orderly and regular direction. Thus, if the magnetic substance is a rod or bar, and the magnet is in this form also and is brought near one en<l of the former, so that its north pole is nearest it. the nmlerules will turn so that their mmtli poles are ird the imrth pole <»f the magnet. Therefore the molec- ular magnets no longer neutralize each other; they now have an external a. tion, and. in fact, the bar which they con- stitute is now a magnet \\itl ;h pole toward the north polr of the magnet i/ini: magnet . The change produced in the ;he nioleeular magnets by the magnet is 598 MAGNETISM roughly indicated in the accompanying cut, where each mole- cule is represented by an elongated rectangle whose ends are shaded differently. (Of course, we do not assume that a molecule has actually the shape of a rectangle.) This explains, then, not alone why a piece of magnetic substance is magnetized by the magnet, but also why the two attract each other. The phenomenon is called "magnetic FIG. 807. — Arrangement of molecules In an un magnetized and a magnetized iron bar. • nduction"; or the former is said to be magnetized by "in- duction." Strictly speaking, these names apply to the phe- nomenon only so long as the magnetizing magnet is kept in its position near the magnetic substance ; when the two are separated, the latter remains a magnet, although a weaker one, for a greater or less t |
ime, as described above ; but its magnetism is now spoken of not as induced, but as " intrinsic " or "permanent." Similarly, when a rod is magnetized by the action of a solenoid, the magnetism is said to be induced ; etc. The reason why a long thin magnet is more permanent than a short one is clear, because in the latter the two ends are closer together and the molecular magnets at one end may disturb the direction of those at the other, and so produce demagnetization. The action of the armature of a PXSMANMNT AM) i\i>r< I;D MA<;.\ /•;///. I y/o.v 599 horseshoe magnet is also easily explained : it keeps the molecular magnets at the ends from changing their positions. It is evident that, \vh«-n a bar or rod is magnetized by the action of a magnet at one end, the molecular magnets in the former will be arranged in an orderly manner at the end near the magnet; but at its other end FIG. 80S. — Magnetization of an iron bar by induction. these minute magnets will not be so systematically distributed. Tha magnetizing of the bar or rod may be made more complete if two mag- nets are used, one at each end of the rod, and turned in opposite direc- tions as shown in the cut. The action may be made still more complete if the two magnets are plan •«! on top of the rod at its middle point — oj >| "»ite poles being in contact — and are slightly inrlinrd to it, as shown in tin- cut (and if then the two magnets are <lra\\ti otf the bar lengthwise in opposite The process should be re- Fio.aw. — Processor mapneti/ peated several limes. If during any of these processes the rod is ham- mered or Jan. .1. tin; magnetization is increased. Experience shows that it is impossible to magnetize a ba» more than up to a certain degree; ii is then said to be "satu- rated." This fact and all those just described in regard tG methods of magnet i/.at ion are explained easily if it is assumed that the molecules of a magnetic substance are themselves ma '_n i.-i N. One can, in fad, make a model of a magnetic substance by placing on a board a great number nail magnets, all pivot.-d BO M t«» be five to turn around \ei-tical axes; and the entire p ..f magnetization can be imitated by placing this hoard inside a large solenoid OF l>rin^inur magnets n.-ar it. Temperature Effects. — It a "permanent" magnet has its temperature raised sufficiently high. it loses its magnetism, and becomes again .simply a magnetic substance. If, how- 600 MAGNETIC. M ever, when magnetism is being induced in a magnetic sub- stance, its temperature is raised, — not too high, — the induced magnetization is increased. These facts are at once explained on the molecular theory of magnetism. (Iron loses its magnetic properties at about 785° C. ; nickel, at about 300° C.) The Poles of a Magnet are of Equal Strength. — Several interesting facts are learned when one studies the force which a bar magnet exerts, either on another magnet or on a piece MAGNET °f magnetic sub- stance. One way FIG. 810. — Action of a large magnet upon of doing this quali- tatively is to place the magnet on a COMPASS NEEDLE , . a small pivoted magnetic needle. horizontal table, and to move near it a small pivoted magnetic needle — such as an ordinary pocket compass. If the latter is short, it may be pushed up close to the magnet ; and it is observed that when this is done, the small needle does not always point toward or away from one of two points, one in each end of the magnet, as it would if there were two centres of force in the magnet. On the contrary, the needle points nearly perpendicular to the FIG. 311. — Diagram representing arrangement of molecular magnets in a bar magnet. surface of the magnet at all points except near its middle ; and this is what we would expect from our knowledge of the nature of magnetization. The molecular magnets near the centre of the magnet are all more or less parallel to its axis ; but near the ends they are turned so that for one half I'KH. M. i. V/..V7- AND i\i>r MM;M-:TIZATION 601 the magnet their north poles are in the surface, while for the other half their south poles are there. We must then regard each point of the surface of a magnet as a centre of force ; and the total force of a magnet on another magnet, or on a magnetic substance, is found by adding geometrically all the forces due to these magnetic poles in the surface. These may. however, be combined mathematically in certain cases in a manner that is instructive. Let a bar magnet be under the action of another magnet perpendicular to it at its middle point, whose south pole is turned toward the former, but which is so far away that all these minute surface forces R. Fio. 812. — Forces acting on a bar magnet owing to a distant magnet. may be considered parallel. (We can, if we wish to, con- sider the second magnet as so long that we need not take into account the forces due to its north end.) All the forces over the north half of the magnet may be added so as to form the resultant Rn : similarly, those over the south half may be added to form Rt. If the magnet is turned so that it is no longer perpendicular to the distant one, these forces still keep their value : and their direction is toward the latter. This condition is practi- cally the ease when a magnet is under the action of the earth, and if a magnet is suspended so as to be free, — not alone to turn, but also to move as a whole, e.g. if it is floated on a cork ivMiiiLr <>n the surface of a tank of water, — it is observed that the mag- net, if not originally in a north OF nth direct ion. turn* hut does not move i 18. — Mom .1.1 a maifiirt wh.-n pliwii! as a whole. ThU provei that the tun foroei /»', and //, are e«|ii.d in amount but opposite iii direction, forming a COUple. MAONET18M (See page 100.) If a bar magnet could be made whose molecular magnets were all parallel to its axis, there would In- magnetic forces only at its ends. So if in the actual magnet just described L is the distance apart of Rn and Rt measured along the magnet, and if this special bar magnet is made of the length Z, it might replace the former so far as action at a considerable distance is concerned. The forces on the two ends of the latter magnet due to the distant one are then equal in amount but opposite in direction. We are therefore led to make the assumption that each molecular magnet has two centres of forces, a minute distance apart, which are equal in amount but opposite in kind ; that is-, if one centre exerts a force of attraction on one end of any distant -Forces magnet, the other exerts on it a force of repul- between two small sion. Thus, if JV^ and N^S2 are ^wo molec- ular magnets, there are four forces acting on each, as shown, arid if S2 is at the same distance from both N! and Sr the two forces acting on it are equal in amount but are in nearly opposite directions. All observed facts in regard to magnetization are in accord with this assumption. We say that each molecular magnet has two poles whose " strengths " are equal but opposite ; or that they have equal but opposite " quantities of magnetism " or " magnetic charges." The same statement in regard to magnetic charges must then be true of any magnet, however large or compli- cated, because it is made upof magnetic molecules, and therefore contains as much south as north magnetism. It is impossible to separate a north pole from an equal south pole and obtain them distinct from each other, because, when a magnetic mole- cule is decomposed into simpler parts, it ceases to be a magnet, and equal amounts of north and south magnetism vanish. Magnetic Field and Lines of Force. — When a small mag- net is placed near a large one, it is acted on by certain forces; I'Ki;MA\i:\T AM) 1XDUCED MAGNETIZATION 603 and, in general, a region in which a small magnet experiences forces is called u " magnetic field." A simple mode of study- ing and describing the properties of such a field is to draw what are called "lines of force." These are continuous lines such that any one indicates by its tangent at any point the direction in which a north pole would move if placed there. Thus, if P is any point on a line of force AB, a north pole, if placed there, would move in the direction of the curve at P, while a south pole would, of course, move in the opposite direction. (If the poles had no inertia, they would actually trace out the whole curve AB.) Therefore, if a small magnet is placed with its centre at P, there will be a moment acting on it which will make it turn and lie tangent to the line of force at P. Fie. 815.— Lines of mig- netic force, and the forces acting on a small magnet. This leads at once to two modes of actually drawing the lines of force in any li'-l.l. One is to move a small magnet from point to point in the fit-Id, noting at each its direction; thus, starting from any point P, the magnet should be moved a short distance in such a direction that in its n«'\v |M.<iti<m it is tangent to the same curve to which it was tangent atP, etc.; a line can be drawn through these points. Then another starting I >oi nt is chosen and another line is drawn, etc. The other method is to place a horaontal sheet of paper or a glass plate in the field of force, .sprinkl.- tin.- inm filings Ov«-r it, ami jar the paper or plate until tin* filings assume definite positions. Under the action of the magnet i< : of the field each minute tiling Kccomcs magnetized, and therefore acts likr a small magnet; and under tin- action of the couple it turns and places ; along the lim- <>f force. Thus tin- action is just as if one had ma ny thousand minute magnetic needles in the field at one time. (Actually in neither of these cases do the forces acting on the small magnet form a couple, unless the inagn-'t which causes the field of force is at a great 'li-tan< .• ; 1,1.1 the resultant force of translation is as a rule minute.) Several Qlostrations of lines of magnetic force, as mapped by means of iron til ings, are shown in theMOOmptnyingOUt |
e] 604 MAGNETISM around a single bar magnet, and between two magnetic poles, alike and unlike. The direction of a line of force is defined FIG. 816. — A bar magnet. m by its being that in which a north pole would move. It follows, therefore, that lines of force start from north poles of magnets and end on south poles. Fur- ther, two lines of force cannot cross. It is interesting to note the effect up- on a field of force of the introduction of a piece of magnetic substance as shown in Figs. 317 and 3170. From the first of these it is appar- ent that the lines of Fie. 816 a. — Two similar poles. force, instead of di- verging away from the end of the magnet, are converged and enter the end of the iron bar, from the sides and other end of which they again emerge. The action is exactly as if it were PERMANENT. AM> LM>r< KD MAGNETIZATION 605 i for the lines of force to traverse a space when it is filled with iron than when it is occu- pied by air ; the iron is therefore said to have a greater magnetic "permeability'' than air. The explanation in terms of force of the ooii\- lines toward the iron is evident. In Fig. 318, let SN be a magnet and I II be a F,o. 8166. -Two unlike poles. under the action of the former it is magnetized with A as a south pole and B a north one. The force which would act upon a north pole placed at any point, P, in the field of the magnet due to the magnet alone might have the direc- tion and the amount PQV while that due to the induced magnetiza- tion in AB might be in- I hy /'Q8; so the resultant force would be PQy Thus, in gen- eral, if a piece of iron i- put in a magnetic li«-ld. the lines of t»n-e crowd into it out of the air. If the poMtion and v : lie lines of force in an the illnst rations '/i\en above be considered, it is 86611 that they may !••• . I.-M ribrd by saying that the lines are distributed MAGNETISM as if lines in the same direction repelled each other sidewise, and as if all the lines were stretched so that they exerted a Qi Fit. . 818. — Explanation of change in di- rvction of a line of force at P owing to the presence of the soft iron. tension. (Attraction of iron and of Unlike poles may be u explained" thus.) Lines of force do not, of course, have a physical existence ; and the above statements are simply descriptions of the appearance of their geometrical curves. By means of these ideas it is often possible to give simple descriptions of com- FIG. 317 a. — Bar of soft iron placed in a uniform magnetic field. plicated cases of magnetic forces. CHAPTER XL MAGNETIC FORCE AND INDUCTION Quantity of Magnetic Charge ; Law of Force. — If the art inn of several magnets upon one which is pivoted is observed, it is seen that the intensity of this action, as measured by the deflection of the magnet from a north and smith line, depends upon many things. It is different if the magnets are inclined at different directions to the pivoted one; and so for purposes of comparison |/>N of different magnets we ft may place them, in turn, Fio. 819.-Force actlnc upon a small S I east and west (magnetic- east or west of it. ally) of the pivoted mag- t , s • pivoted iniiLMirt l.y a bar magnet placed net in a hori/.ontal plane, with their north poles pointed toward the latin. As a result, the latter will be deflected and will come to rest, making a definite angle with its origi- nal north and smith position, which will vary with the dis- tance of the magnet from it, and also, in general, with different magnets when placed in the same position. This indicates that the magnetic forces between poles vary with their distance apart and with different magnets. Let us assume, as an ideal case, that the magnets have all magnetic charges at their ends (see page 602), and let us assume that we can assign a numerical value to this mag- netic charge, so that the forces it exerts are proportional to it. Thus, if /// is the magnetic charge of the north pole, — 9ii is that of the south pole; and the forces which each 607 608 MAG. \KTISM exerts are proportional to m. Then, if there is another magnet which under similar conditions exerts different forces, its poles must have different magnetic charges, which may be written ml and — mr Therefore, if these two mag- nets are acting on each other, the force of the north poles on each other is proportional to the product mmr Experiments show that this force varies with their distance apart, being less for a great than for a small distance. Coulomb made the assumption that, so far as distance was concerned, the force varied inversely as its square. So calling this distance r, the law of action of two poles is assumed to be that the force between them is proportional to — ^> This is known as Coulomb's Law, and it was verified by him (1785) so far as was possible with the instruments at his command. It was verified also by Gauss, and to a greater degree of accu- racy ; but our main reason for believing that the law is exact is that all of its consequences are found to be in accord with the varied facts of electrical engineering, into which enter so many questions connected with magnets and magnetic fields. In order to assign a number to the magnetic charge of any magnet, it is necessary to define a unit charge ; and in doing this it must be remembered that magnetic forces are different in different media. (See page 595.) Making use of the C. G. S. system of units, a " unit magnetic charge " is defined to be such a charge that, if placed at a distance of 1 cm. in air from another equal charge, the force between them is 1 dyne. Then, if a charge equal to m of these units is placed at a distance r cm. from a charge m1 in air, the force between them expressed in dynes is given by the equation f=—^- In any other medium the force is proportional to this, and therefore, following the accepted system of symbols and writing as a factor of proportionality -, the force in any f* MAGNETIC FORCE AND INDUCTION 609 medium is/ = From what has just been said, the value of fj, for air is one on the C. G. S. system of units and using as a unit magnetic charge that defined above; natu- rally, if another unit charge were adopted, the value of this constant for air would be different. The factor /A is a quan- tity which is characteristic of any medium — it is not, how- ever, necessarily a constant. It is called the " permeability," for reasons which will appear later. Intensity of Magnetic Field ; Magnetic Moment. — When a magnet is placed in a magnetic field, it is acted on by two forces, one at each pole, they being the resultants of all the forces acting over the surface, as explained on page 601. Let us assume the simplest case, viz., that the charges are entirely at the ends ; then, if the magnet is short, the forces at the two ends are equal in amount, although opposite in direction, because the two ends are at almost the same point in the field. The "intensity" of the field at any point is defined to be the value of the force which would be exerted (»n a unit north charge if placed at that point. Therefore if the inten- sity of the field is R, and if m is the (1 large on either pole of a short ni;i'_rn«-t which is placed in the field, there is a force Rm acting on each pole. If I is the length of the magnet and if it is in such a posi- ti"M that it makes an angle N with the direction of the field, the per- pendicular distance between the Fio. MO.— Moment acting on a tor magnet when placed In a field of intensity R. two i'"ives is I sin N i so the strength of the couple acting on the nii_rii.t is Rml sin N. If the magnet is pivoted AMES'S PHYSICS— 89 f>10 MAGNETISM around an axis perpendicular to a plane which includes the magnet and the line of force at its middle point, it will turn under the action of this couple toward the direction of the field ; so the moment of this couple should be written — Rml sin N. The product ml is evidently a property of the magnet itself, and it has received the name " magnetic moment of the magnet." In the general case of any kind of magnet, the magnetic moment around any axis is defined to be the maximum value of the moment of the forces acting on the magnet when it is placed in a field whose inten- sity is one, with this axis at right angles to the field. If a magnet is broken up into parts, these are found, as a rule, to have different magnetic moments ; and the value of the "intensity of magnetization" at any point of the magnet is defined to be that of the magnetic moment per unit volume around that point. Thus, if M is the magnetic moment of a portion whose volume is V, the ratio — in the limit, as Vis taken smaller and smaller, is the value of the intensity of magnetization. If the magnetic charges are entirely at the ends, the intensity is the same throughout the magnet ; so if this is a cylinder of length I and cross section A, and if the charges on each end are m, the mag- netic moment M is ml and the volume V is IA. So the V J\. intensity of magnetization — = m-\ or, it has the same value as that of the charge per unit area on the ends ; this is called the " surface density " of the charge. Magnetic Pendulum. — Measurement of RM. It has been proved in the preceding paragraph that, if placed in a field of intensity R, a magnet of magnetic moment M is acted on by a couple whose moment is — RM sin JV, when it makes an angle N with the direction of the field. If, then, it is pivoted so as to be free to turn, and if I is its moment of inertia about the axis of rotation (see page 88), its angular MA<,M-:n<- FOBCA AM) 1M>\ <HON 611 acceleration due to the magnetic field is - 8in — . There- fore the magnet will make oscillations to and fro, through the direction of the field of force. If the angle of oscillation is small, the acceleration is -- T~^> au(^ ^ne vibrations are harmonic with a period 2 TTA— — . (See page 91.) The • n moment of inertia may be calculated from a knowledge of the dimensions of the magnet, and its period of oscillation may |
be measured ; so calling this last T, we may write I o y RM= , and therefore the value of the product RM may be determined. (We shall show in the next paragraph how the value of the ratio — - may be determined ; and so JxL the values of both R and M may be obtained.) A method is thus offered for comparing the intensities of different fields of force. Call th.-se Rl and #8; and using the same magnet in the two cases, let 1\ and Ts be its periods of oscillation in the two fields. Th<-ii .17 ami / In-ill^ constants for tin- magnet are the same in both ex- l we have the two equations *W =' /f"K=-- n \f 47T2/ nm n . f» 1.1 Measurement of — . — If the same bar magnet whose M n period has been determined when vibrating freely in a uni- form field of intensitv // is plaeed at rest at right angles to the field, it will itself j.ro<lu< < a I'M -Id of force which at some distance away is nearly uniform, and at right angles to the existing field R. So, it a >////// magnetic needle is suspended at a point BOOM dfetanoe awaj in the direction of the length of tin- l»ar nia^nrt. in Midi .1 manner as to be free to turn around i perpendicular l»<.tli t<> the Held of force R 612 MAGNETISM and to the bar magnet, it will be under the influence of two fields of force at right angles to each other. It will there- fore place itself at such an angle that the moments due to the two fields are equal but opposite, so that they neutralize each other. Call, for the time being, the intensity of the field due to the bar magnet,/. The moment acting on this mag- netic needle due to the field whose intensity is 72, when the l j S N f FIG. 821. — Diagram representing mode of measurement of MjR. magnet makes with its direction the angle JV, is EM' sin N, if M1 is the magnetic moment of the magnetic needle. But the magnet makes the angle 90° — N with the field of in- tensity / ; hence the moment due to it is fM' cos N. The moments are in opposite directions, and must be equal, if the needle is at rest. Hence or RM' siuN=fM' cosN, tan N = £-. R The value of/ may, however, be easily expressed in terms of MI the magnetic moment of the large bar magnet, and r, the distance from the centre of this magnet to the centre of the magnetic needle. / is the intensity at the point 0, the centre of the needle ; i.e. it is the force which would act on a unit north pole if placed there. The large bar magnet, as placed in the diagram, has a pole of strength + m at a dis- MA<;.\I •://' l'<HH'E AND INDUCTION 613 - ^ tiim-e r — j ;i\viiy from 0, and another of strength — m at a distance r + -. Hence the force on a unit north pole at 0 is 2 - m m 2 m/r But if r is very great in comparison with Z, /=— ::ir- , 2 m/r 2 A/ Consequently, substituting in the formula for tan JV, M r* tan N Various precautions and modifications for this experiment are explained in laboratory manuals, but it is evident that * = — 2— both r and N can be measured ; and so — may be determined. R Measurement of R or M . — By a combination of the two formula' for RM and — -, it is seen that R B>= »«V r«r»tanJV* and so both R and M may be measured. It /! i> known fur any one fu-ld, it has been explained how its value for any other field may be determined by means of a \ ilir.it in-_r magnet whose period can be measured. Magnetic Tubes. - If one refers to the illustrations of lines of force given on page 604, it is evident that these lines are most crowded together at those places where the intensity of the tirlil is the greatest, and are the farthest apart at those »'.U MAGNETISM points where the intensity is the least. This suggests a systematic mode of drawing lines of force. We can describe a small closed curve at some point near the magnet, and can imagine lines of force drawn through each point of this curve ; these lines, if continued, will of course be found to start from a north pole of a magnet and end on a south pole ; so they thus form a hollow tube leading from one pole to the other, whose cross section is small near each end, but greater at a distance. If the initial small curve is taken of exactly the proper size, this tube will inclose at its two ends a unit magnetic charge. Such a tube is called a "unit tube"; and, if the magnet has a charge m at each end, tnere are m tubes leaving the north pole and returning to the south pole. It is evident that where the cross section of a tube is least, 'the intensity of the field is greatest ; and vice versa. Similarly, if A is the area of any small surface in the field at right angles to the force, and if there are JV tubes passing through this surface, the intensity of the field at that point is pro- portional to the limiting value of the ratio — , as A is taken -ZV A smaller and smaller. In words, the intensity of the field at any point is proportional to the number of tubes per unit area at that point. Magnetic Induction. — As was explained on page 605, and as is apparent from the cuts on that page, the effect of intro- ducing a piece of iron or other magnetic material into a field of force in air is to cause the lines of force to change their direction and enter the iron. If the iron is in the form of a rod, and if its cross section is A, more tubes pass through it than did through the same area of air before the iron was substituted for it. If the original field of force in the air is uniform, so that the intensity is the same at all points, the lines of force are all parallel, and the tubes are all of the same cross section. If the intensity of the field is 72, the number of tubes per unit area is proportional to this. If, MA<; \KTH- FOWK AM) IMK'CTION 615 now, a long iron rod is introduced parallel to the field, the number of tubes per unit area of its cross section is greatly increased; and it may be proved by methods of the infinitesi- mal calculus that the ratio of this number to the previous one equals the value of the quantity ft for iron, as defined on page 608. It is for this reason that- p is called the permea- bility. (For different kinds of iron, and for different con- ditions. p may have values as great as 2000.) It is thus seen that for any magnetic substance /A is greater than for air. The number of tubes per unit area in the iron (or other magnetic substance), when in a field of intensity 72, is, then, proportional to the product pR ; and this quantity has re- ceived the name of the "magnetic induction" at the point where Ii is the intensity of the field and p is the permeability. The fact that the tubes do not simply end on the iron rod, l>ut must be considered as passing through it, may be proved by certain phenomena of electric currents which will be dis- cussed in a later chapter, and also by the simple experiment of cutting the rod into two pieces by a transverse section and separating tin-in slightly ; the field of force in the crevasse is found to be much more intense than in the original fit-Id. When the magnet causing the field of force is removed, the iron remains magnet i/.«-d. and therefore some tubes still remain passing through it. So we are led to believe that in tin- case of an ordinary magnet these tubes do not end at its poles, but continue thron-h it, forming endless closed ri exactly as if < ..... wen- to take a piece of rubber tubing and bring its two ends together. This conception of tul» M-tie induction is due to Faraday. Diamagnetic Bodies. — The case of a diamagnetic body ma\ ited in a similar manner. Kxperiments .show that if such a hody is made in the form of a 1, n. and brought near a l> net, it is magnet i/ed 1>\- induction, hut in a direction opposite t«» what it would he if it \\eiv iron. In the cut, where AB is the diamagnetic body, the end .1, 616 MAQNBT18M F,o. ^.-Action nearest the north pole of the magnet, is thus found to be a north pole, while B is a south one. There is repulsion, therefore, between the magnet and the diamagnetic body. jf the ^ Q£ force are drawn, they are found to be as shown in the cut, showing that, instead of being crowded together in the diamagnetic body, they avoid it. There is therefore a smaller number of tubes in the bar than there was in the original field before the bar was intro- DIAMAGNET.C BAR bod,: A Fio. 828. — Effect of introducing in a uniform magnetic field: (1) a sphere of magnetic sub- stance ; (2) a sphere of diamagnetic substance. duced. This proves that the value of the permeability JJL for a diamagnetic substance is less than for air. Energy Relations. — Since all motions in a system of bodies take place in such a manner that the potential energy of the system becomes less (see page 114), it must be possible to explain from this point of view the attraction and repulsion observed with magnetic and diamagnetic substances. If we consider the formula /= — -£ for the forces between mag- netic charges, it is evident that there are changes in the potential energy whenever magnets are moved with refer- ence to each other ; because work is done in overcoming the forces, or by the forces. The magnets themselves are not changed; and so we are led to believe that the energy is MAGNETIC FORCE AND INDUCTION 617 located in the surrounding medium where the magnetic field exists. It follows from the formula that / is small if //. is huL,re, or, in words, the forces are small if the permeability of the medium is large ; and consequently in such a medium the energy per unit volume is also small, since small amounts of work are involved in any changes, other things being equal. (Magnetic forces can be felt through a vacuum, and so the energy of a magnetic field is, in the case of any material medium, both in the ether and in the matter.) Attraction and Repulsion. — Therefore, if a piece of iron — for which ft is greater than for air — is introduced into a field of force in air near a magnet, the energy in the space occu- pied by the iron is less than when it was occupied by air; and the decrease in the energy is greater if the field of force is intense than if it |
is feeble. In other words, if a piece of iron is moved up gradually toward a magnet, the potential energy of the field becomes less and less ; therefore, if a mag- net and a piece of iron are left to themselves, there is a force of attraction between them, and they will approach each other. Similarly, a magnet will attract a piece of any magnetic sub- stance in air. Conversely, and for obvious reasons, a magnet will repel a piece of any diamagnetic substance in air. In general, if /z for any substance is greater than for the sur- rounding medium, it is attracted by a magnet; while, if it is less than for the medium, there is repulsion. The obser- vations described on page 595 are therefore explained. An exactly analogous phenomenon in mechanics is afforded by the motions of a block of stone and a block of wood wln-n immersed in a tank of water: the fonwr will be attracted by the earth and will sink; tin- latt.-r will be repelled and will rise. The explanation in both canes is that the motion takes place in such a direction as to make the potential energy of the system less. The stone sinks because it is heavier than the water; and therefore by replacing an equal volume of water closer to the earth, the potential energy of gravitation is decreased. The wood rises because it is lighter than the water; ami. th<-i< !•»• . it it moves up iin.l \\at.-r rrj.Isu-."* it. tin- potential energy is again decreased. CHAPTER XLI MAGNETISM OF THE EARTH Magnetic Elements. — The fact that there is a magnetic field of force on the surface of the earth is proved by the ob- servations on the motion of a suspended magnet, which were referred to on page 596. If a bar magnet or a magnetic needle is suspended in such a manner that it can turn freely in all direc- tions, it will finally come to rest in a position such that its axis is inclined with reference to a horizontal plane and lies in a vertical plane which nearly, if not quite, coincides with the geographical meridian at the point of suspension. This vertical plane is called 'the " magnetic meridian " at the point ; and the angle it makes with the geographical meridian is called the " magnetic declina- tion " or the " variation." The angle which the axis of the needle makes with the hori- zontal plane is called the "magnetic inclina- tion " or udip." The earth's magnetic field at any point is then completely denned by its intensity, the declination and the inclination. These three quantities are called the "magnetic elements." FIG. 324. — A mag- netic needle suspended free to turn in any di- rection. The dip can be measured by observing the angle which the needle makes with the horizontal plane. The variation is most easily determined by mounting the needle so that it is free to turn about a vertical pivot, and noting the angle it makes with a true north-and-south line, which may be 618 MAVXETISM OF THE AM/,' 777 619 1 orated by astronomical methods. It is convenient for most purposes of measurement to consider the earth's magnetic force as resolved into two com- ponents, one horizontal, the other vertical. Thus, if OD is the direc- tion of the field of force at 0, and if OA and OB are horizontal and vertical lines through 0 in the same plane as 0/>, the angle (AOD) is the dip ; and calling it N and the intensity of the force R, the horizontal com- ponent is R cos N, and the vertical one R sin N. The former can be meas- ured with great accuracy by the method described on pages 610-613. Therefore, since the angle of dip, N, can be measured directly, the value of R may be deduced. Further, if the ratio of the horizontal and vertical components can be meas- ured, the dip may be calculated ; for, calling these H and V* "26. — Diagram rep- resenting the horizontal and vertical components of the earth's magnetic force. \, A dip circl.-. Y=-~. Variations in the Elements. — Observations show that the values of all three of these elements at any one point are con- tinually (hanging. So far as is known, these changes are pei i(,di( , that is, for instance, the dip makes a pendulum-like oscillation during the twenty-four hours, increasing slowly, t hen decreasing, etc. ; further, the mean value for any one day iges slightly the next day, and so on, having an oscilla- tion whose period is a year; and, again, the mean value for any one year is not the same for the next year, but changes slightly; but the period of this change is not known, for 620 MAGNETISM since regular observations began to be taken — about the year 1540 — this oscillation in the mean annual value of the dip has not been completed. Similar statements may be made in regard to the other two magnetic elements ; there are daily, yearly, and secular changes, so called. (There are other periodic changes than these, but they are the most important.) It often happens that there FIG. 827. — Chart showing secular change In IG. . — r w cr cg . -, •, j the earth's magnetism, from observations made 1S a Sudden and Unexpected at London ; the black line indicates both the in- disturbance of the magnetic clination and the declination. elements of a magnitude far greater than the regular changes ; this constitutes a " mag- netic storm." The explanation of such phenomena is not known ; but observations have shown that they occur most frequently when the spots on the sun and when aurorse in our atmosphere are most numerous. „ ., , .. Magnetic Maps. — The magnetic field over the earth's sur- face may best be described by drawing on a map of the earth certain lines which indicate the values of the elements at any one epoch. Thus lines can be drawn such that at each point of the earth's surface, through whose position on the map any one line passes, the value of the declination (or variation) is the same. Such lines are called "isog- onals," and are of the greatest possible assistance to mariners and surveyors. They are shown in the cut for the year 1900, and each one is marked with a certain number, e.g. 5°, which indicates the value of the variation for all points on that line. These lines run approximately north and south ; and it should be observed that for two lines the variation is zero, i.e. at points on them a mag- netic needle points true north and south: they are called "agonic" lines. One of these is approximately a great MAGNETISM OF THE EARTH 621 circle of the earth ; the other lies in northern Asia, and is called the "Siberian oval." Again, lines can be drawn which indicate in a similar man- ner the inclination or dip ; they are called " isoclinals," and are approximately parallels of latitude. The line of zero dip is called an "aclinic" line, or the "magnetic equator." There are two points in the earth's surface where the dip is 90°; these are often called the "magnetic poles." The lines for the year 1900 are shown in the cut. Other lines, giving other information, may also be drawn ; but they need not be described here. Conclusion. — The explanation of the magnetic action of the earth is not known. It has been proved, however, that it is due almost entirely to causes which are within the earth itself. Certain of the periodic changes are occasioned, how- ever, by external causes, such as electric currents in the atmosphere. Historical Sketch of Magnetism The property which the lodestone possesses, of attracting iron, was known centuries before the beginning of the Christian Era, because it is mentioned by Thales, who lived from the year 640 to M»; i..< . The Greeks and the Unmans were acquainted with the fact that the intervention of other bodies, like brass, does not destroy magnetic effects. That like poles repel and unlike attract, and that a lodestone possesses the power to eniniminicate polarity to inert iron, were known at least as rarly as the twelfth century. The compass was in daily use in Kumpo also as early as this, l>ut the disenvrries nf magnetic declination and its variation t'mm place to place were made by Columbus in Hart man is reputed to have di.senvn-ed the dip in 1644. He obtained a value of 9° when- he should have obtained 624 MAGNETISM 70°. This fact was not published, and Norman, in 1576, independently discovered it in London, obtaining a value of 71° 50'. Norman was probably the first to suggest that the source of attraction is in the earth, and not in the heavens as gen- erally supposed. He also showed that the earth's field is simply directive and produces no motion of translation, by floating a needle on water. The variations in the magnetism of the earth were discovered by Gellibrand in 1636. The first systematic treatise on magnetism was William Gilbert's De Magnete. It was published in 1600, and con- tains a complete account of what was known as magnetism up to that time, as well as a great number of new ideas and experiments which are due to Gilbert himself. Gilbert was the first to recognize the difference between temporary and permanent magnets ; to detect the effect of a change in temperature ; to show that the fragments of a magnet are themselves magnets; to observe the effect of hammering, etc. ; to make use of the idea of lines of force, although in an imperfect manner. The fact that iron was not the only magnetic substance was shown by Brandt, who proved in 1733 that cobalt was magnetic. The diamagnetism of bis- muth was recognized by Brugmans in 1778, but the first systematic study of the subject was made by Faraday in 1845. It was he also who made the great discovery that the forces of attraction and repulsion depend fundamentally upon the surrounding medium. ELECTROSTATICS CHAPTER XLII 1 IXDA.MKNTAL PIIKNOMEXA Introduction. — It is observed if a piece of silk is rubbed against a glass rod and is then separated from it that both now have the power of attracting small fragments of paper, of metal foil, of thread, etc., toward those portions of their surfaces which had been in contact, and that, further, if either one of them is suspended so as to be free to move, it may be attracted by t |
he other. The silk and the glass are said to be " electrified," to have on them " electrical charges " or " charges of electricity," or, more simply, to be u charged." The same phenomena may be observed with any two por- tions .)f different kinds of matter; but with certain kinds the forces of attraction are manifested, not alone by those portions of the surface where they were in contact with the other body, but also over all their surface. This is true of metals, for instance. So, if one end of a long metal wire is charged, the forces are evident over all its length; the wire is said to "conduct" the charge, and it and similar bodies are called "conductors." l'»\ suitable means the charges on the other end of the wire may be removed; but, if the charge on the first end is continually renewed, charges will appear again at the former one, etc. This may be called, then, an electric "current," and while it is going on, many interesting phenomena occur both in the wire and ide it. We are thus led to divide the subject of Elec- AMBS'S PHYSICS — 40 ''25 626 ELECTROSTA TICS tricity into two parts : one deals with electrical phenomena when the charges are at rest, it is called " Electrostatics " ; the other with the phenomena of electric currents, it is called " Electrodynamics." We shall begin with the former. Electrical Charges. — As said above, when two portions of different kinds of matter are rubbed together and then separated, they are charged, and can produce forces which they could not when in an uncharged or neutral condition. The act of friction is not essential ; all that is necessary is that the two pieces of matter should be brought closely in contact. We distinguish, too, as stated above, between "conductors" and " non-conductors." The following bodies are the commonest illustrations of conductors : all metals, either solid or liquid; water containing in solution almost any salt or acid ; the human body ; the earth. The follow- ing are illustrations of non-conductors : glass, silk, paper, cloths, dry wood, porcelain, rubber, sulphur. In order to produce any appreciable charge, therefore, in a conductor, it must not come in contact with the hand, but must be " insulated " by holding it in a piece of paper or cloth. Energy of Charges. — The fact that forces are exhibited near charged bodies and that therefore work can be done by producing motion, proves that there must be energy associ- ated with charges. This is evident also, because, as stated above, when two bodies are charged by rubbing them against each other and then separating them, one attracts the other, and this proves that in order to separate them work was required. In other words, work is necessary in order to have electrical charges. This energy which is associated with charges is not in the bodies themselves: it is in the medium which surrounds them wherever the electrical forces may be felt, that is, throughout the "electric field." This fact is proved by the phenomena of electric sparks. It is known to every one that if the electric charges are too intense, sparks take place in the medium (e.g. ordinary / •'[ • \ DA M /•; \ T. 1 /. I'll h'\<>M i:\ . 1 627 all(l these are due to the breaking down of the material structure of the medium. It' a spark passes through a sheet of paper or a pane of glass, a hole is made in it : it the spark is in air, the molecules of its gases are broken into parts. This proves that the medium must have been greatly >t rained just before the sparks passed ; and, if it was strained, it must have possessed potential energy. Electric forces may be shown in a vacuum; and therefore the seat of the energy of electric charges is in both the surrounding ether and the material medium immersed in it. The importance of the nature of this medium in all electrical phenomena is thus established. r Positive and Negative Charges. — If two rods of the same kind of L,rlass are charged by means of a piece of silk, and if • >nr is suspended horizontally in a paper sling so that it is free t<» turn, it may be seen, "ii bringing the charged por- tion of the other rod near it, that one repels the other. Whereas, if the piece of silk \vhieh was used to charge the glass rods is brought near, there is attraction. Simi- larly, if other charged bodies are brought near the sus- P'-nded glass rod, some repel it and the others attract it. All those charged bodi« •> whieh repel it are said to be " positively " charged ; while those which attract it are said to be "negatively' d. Thil unonntfl definition of positive cr plus ( 4- ) and negative or minus ( — ) ehurijes. Thus the experiim-nt s jn that 628 ELECTROSTATICS glass rubbed with silk is charged positively ; and that the silk is charged negatively. Similarly, in all cases, experi- ments show that when any two bodies are brought in con- tact and then separated, they are charged oppositely. If different charged bodies are suspended in turn, it is observed that it is a general law that a positive charge attracts a negative one but repels another positive charge, and that a negative charge repels another negative one. "Like charges repel; unlike ones attract." It is found, further, that the force becomes less as the distance apart of the charges which are acting is increased. A body which is charged positively when rubbed with some definite body may be charged negatively when rubbed with another one. And, further, the character of the charge received by a body often depends upon the condition of its surface, whether it is smooth or scratched, etc. Thus, glass is charged positively by a piece of silk, but negatively by a piece of flannel ; and smooth glass may be charged positively, while, if it is rough, it may be charged negatively. By a careful study of the character of the charges produced on different bodies when rubbed with other ones, it is found that it is possible to arrange all bodies in a series, A, B, (7, etc., such that if B is rubbed with A it is negatively charged, whereas if it is rubbed with C it is charged positively. Such an arrangement is called the "electrostatic series." A few of its terms are : cat's fur, flannel, glass, cotton, silk, wood, the metals, rubber, sealing wax, resin, sulphur. Conductors. — We say that a body is charged at any point if electrical forces are exhibited when a small piece of matter is brought near that point. If the charged body is a con- ductor, there are no forces shown in its interior ; if it is a hollow solid, — like a hollow ball, — there are no forces in the interior region ; in other words, if a conducting body is charged, the charge is entirely on its surface. This phenomenon may be considered as due to the repul- l--l-\l>.\Ml-:\TAL PHENOMENA 629 sion of a charge by a similar charge ; the charges distribute themselves as far apart as possible, and, since a conductor allows charges to flow, they will all be on the surface. (This is true only after the charges have come to rest ; it does not hold when there are currents.) This fact may be proved by direct experiment in many ways. Faraday made a metal box large enough to allow him to enter it and carry with him his instruments ; and he showed that, however the box was charged, there were no effects inside after the charges came to rest. (Similarly, he showed that whatever electrical charges or changes he produced inside, there was no elec- trical force outside. The explanation of this will be given later.) Lines of Force. — The region around charged bodies in which electrical forces may be shown is called the " electric field " ; and a " line of electric force " is a line in the field such that at each of its points its tangent is the direction in which a minute body charged positively would move if left to itself. (A negatively charged body would, of course, move in the opposite direction.) If a line of force is con- tinued, it will be found, therefore, to start from a positive charge and to end on a negative one. Two lines of force < iiiniot cross, for that would mean that at the point of inter- section a charged body would move in two directions. 'I 'here are no lines of force inside a conducting body; they all end at its surface. In I i^. :>:51, lines of force are drawn for several special cases. It is seen that the phenomena of attraction and re- pulsion and the distribution of the lines themselves may be described by saying that lines in the same direction repel each other, and that there is a tension in the lines tending to m.ike them contract. It may not be unnecessary to state the obvious fact that these lines have no physical existence, but are merely geometrical constructions. The lines of force may be mapped by a method exactly 030 ELECTROSTA TICS Two unlike charges, the positive one four times Two similar charges, one four tiim.« as great as the negative one. as great as the other. FIG. 381. — Lines of electrostatic force. FUNDAMI-:M.\L ni I:\OMENA 631 similar to that described for a magnetic field. It will be shown in the next paragraph that when any piece of matter is put in an electric field it becomes electrically charged, some portions with plus electricity, others with minus. If it is an elongated body, its two ends become charged oppo- sitely : and so, if it is short and is suspended or pivoted at its middle point, it will turn and set itself along the line of force. It plays, then, the same part in an electric field as does the short magnet in the magnetic case. Induction. — If an uncharged body is insulated from the earth and is brought near a charged body which is also in- sulated, the former will exhibit electric forces. On the side nearer the charged body, it will apparently be charged with the opposite kind of elec- tricity to that of the latter ; while on its more remote side it will be charged with the >ame kind. Thus, if in the FIO. 882. — An electrified bo<ly is brought near cut the charged body has a positive charge, the end A of the other body which was orig |
inally uncharged will exhibit the properties of a negative charge, while the end B will have those of a positive charge. If th<- charged body is withdrawn, these charges on the other disappear. The phenomenon is called "electrostatic induc- tion" ; and tin; charges are railed •• induced " ones. If the uncharged hody which is brought near the chared one is a piece of a non-coin! net or, — <•.//. of L^lass, of sulphur, . — its molecules are affected by the electric fours and the whole body is strained, if is said to he -polari/ed." The trie charges are distributed «• \actly as are the magnetic in tin- case of ,i pi. MM- <.f imn which is brought near a magnet. ( See page 597.) There is, in fact, almost complete analogy between the case of a piece of non-conductor put in an electric tidd and a piece ,,f magnetic substance put in a magnetic field. There may be one ditVerence; after the 632 ELECTROSTATICS latter is removed from the field, it remains a magnet, while in the former case the electric charges in general disappear. Cases have, however, been observed in which the electric charges remained; and so the analogy in these cases is exact. I A FIG. 888. — Induced charges on a conductor. JL If the uncharged body which is brought near the charged one is a conductor, it becomes charged, oppo- sitely on its two ends or faces, as described above. There is an essential difference, however, between this case and that of a piece of non-conductor, owing to the fact that lines of force do not pass through a conductor and, there- fore, end on its surface, while they can and do pass through a non-conductor. This difference will be explained more fully in the next chapter. These induced charges on a conductor are caused by the attraction of the charge on the charged body for an unlike charge and its repulsion of a similar charge ; it being borne in mind that these forces are due to the fact that when unlike charges approach each other, or when like charges recede from each other, the potential energy in the medium becomes less. Thus, if a conductor is joined to the earth by a con- PIG. 884. — Charging a conductor by induction. ducting wire, and if a positively charged body is brought near it, a positive charge is repelled to the earth and the rr.\i>.\Mi-;\ i AL 633 conductor itself has a negative charge ; if now the conduct- ing wire is removed, the conductor retains its charge. The distribution of the lines of force is shown in the cut. This process of charging a conductor is known as "charging by induction." Fio. 884 a. — Charging a conductor by induction. Experiments show that, if a charged body has points on its sur- face, the electric force in the air is greatest near them ; and, in fact, if such a charged conductor is carried into a darkened room, faint sparks will be seen at the points. The charges are passing off to the particles of dust and to other small portions of matter in the surrounding air. These thus become charged with the same kind of electricity as that on the body, and are, there- fore, repelled by the latter, forming a current in the air, or a wind. This is often sufficient to be felt by the hand or to blow out a candle flame. If, then, a pointed conductor is brought near a « haiLced body, so that its points are to wanl the latter, which may be either a conductor or a_ non-conductor, t In- latter will induce charges on the for- mer ; and those on the points turned toward the charged body will escape, Fw. 886. -Action of ,K,int» by induction. be drawn to the latter, and "discharge " it by neutrali/in^ the charges on it; the other induced charges, which are like tlmse on the body originally charged, will remain on the eondiu -tor. The final action, therefore, is as if the obi 634 ELECTRO 8T A TICS were bodily transferred to the pointed conductor. This action of such a pointed conductor, or a "comb," is made use of in many electrical machines. (Its importance was first recognized by Benjamin Franklin (1747). It is the reason why lightning rods are always made with sharp points.) Electroscopes. — It may be well to explain at this point one or two simple instruments which are used in the study of electric phenomena. One of the most useful of these is the "gold-leaf electroscope," which consists essentially of two vertical slender strips of thin gold foil connected at their upper ends to a metal rod which is attached to a metal plate or ball. The gold leaves are, as a rule, inclosed in a glass bottle so as to prevent any action of draughts of air. If the plate or ball is given a charge, this will spread over the leaves, and since they are now charged alike, they will repel each other, and will diverge. The angle of divergence will vary with the intensity of the force of repul- sion. Further, if a charged body is simply brought near the plate (or ball), charges will FIG. 886.— Gold-leaf electroscope. be induced on the leaves and they will diverge. In most gold-leaf electroscopes there are thin strips of tin foil fastened to the walls of the glass vessel and attached to the metal base of the instrument, so that if the gold leaves are diverged too far they will not communicate their charge to the non-conducting glass walls, but to the conducting strip, which will carry the charges to the outside of the instrument. Another simple instrument is the "pith-ball electroscope." It consists of a small pith ball covered with a thin layer of metal foil and supported from a vertical metal rod by a fine wire or other conductor. If the rod is charged, it will trans- fer some of its charge to the pith ball, which will be repelled. The angle its supporting wire makes with the rod is a meas- FUMtAM K.\TAL I'll K.\< >M KNA (J35 urc of the force. It is obvious that a single gold leaf could be used in place of the pith hall, or that two pith halls could be used in place of the two gold leaves in the former instrument. Electrical Machines. — As we have seen, electrical charges may be produced by two independent methods: by friction or contact between two different bodies, and by induc- tion on a conductor. Corresponding to these are two types of machines for producing (barges continuously. a. Friction Machine. — There are various forms of these so-called friction machines ; but a description of the one shown in the cut will apply to all. There is a large glass plate pivoted on an axle, which is clasped at one point by two metal clips lined with leather; so that. as the wheel is turned, the glass becomes charged positively and the clamps negatively. The charges are removed from Fi... M, the latter by joining them to the earth, and from the former by the use of a point. -d conductor or "comb." A positive 036 ELECTROSTATICS charge is thus accumulated on the large conductor which is joined to the comb. b. Induction Machine. — The simplest form of instrument for producing charges by induction is the so-called "elec- trophorus," which was invented by Volta about 1775. It consists of a thick plate A, of some non-conducting sub- stance such as glass or hard rubber, which rests in a metal base B ; and of a loose metal cap (7, provided with an insu- lating handle D. In using the instrument, the cap is removed and the upper surface of A is charged by friction with a piece of -flan- nel or cat's fur ; let it be assumed that it is thus charged negatively. This charge will induce a plus charge on the upper surface of the metal base B, and the induced minus charge flows off to the earth. (The function of this in- duced charge on B is by its attraction for the charge on A to prevent the latter from escaping or leaking.) The metal cap 0 is now lowered on A. Actually, it touches it at the most in only a few points and so does not receive any appreciable charge from A directly. But the charge on A induces a positive charge on the lower side of C and a nega- tive one on the upper side. This cover 0 is now touched with the finger or otherwise connected to the earth ; so the negative charge is removed, and only the positive one re- mains. Connection with the earth is now broken, and if the cap is lifted by its handle, it will carry with it its positive charge. This charge may be transferred to some conductor ; and the cap being discharged may be replaced on the plate A, which still retains its charge. So the process may be repeated indefinitely. FIG. 889.— Electrophorus. Machines have been made by which these various steps are M i-;.\ i . i L /'// I-:.\»M I:.\A 037 carried out automatically. One of these is shown in the cut. The explanation of its action is simple, but is so long that it need not be given here. It may be found in almost Fio. 840. —Induction electrical machine. any special treatise on Electricity, such as dimming, Elec- tricity ; Perkins, J5Y«///W/v <tu<l Magnetism; S. P. Thomp- son, Elementary Lessons in Electricity and Magnetism, etc. CHAPTER XLIII ELECTRIC FORCE; MEASUREMENT OF ELECTRIC QUANTITIES Quantities of Electricity. — The fact that an electric charge is a quantity to which a numerical value may be assigned is suggested by many experiments. If a hollow metal vessel, like a can, is placed on top of a gold-leaf electro- scope, and if a charged body is lowered into it by means of a silk fibre (or other non-conductor), the leaves diverge owing to induction ; but the amount of the divergence is found to depend upon the charge lowered, not upon its position inside the vessel. Further, if another similar charge is lowered into the vessel, the leaves diverge still more, but the amount of this does not change if the two charged bodies are brought in contact, or even if they touch the walls of the vessel. We are thus led to speak of the "quantity" or "amount" of charge, or of electricity. It should be noted that the last experiment described shows that the total quantity of electric- ity on two or more conductors is the same before and after they touch. Thus we speak of the " Conservation of Elec- tricity." Equal Quantities of + and -- |
always produced. — If two uncharged bodies are lowered into the vessel, e.g. a piece of silk and a piece of glass, the leaves do not diverge, even when these two bodies are touched or rubbed together and then separated. But, if one of them is now removed, the leaves do diverge, showing that the two bodies were charged, but with exactly equal amounts of opposite kinds of electric- ity. Similarly, if an insulated conductor is lowered into 638 ELECTRIC FORCE the can in which there is already a charged body, there is no change in the di\ er^vnce of tlie leaves, thus proving that the two induced charges are of exactly equal amounts of opposite kind. Thus it can be stated as a general law that whenever a charge of any kind is produced, an equal charge of the opposite kind also appears. Faraday's "Ice-pail Experiment. " Dielectrics. — An in- teresting experiment in this connection is one due to Faraday. If a charged body is lowered by means of a silk cord into the interior of a nearly closed hollow conducting v< which is joined by a wire to an electroscope, the leaves of the latter will diverge ; but, as said above, the amount of the divergence does not change as the charged body is moved about inside the vessel, or even if the two touch. aday in his original experiment used a metal "ice-pail " as the vessel.) If the . charged body is a conduc- tor, it will lose its cha when it touches the metal vessel, because the charge will all go to the outside of the hollow conductor. (See page 628.) Hut since the divergence of the leaves is not affected, there can hare been no change in the ce*p*11 e*p6r"nent charge on this ronihietor. The explanation is that when the charged body is lowered into the hollow vessel, it induces an equal amount of electricity on the inner wall of the vessel of a kind opposite to its own — and therefore also an equal amount of the same kind as its o\\-n on the outside of the wall of the vessel; so, when the charged conductor touches the inner wall of the vessel, equal amounts of plus and minus charges pass to the outer surface, and there is con- sequently no change in the external conditions. It is "I, 640 ELECT If <>* T. I TICS served, further, that when the charged body is lowered into the hollow vessel, the divergence of the gold leaves, is the same whatever non-conducting medium is used to fill the vessel or is present in it: air, oil, sulphur, etc. Thus, electri- cal effects are transmitted through these various substances; and for that reason Faraday called them "dielectrics." (Actually there is a distinction between the idea of a dielectric and that of a non-conductor or insulator, but it need not be emphasized here.) Law of Force. — The force between two charged bodies is found to depend upon the amounts of their charges, their distance apart, and the nature of the surrounding medium or dielectric. So far as distance is concerned, Cavendish proposed the relation that the force varied inversely as the square of the distance. He showed by most ingenious mathe- matical reasoning that, if this were true, the charge on a spherical conductor must be entirely on its outer surface, even if there were bodies in its interior which were joined with it ; and he then proved by direct experiment that the charge was entirely on the outer surface. (This was pre- vious to 1773.) This same suggestion as to the law of force was made independently by Coulomb (1785) ; and he verified it by direct experiment, placing charges at different distances apart. A unit electrical charge may be defined in a manner simi- lar to that used for a magnetic charge. On the C. G. S. system of units a unit charge is defined to be such a one that, if it is at a distance of 1 cm. in air from an equal '•- charge, the force is 1 dyne. This is called the "C. G. S. Electrostatic Unit." Then, if a charge whose value is e is at a distance of r cm. from - a charge e^ in air, the force in dynes between them is —• But it is found that the force depends upon the surrounding medium ; and this is expressed by writing the value of the force in ELECTRIC FORCE 641 dynes as /= *,. where K is a quantity which is character- istic (and constant) for any one dielectric. It is called the '•dielectric constant." Using the system of units defined ahove, the value of ITfor air is one. Its value for all other dielectrics (with the exception of a few gases) is greater than for air, as may be proved by directly measuring forces in different media. Tubes of Induction. — The "intensity" of an electric field of force at any point is defined to be the value of the force \vhieh would act on a unit positive charge if placed at that point. Kxactly as in the case of a magnetic field, too, tubes hounded by lines of electric force can be drawn; and if they are of the proper size they will end on unit plus and minus charges. They are called " tubes of induction." Since there is no force inside a closed conductor, even if it is charged, these tubes must end on its surface, not traverse it. They do, however, pass through a dielectric, as is shown by i day's experiment described on page 639. In this the tubes all start fn>ni either the charge which is introduced or from the inner wall of the vessel, and end on the other charge. We can, in fact, define a real charge as one which origi- nates tubes of induction. Thus, in the cases of induction described on page 6tfl tin- tubes from the charged body pass into and through the dielectric bodyj'but they end on the iiictiiiLT on,-, and an e.jual number leave the other end. Thus, the charges on the latter body are /v/// ; while in the former case they are only <//'/»//v///, the forces manifested bein'_r di; iv be sliown by methods of the inlinitcsimal calculus, to the fact that the tubes are passing from one dielectric into another. The number of unit tubes per square centimetre at ri^ht angles to the Held at any point is, as in the case of magnet- — 41 ism, prop.M-ii.mal to the intensity of the field at that point. 642 ELECTROSTATICS Explanation of Attraction and Repulsion. — The explanation of electric attraction and repulsion may be given in terms of energy exactly as was done for mag- netic forces ; the constant K taking the place of /*. Since K is less for air than for all other dielectrics, a small piece of any such dielectric is more permeable for tubes than is air, and is attracted by a charged body if air is the surrounding medium. Further, since there is no force and therefore ™ energy inside a conducting body, a small piece of a conductor is attracted even more than a piece of non-conductor. The action of charges on each other has already been discussed. Electric Potential. — The properties of electric charges and the condition for their being in equilibrium may be expressed in a different manner. When a charge is moved in an elec- tric field, work is done, either at the expense of the energy of the field or against the forces of the field. Thus, if a plus charge moves in the direction of the field of force (or a minus charge in the opposite direction), the field loses energy and the charged body gains kinetic energy as it moves ; if a plus charge is moved against a field of force (or a minus charge in the opposite direction), work is done by some outside agency i and the energy of the field is increased. We have a mechanical analogy in moving a body toward or away from the earth ; if it is raised, work is done against gravitational forces and the potential energy is increased ; if it falls, it gains kinetic energy at the expense of the potential energy. We may define the potential energy of a body of unit mass with reference to the earth when placed at any point as the " potential at that point," or we may say that the potential at a point is the work required to raise a body of unit masft from the earth to it ; and we may describe /;/./,' ntic FORCE 643 the gravitational forces in terms of this quantity. The potential is evidently constant at all points of any horizontal plane; and the higher the plane is from the earth, so much the greater is the potential. Such a surface of constant potential is called an " equipotential " one. Evidently there is no change in energy as a body moves along such a surface ; but the change as a body of unit mass is moved from a point Pl in an equipo- tential surface, the -£ V, potential of whose points is Vv to a point P2 in a second equipotential sur- face whose potential I .2 - J r and FIQ m _ p. and F> ^ two horlzonul planes is entirely independ- ent of the path of the motion. (See page 108.) The line of action of the force at any point is perpendicular to the equipotential surface through that point ; because, if it were inclined to it, there would be a component force in the surface, and work would be required to move a body against it, \\hich is contrary to tin- idea <>t' an equipotential surface. The direction of the force is, obviously, from points of high to those of low potential. Similarly, in the case of electrical phenomena, we may >se the earth as our standard body, since it is a con- ductor, and is so large that its electrical condition ma\ In- regarded as permanent, and may define the "electric pot en - tial " at any point in an electric field as the work required to a unit plus charge from the earth to that point. I not necessary to specify any particular point on tin- earth, because tin- potentials of all points of a conducting body are the same, if tin' charges are at rest. If this were not so, it would require work to carry a charge from one point to another in the conductor; this would presuppose that there was an elec- 644 EL EC TROSTA TICS trical force in a conductor ; and, as we know, this is not true. So, since the earth may be regarded on the whole as a con- ductor, all points of its surface are at the same potential, whose numerical value is zero in accordance with the definition of potential given above. (This does not imply a zero amount of anything; for p |
otential is not a quantity which can be measured. See page 11. We give it a number, just as we give temperature a number. 0° temperature does not mean a zero amount of anything, but indicates a temperature which serves as the starting point of a thermometer scale. So the potential of the earth is 0, because the earth is the body of reference. Actually the earth is not a good conductor, and there may be local differences of potential.) Similarly, the potential at any point far removed from the electric charges, that is, at " infinity," is zero, because no work would be re- quired to move a unit charge from such point to one on the farther side of the earth, where by definition the potential is zero. We can draw equipotential sur- faces in the field of force ; the lines of force are at right angles to them ; and the direction of the lines is from high to low potential. Thus, if the field is due to a charged spherical conductor, whose complementary charge is at a great no. 844. -Lines of force and equi- distance, everything is symmet- potenttal surfaces around a charged rical with reference to its Centl'6 ; spherical conductor. „ the equipotential surfaces are con- centric spheres, and the lines of force are portions of radii starting from the spherical conductor. If the charge on the conductor is plus, the potential at points near it is higher than that at those more distant ; if the charge is negative, just the reverse is true. Thus, if a plus charge is put at any point, the potential of all points near ELECTRIC FORCE 1>\ is raised ; while the contrary is the case if the charge is negative. Induction. — We can thus explain the appearance of in- duced charges on a conductor. Let a positively charged body be brought near an insulated uncharged conductor AB. All points of this must be at the same potential since it is a conductor ; but if tlu' conductor were absent, the potential at a point A near the charged body would be higher than at a point B which is more remote ; consequently, if the poten- tials at A and B are to be the same, a negative charge must appear at A so as to lower its potential, and a positive charge at 5 so as to raise its potential. Or, again, since when the conductor is absent, the potential at A is greater than at B^ the electric force is in the direction from A to B\ and, when the conductor is introduced, a plus charge moves in tin- direction of the force toward B, and a negative charge moves in the opposite direction toward A. T I A FIG. 845. -Electrostatic Induction. Further, if the conductor is joined to the earth by a wire, its potential must be zero; but 'under the influence of the FIG. 846.- Effect upon line* of force and c.|ii1poti>ntUt Mirftrr* ,,f introducing a uphoric*! dttctor in the field and HIM. joining It to the earth. charged body alone it would be some positive amount ; there- fore, in order to lower it to zero, a negative charge must appear on it. (The same explanation can be given of the inducing action of a negative charge.) Distribution of Charges. — The fact that the potentials at all points of a conductor on which the charges are at rest are the same is a consequence, as was shown above, of the fact that there are no forces in a closed conductor. This may be expressed in a different manner : the charges on a conductor are all on the surface, and they are so distributed that the inten- sity at any point inside is zero, or, what is the same thing, that the potential at all points is the same. Thus, consider a closed FI«. 347. -Diagram illustrating the conductor of any shape, and let P fact that the force inside a closed con- be any point in its interior and Qv ductor is zero. r\ r\ i • ± f • L Qv Qp etc., be any points of its surface. The charge at Ql is at the distance rl from P ; that at Qy at the distance rv etc. So, if Av A2, Ay etc., are small areas at Qv Qv $3, etc. ; and if dv d%, c?8, etc., are the values of the surface density of the charges at these points, i.e. the charge per unit area, the intensity at P, or the force acting on a unit plus charge, if placed there, is the geomet- rical sum of 1 21, 2 22, 3 23, etc. This sum must be zero. ri rz rz By considerations of this kind it may be proved that the sur- face density at a point is greater than over a plane surface. (See page 633.) Sparks. — One of the commonest phenomena associated with electric charges is that of sparks. They are occasioned, as has been explained, by the mechanical rupture of the mate- rial medium in an electric field ; and they prove the existence of a great strain due to the electric forces. The intensity of the field at any point may be expressed in terms of the poten- ELECTIUC FORCE 047 tiul. If F"and V + AF"are the potentials at two neighboring points at a distance apart Aa?, the electric force is in the direction from the second point to the first ; and if R is its numerical value R&x = &V, because each member of the equation expresses the work required to move a unit charge from one point to the other. Since R is actually in tin* direction in which V decreases, the exact formula is R&x = AF — AF; or 72=——. Consequently, if the intensity is great, there must be a great fall of potential in a small dis- tance. Thus, if the difference of potential between two con- ductors is high, there is danger of a spark passing between them, and a connection may be found by experiment between the potential difference and the spark length in any dielectric under definite conditions. A limit is therefore fixed by the electric properties or " strength " of the air for the value to which the potential of a conductor may be raised; for, if it is exceeded, a spark will pass to the earth or to particles of foreign matter in the air. When a spark passes between two conductors, its path through the air is an excellent conductor ; and therefore both bodies are brought to the same potential. The pot. n- tial of one is raised by the passage to it of a certain amount of positive electricity, or by the \\ ithdrawal from it of a cer- tain amount of negative electricity; and that of the other is lowered by the opposite process. (See Electric Current*. page 663.) The luminous character of a spark or dischar^1 in any gas is due to the luminosity produced by the electrical changes \\-hich accompany the disruption of the molecules and the conduct ion of the current. Capacity of a Conductor. — If we consider an insulated eondiirt.ir 1.;. i ii space, it is eviden t that if it is charged pnMt ivdv. it \\ ,11 itself liave a posit ivi- potent ial. and that if it* ( -hai-irc is increased, so is its potential, because a greater amount of \\«.ik would be required to bring up to the 648 ELECT HOST A TICS conductor a unit charge from the earth. If the charge is doubled, so is this amount of work, and therefore so is the potential, etc. We may express this fact in a formula, writing e for the charge, V for the potential of the con- ductor, and 0 as a factor of proportionality, viz., e = OV. This quantity O is called the " capacity " of the isolated conductor ; it may be defined, as is seen from the formula, as equal in value to that charge which would raise the poten- tial of the conductor by a unit amount. It is evident from general considerations that 0 must depend upon the shape and dimensions of the conductor, and upon the dielectric constant of the surrounding dielectric. If air is the dielectric and if the conductor is charged with a quantity, e, the potential VA is, by definition, the work required to carry to it a unit plus charge from the earth; while if the dielectric has the value jfiT, the forces are dimin- ished JT-fold (since the electric forces vary inversely as 7f ), and the potential of the conductor, F^, or the work now re- quired to bring up a unit plus charge, is less than VA in the ratio 1 : K\ or, VA=KVK. So if CA and CK are the capaci- ties in the two cases, e = CAVA=CKVK\ and hence OK=KQA. In words, the capacity of a conductor varies directly as the dielectric constant of the surrounding medium. The connection between the capacity of a conductor and its shape and size may be deduced in certain simple cases by means of the infinitesimal calculus. Thus, it is known for a sphere, an ellipsoid, a cylinder, etc. The capacity of a sphere of radius a in air is numerically equal to a ; and, there- fore, in any other medium it is Ka. If a charge is distributed over two spherical conductors of radii rl and r2, which are in contact, their potentials are the same, but their charges are different. If the dielectric is air, we have, writing el and e2 for the charges and V for the common potential, e\ = r\V, e2 = ?'2F; and the surface densities of the charges on the two spheres are . * a and 2 2> (/" we assume the distribution over each to be uniform. Calling these d\ and ELECTRIC FORCE 649 rfj = F orrf,:rf,= I:I. This in- 4?rr2 rt r2 dicates that if n > r2, </i < </2- So, if the curvature is great, the surface density is great. A point on the surface of a sphere may be compared roughly with a small sphere attached to it ; and so we see why the sur- face density of the charge on a point is so great. Energy of a Charge. — The energy of a charged conductor is located in the surrounding dielectric ; but its numerical value can be expressed in terms of the charge, e, and the potential, F, of the conductor itself. We can imagine the conductor as being originally uncharged, and the process of charging as consisting in the bringing up to it from the earth a series of minute charges. In this manner the charge grad- ually increases from 0 to its final value, e ; and the potential rises from 0 to F". Since the potential at an instant varies directly as the charge at that moment, the mean value of the potential during the process is \V\ that is, the work done in charging the conductor is the same as if the whole charge, «, were brought up against this potential of JF", in- stead of the small amounts having been brought up agains |
t the continually increasing potential. By definition, the po- tential is the work required to move a unit plus charge from the earth up to the conductor, and so tin- work required to move the charge e is the product of e and the potential. In the present case, then, the \\.uk don.- is the product of e and JFi or \eV. This is the value of the energy of the field. It should !M» noted that in this mode of considering the charg- ing of the conductor, an equal charge, — e, of the opposite kind, is left on the earth, whose potential is /ero. Similarly, the energy due to a charge — e whose potential -%cV. (This does not mean that there is such a thing as negative energy ; for if a charge — e is by itself in space, its potential T has a negative value. » S... in general, if there are two conductors with equal and opposite eharges, + e and — e, at potentials /'..and \'r the energy in the di- electric surrounding them I'., l\i 650 ELECTROSTATICS The energy of an isolated conductor can be expressed in another form, which is often useful. Calling it W, the formula is W = \eV\ but e = CV, so we may write W- \ ?- or W= J C V\ The capacity is a constant C for a given conductor in a given dielectric, and is independent of the charge or its potential ; so these formulae show that the energy is inde- pendent of the sign of the charge, depending simply upon its numerical value. Mechanical and Thermal Analogies. — An analogy may be drawn between electrostatic potential and fluid pressure, which is useful. A fluid, either gas or liquid, always flows from points of high to those of low pressure : a positive elec- tric charge moves from points of high to those of low poten- tial. When a gas is compressed by a pump into a vessel of any kind, the pressure continues to increase until a point is reached at which the vessel breaks or the gas leaks, and this maximum pressure does not depend upon the size of the vessel, but upon its strength, etc. ; when the charge of a conductor is increased, the potential rises, and a condition is finally reached when a spark passes or the charge leaks off, but this maximum potential is determined by the " strength " of the surrounding dielectric, not by the size or capacity of the conductor. Similarly, heat energy always flows from bodies at high temperature to those at low ; and, if a small flame is main- tained at as high a temperature as a large one, it is just as useful. Condensers ; Capacity. — Owing to the liability of a charged conductor to lose its charge if its potential is high, a method has been devised by which the conductor may keep its charge unaltered, but may have its potential lowered. When this is done, its charge may be increased before there is again danger of its escaping. The apparatus is called a "con- denser." The general principle, then, is to make use of any processes which will decrease the potential of a charged conductor. ELECTRIC FORCE 651 If the conductor is a plate and is charged positively, the charge will he distributed the same on its two sides, if it is isolated; but if, as shown in the second figure in the cut, another conducting plate is brought near it, minus and plus charges will be induced on this, and an additional amount of the plus charge on the first plate will be attracted around to the face opposite the second one. As a result, the poten- tial of the first plate is lowered ; because if a unit plus charge is brought up to it from the earth, less work is required than before, owing to the action of the induced negative charge on the second plate. The induced plus charge on this plate serves to keep the potential hi<_rh ; and if it is removed by join- in- the plate to the earth, the potential is lowered still more. This potential is now the work required to carry a unit plus charge across from the second plate, whose potential is zero, to the first one; and it may be decreased n if a dielectric, such as glass, is substituted for the air between the plates; for if K is increased, the force required to move a charge is decreased. Therefore, in the end practically all the charge on the first plate is on the face toward the second one, and there is an equal amount ot electricity of the opposite kind on that lace of the second plate which is toward the first one; and the potential of the iir>t plate is greatly below that which it was originally. Fio. 848. — Different steps in the construction of a condenser. "EARTH If the connection with the earth is removed, and the whole apparatus is moved elsewhere, possibly near some other charged bodies, the potentials of the two plates will change, but f/i.-ir difference remains constant, because it equals the \\ork re.|niivd to move a unit positive charge from one plate 652 ELECTROSTATICS to the other; and we may assume that the two plates are so close together that this work depends simply upon the charges on them. If, then, -f- e and — e are the charges on the two plates, and V^ and Vl are their potentials, F^ — V\ is a constant so long as e does not change. If it varies, so does V<i — V\\ and one is proportional to the other. We may therefore write e — G(V^— Fi), where 0 is called the "capacity of the condenser." It is evident that this quantity is a constant for a given combination of two conductors of definite size and shape separated by a definite dielectric of a definite thickness ; and it has, of course, no connection with the similar constant for a single isolated conductor. The numerical value of the capacity may be calculated for many simple cases. A condenser consists essentially of two similarly shaped conductors placed close together and sepa- rated by a dielectric, such as glass, mica, etc. The com- monest forms are those in which the conducting plates, or "armatures," are parallel plates, concentric spheres, or coaxial cylinders. A few facts in regard to the capacity of these condensers are evident from the formula of definition : The capacity must vary directly as the dielec- tric constant of the two dielectrics, because for a given value of e, V^ — V\ varies inversely as K\ the capacity must vary directly as the area of the armatures, because for a given "value of V% — F^, e varies directly as this area ; the capacity must increase as the armatures are made to approach each other, because for a given value of e, V%—VV varies in- versely as the distance apart of the armatures. Exact calculations by means of the calculus show that the capacity, on the C. G.S. system, of two parallel plates of area A and at a distance d apart is (7= ; that the capacity of two con- wd Kr r centric spheres of radii rl and r2 is O= — — ; and that the r 1 KLi:< ilUC FORCE capacity of two coaxial cylinders of radii rl and rv per unit length, is C = 2 log 5 This formula for two parallel plates holds, of course, for those por- tions of the plates only which are not near the edges ; for it is only over 19. — Lines of electrostatic force in the case of a plate condenser, the dielectric being air. those portions that the conditions are uniform. This is shown in the cut, which represents the lines of force. Therefore if such a condenser is to be used for purposes of measurement, a device is employed, invented i. \\hii-li consists in having a 1 one plate near its centre cut out from the rest of the plate and supported independently of it. The rl this disc are uniformly distributed, and the formula given above applies to its capac- |.!.-it- -f it One form of con- denser most often used for qualitative experiments is the so-called whos. iuii.-r mil outer surfaces, except for t li. ii of a glass bottle ""^^^•••T""' 654 ELECTROSTATICS upper portions, are coated with tin foil; the bottle is closed by a wooden stopper through which passes a brass rod tipped with a knob and ending below in a chain which makes contact with the inner coating. The capacity of such a condenser may be calculated fairly closely from the formula for two parallel plates. Discharge of Condensers. — If a conductor carrying a knob is attached to each armature of a condenser, and if these two knobs are gradually pushed toward each other by means of some insulating rod, a limiting distance will be reached at which a spark will pass between them. This distance varies with the difference of potential V^—V^ the greater it is, the longer the sparking distance. When the spark takes place, the potentials of the two armatures become the same, and since the plates had equal and opposite charges, the final charge on both plates is zero : the condenser is said to be "discharged." Experiments show that there are two types of discharge. In one, the charge on each plate becomes gradually less, the potentials gradually approach the same value ; and in the end the charges have disappeared and the potentials have become equal. This is known as a " steady " discharge, and occurs if considerable opposition is offered to the discharge ; for instance, if one of the knobs between which the discharge takes place is joined to its armature by a poor conductor, such as a wet thread. In the other type of discharge, the charge on either plate disappears rapidly, then it becomes charged again with electricity of the opposite kind, this dis- appears, it becomes charged again as it was originally, etc. ; each successive charge being less in amount than the previous one. This is called an " oscillatory " discharge, for obvious reasons; and it occurs if the opposition to the discharge is small. (We have mechanical analogies of a steady discharge in a pendulum moving in a viscous liquid such as molasses, and of an oscillatory one in a pendulum vibrating in air.) If the spark of the discharge is viewed in a revolving mir- ELECTRIC FORCE ror, that of a steady discharge appears like a broad continu- ous hand, gradually fading away, while that of an oscillatory one is seen to consist of a series of distinct sparks, showing that the electric current is intense, then vanishes, rises aLCain. etO. Electric Waves. |
— It is proved by experiment that, when an electric oscillation of this kind takes place, there are dis- Poles between which the spark passes. Flo. 8M. — Photographs of an oscilUtlug «|.ark when viewed In a revolving mirror. turhanees of th<- natmv ••!' \\a\es prndurrd in the sur- roumliiiL: medium, llet'mv the disehari^ occurs the electric intensity at any point m-ar 1»\ has » definite value. hu( as the <li>.-harge goes on, theeleotrio intensity varies in amount and diivetinn. ami th.- oseillat ion jirudiicus waves. Kxperim- lia\i- .sliown that ihe^- waves tra\«-l in air with the vel<> of li^lit ami that thev are t rans\ ers.-. I IP ean In- relli-j-ted, • ;:>• ; ELECTROSTA TICS refracted, diffracted, polarized, etc. Their wave lengths can be measured, and waves as short as a small fraction of a cen- timetre have been obtained; as a rule, however, they are much longer. These waves may be detected by means which will be described later. One method may, however, be men- tioned here ; if they fall upon two conductors which are close together, they will — under suitable precautions — cause minute sparks to pass between them. These " electro-magnetic " waves, so called, were first in- vestigated by Joseph Henry, in 1842, but were rediscovered many years later by Hertz. They serve a commercial pur- pose in the various systems of wireless telegraphy which are now in daily use. Since the waves travel in air with the velocity of light, it is proved that they are ether waves. One would not expect them to travel in solid dielectrics such as glass with the same velocity as does light, because their wave lengths are so different, and it has been shown in Chapter XXX that the velocity of waves varies greatly with the wave length in all solid or liquid media. The medium, then, which serves as the means by which magnetic and electric forces are manifested, which is the " carrier" of the tubes of induction, is the luminiferous ether. This fact was first suspected by Faraday, but was proved by Maxwell by an indirect method. Condensers (continued). — The energy of a charged con- denser is, from what was proved above, Je(F^ — Fj). This may be written \^ or £tf(F2- F^2. Since the field of force is confined almost entirely to the space between the two armatures, as is shown in the cut for a parallel plate con- denser, the energy is located there also. Condensers are often joined together so as to increase their action. There are two general methods of doing this. Let the two plates of the first condenser be called Pl and §1 ; ELECTRIC FORCE 657 those of the second, P2 and Qv etc.; and let them be always charged in such a manner that Pp Pa, etc., are positive, and Qv Qv etc., are nega- tive Then, if Px, Pv p,. f . P. 1 r . p3( etc., are connected by wires, and (>r 0* etc., are also connected, the FIG. 858. — Three condensers joined in parallel. condensers are said to be "in parallel." Whereas, if Ql is joined to P2, #2 to P8, etc., they are said to be "in series." Let the condensers all have the same capacity and all be charged alike before they are connected ; then their differ- ences in potential are all equal, but the potentials of any two plates, e.g. Pl the be and P, need same. the potentials of P1 and Qv V, and U2 those of P2 and Qr etc. Then in all cases not be and •I.— Three condensers joined in series. Let FJ- tf1= F2- Z72 = etc. ' 1 1 the condensers are joined in parallel, Vl = V^ = Fg = etc., and Ul = U^ = Us= etc. ; so, it is exactly as if the condenser were made up of two large plates, one, Pv Pv P8, etc., the other, Qv Q2, Qy etc. The difference of potential 2, is Fj — Ur and the total charge on either " plate " is ne^ \\ here n is the number of condensers connected and e is the charge on each plate; so the capacity is increased n tim« •>. Thus, joining in parallel gives an increased quantity^ but does not change the difference in potential. 1 1 the condensers are joi m-d in series, J7j = Vv U^ = V# etc. ; t there are n condensers, V± — Un^n(Vl— U-^). Thus if P! and Qn are connected so as to discharge the condenser, the difference of potential is increased n-fold; but the quantity of electricity discharged is the same as for a single con- d. Miser. Since the distance between two conductors at which a spark will lak«- place is increased if their difference AMES1* PHT81C8 - 42 658 ELECT It OSTA TIC 8 of potential is incrt-usi'd, joining condensers in series in- creases their sparking distance. (When two or more con- densers are joined in series, the minus charge on Ql does not combine with the equal plus charge on P2, etc., until Pl is joined to Qn. Before this, the minus charge on Ql is held in place by the attraction of the plus charge on Pv etc.) Electrometers. — Before we can explain how the various electric quantities are measured, it is necessary to describe an instrument which enables us to measure differences in Principle of quadrant electrometer. potential. Such an instrument is called an "electrometer." There are many forms which may be used to measure the ratio of two differ- ences in potential; so that, if one is known, the other may be calcu- lated. The best of these is the Km. 856. — Thomson's quadrant -l-'tP.meter; one of the quadrants is removed so as to show the " needle." " quadrant electrometer," which was invented by Lord Kelvin, then William Thomson. It consists, as shown in the drawing, of a cylindrical metal box which is divided by two trans- verse cuts into four " quadrants," and of a horizontal metal " needle " shaped like a solid figure eight, which is sus- pended by a fibre. The pairs of diagonally opposite quad- ELECTRIC FORCE 659 rants are connected by wires, and the needle is raised to a high potential by some electrical machine. If the difference of potential of two plates of a condenser is to be measured, each is joined to a pair of quadrants ; and the needle, which takes a symmetrical position with reference to the quadrants when they are not at different potentials, will now move so as to enter one pair, until it is brought to rest by the torsion in the fibre. The needle forms with the two plates of a quadrant a condenser, and the motion takes place in such a direction as to make as small as possible the energy of the i ondeiisers it makes with the four quadrants. It may be pn»ved by methods of the infinitesimal calculus that the angle through which the needle turns varies directly as the di Hen- nee of potential of its two sets of quadrants. Thus, two differences of potential may be compared by measuring the corresponding deflections of the needle. In order, however, to measure any one difference of poten- tial, a different instrument must be used. This is the "absolute electrometer," which was also invented by Lord Kelvin. As shown in the cut, it consists of a paral- lel plate condenser, with a disc cut out of the upper plate as described on page 653. In practice this is suspended from one arm of ilance. The two phltCS of the condenser are joined tO the tWO OOndUOtOn WHOM Fl<1 ***• -Thomson'* original form of n1 difference of potential d.-sired: the plates are ihns charge. 1 with opposite kinds of electricity, and the !'<.!•.•<• ,,f attraction on the movable disc may he n. putting vreightfl in the balance pan. It the area of this i the distance .ip.u-t of the plates, d, the difference of potential, f^_ V^ the dielectric con- 660 ELECTROSTATICS stant, JT, the force of attraction on the disc is given by the formula (V.-V^AK B««P zL If the plates are, as usual, in air, K= 1, and ( F^— F1)2=— F, rf, and A can all be measured ; and so V^—V^ is known. Measurement of Electric Quantities. — The four electrical quantities that have to be measured are quantity, potential, capacity, and dielectric constant. We have just shown how differences in potential may be measured ; and, if the poten- tial of a conductor is to be measured, it may be joined to one plate of an electrometer, while the other plate is connected with the earth. The capacity of a sphere or of a simple form of condenser may be calculated from a knowledge of its dimensions, as explained on page 652. But there is a simple method, due to Cavendish, for determining when the capacity of two con- densers is the same ; and so, if the capacity of any condenser is desired, it may be compared by this method with a con- denser whose capacity may the capacities of two condensers. B, known, e.g. a parallel plate FIO. 867.— cavendish's method of comparing condenser the distance apart of whose plates can be varied. The method is as follows : let Av B1 and Av J?2 be the two condensers ; charge them by joining A1 and A% to some electrical machine, while Bl and B% are joined to the earth ; then disconnect B2 from the earth, and A1 and A2 from the machine and from each other ; join Bl to A2 by a conductor, and A1 to the earth (or to B^). If the capacities are equal, an electroscope in contact with the wire joining Bl to A2 will show no effect when Al is earthed ; for let O1 and (72 be the two capacities, and let V be the potential given Al and Az by the machine ; the charge on Al is then + 01 V, on Bl is — Ol F, on A2 is + C2 V, and on B is ELECTRIC FORCE 661 - C'2 r. \VlKMi Bl and A2 are joined, the two charges, — 6\T and -f-(72F, do not combine until Al is joined to the earth. Then they do, and the final charge, which is distributed over Bv AT and the wire joining them, is V^C^— C^)\ and this will utYi'L-t an electroscope unless Ol=Cy This method also permits one to measure K for any di- electric, and was so used by Cavendish. The capacity of the second condenser may be measured when air is the dielectric and again when glass, or sulphur, etc., is substituted. The ratio of the latter capacity to the former is the value of K. In order to measure a charge, the accepted method is to place it on a condenser whose capacity is known and to meas- ure the resulting potential. Then, since e = C ( V^ — Fi), the value of e is known. There may be a difficult |
y in making the charge pass to the condenser, but the method described on page 639 may always be used. This is to put the charge inside a conducting vessel which is nearly closed; an equal charge will appear on the outside and this may be measuivd. An ingenious method was devised by Lord Kelvin for the measurement of the potential at any point in the atmosphere. Let A be the point of a conductor which is joined to an electrometer, and let some means be adopted to have a continuous current of small conducting particles leave it. Let B be such a particle. Then if the potential of A is higher than that of points in the air near it, a plus charge will he induced on B and a minus one on A ; B will carry this charge off as it moves away; and the process is repeated as the stream of particles is maintained. Finally, the potential of A will be lowered by this accumulation of negative charges until it is the same as that of the surround- ing air. Similarly, if the potential of A is lower than that of the air near by, it will be raised until it is the same. There- fore, when the potential of A ceases to change, it gives the potential <>! tin- air at that point, and may be measured by the electrometer. One means of causing a pointed conductor to t ; 1 5 '2 ELECTROS! A TICS oil particles is to use a small flame, because a burning gas is a good conductor. Another method is to have as the conductor A a vessel of water ending below in a small fun- ik-1, so that drops of water are continually forming and break- ing away. In this manner many interesting facts in regard to atmospheric electricity have been learned; one at least should be noted ; the potential of the lower layers of air is as a rule always higher than that of the earth, and its value is continually changing. Strains Due to Electrification. — The fact that the main phenomenon of electrification consists in a strain of the dielectric is shown, as has been said before, by the formation of sparks, and in many other ways also. One of the most direct proofs is furnished by what is known as the " residual charges " of a glass condenser. If one is charged to a high potential and then discharged, a second discharge may be obtained after the lapse of a short time; then a third may be obtained, etc., each one being feebler than the preceding one. These are said to be due to residual charges. They depend upon the fact that glass is non-homogeneous ; for they cannot be obtained with a homogeneous dielectric. Their explanation is as follows : When the condenser is first charged, the glass is mechanically strained, and when it is discharged, certain parts of the glass lose their strain and, owing to inertia, are strained again in the opposite manner, while other portions of the glass do not relax completely; these two portions, however, balance each other for the moment, and there is no resultant strain ; as time goes on, however, these strains, not being maintained by any force, gradually relax, but not to the same degree, so there is again a resultant strain ; this causes the second discharge when the armatures are joined, etc. If it is remembered that there is no field of force inside a conductor, so that such a body cannot maintain a strain, all the phenomena of induction, etc., may be at once explained. ELECTRODYNAMICS CHAPTER XLIV PRODUCTION OF ELECTRIC CURRENTS Definition of Terms. — The simplest case of an electric current is furnished by the steady discharge of a condenser. (See page 647.) In this, two plates having a difference of potential are joined by a conducting wire; and, as a result of the change, the charges of the two plates disappear. It is noticed further that the temperature of the wire is raised, and certain magnetic effects are produced in the region around the wire. All these phenomena constitute the elec- tric current. \Ve speak of the current as beni- in the wire; Imt this is only a mode of speaking. As the discharge begins, the plus charge on the plate of higher potential decreases, and so does the minus charge on the plate of lower potential : if by some means these charges may be maintained constant by adding continually the nec- essary <|iiantitics of plus and minus charges, the potentials of the plates will remain unchanged: and the current i> said to be "steady." The phenomenon in the Conducting wire, which constitutes the current, consists, as will be shown in the next chapter, of a motion of a stream of positively charged particles in the direction from hi^h to low potential in the wire, and of a stream of negatively charged particles in the opposite direction. lly •/•_///////"// the former direction is called that of the current. If t\ is the cjuantity of plus elec- tricity that passes through the cross section of the wire at 688 664 ELECTRODYNAMICS any point in a unit of time, and i2 is the quantity of minus electricity that passes at the same time in the opposite direction, the quantity i\ + e'2 is called the " strength of the current" or, more often, "the current." If the current is steady, the quantities ij and izt pass in an interval of time t ; and (tj + iz)t is called the " quantity of the current." If the current is not steady, and if in any interval of time the quan- tities of plus and minus charges that pass are e1 and ez, the quantity of current is (^ 4-02). (Thus, in the discharge of the condenser whose plates are charged with + e and — e, the quantity of the current is e, because the plates will be dis- charged if e± + 02 = e. If H- e passes from one plate to the other ; or if — e passes in the reverse way ; or if 4- ^ e and — $ e pass in opposite ways ; etc., the plates are discharged.) In order, then, to produce a current in a conducting wire it is necessary to have a difference in potential between any two of its points. This difference is called the "electro- motive force" (E.M.F.) between the two points. Work done by a Current. — The passage of a current evi- dently involves the idea of work. If V^ — V\ is the differ- ence of potential between two points in a wire, and if (il + i'2) is the current strength, the quantity of positive electricity il moves from a point of high potential, Vv to one of low, V-p and therefore the electric forces do the amount of work i\(y<i— Fi) ; and similarly, owing to the motion of a quan- tity of negative electricity in the opposite direction, the same forces do an amount of work ^'2( V2 — V\). So the total amount of work done in a unit of time by the electric forces is (t'i + *2) (^2— V\) 5 or, calling the current strength i and the difference in potential, E^ it is iE\ and the work done in an interval of time, £, if the current is steady, is iEt. Or, in general, if e is the quantity of current, the work is eE. Ordinarily this work is spent in raising the temperature of the conductor which carries the current ; and the necessary amount of energy is furnished by whatever produces the cur- PRODUCTION OF ELECTRIC CURRENTS 665 rent. (The heat produced in the conductor may be meas- mvil if it is in the form of a wire by coiling it in a calorimeter of water. See chapter XII. If the C.G.S. system of units is used in defining the unit quantity of electricity, the product iEt is a certain number of ergs; and so the heat produced must be expressed in ergs.) Heating Effect of a Current. — This heating effect of a current is, of course, greatest where there is the greatest amount of work done ; that is, where the electromotive force, or drop in potential, is the greatest. This is illustrated in various forms of electric lights, in the electric furnace, in electric heaters, etc. 1 Fw. 868. —The electric arc between two oarb< < The arc light, as used for illuminating purposes, consists of two rurbon rods \\lnrh ;uv connected to some source of a 666 ELECT ROD YNAMICS current, and which are so controlled by automatic mechanism that when no current is flowing they are in contact, and then as soon as the current begins they are slightly separated. The two rods when loosely in contact offer great opposition to the current, so the temperature rises at the points of con- tact ; this makes the surrounding gas a conductor, and now the rods are drawn apart. The current passes off one rod to the gas, and from this to the other rod. There is great resistance to the current passing off or on a solid ; and the temperature of the tips of the rod is raised to a " white heat," if the current is sufficient. This produces the light. Experiments show that more heat is produced at the end of the rod from which the current proceeds to the gas than at the other; this is called the "positive pole." In the ordinary "incandescent light" there is a glass bulb into which enter two platinum wires connected inside by a fine filament of carbonized wood fibre, and from which the gas has been exhausted as completely as possible. A current is made to flow through the filament, and its temperature is raised to white heat. It does not burn up, because there is no oxygen left inside the bulb. Fro. 859.— Incan descent lamp. In the Nernst lamp there is a small filament whose constitution is a commercial secret, which ends in two metal wires ; this filament is not a conductor unless its temperature iy high, and even then under the action of a current in one direction it decomposes and breaks down. Therefore the process of using the lamp is first to raise the temperature of the filament until it becomes conducting, and then to have it traversed by a current whose direction is reversed at short intervals. If this is done, the filament gives out a brilliant light; and, as it does not oxidize, it may be used in the open air. PRODUCTION OF ELECTRIC CURRENTS 667 In an electric furnace use is made of the high temperature of the arc ; and the carbon rods are inclosed in a space whose walls are non-conductors for heat, and in which the pressure of the i^as may be increased. Direction of Current. — In order to determine by experi- ment the direction of a |
current it is necessary to ascertain which of the two conductors between which the current flows has the higher potential. The simplest mode of doing this is one invented by Volta. The two conductors 0 whose potentials are F^ and Fi are joined by wires to the two plates of a condenser, A2 and Ar if ra>rr the plate A% becomes A8|A, charged positively, and Ar negatively ; be- cause lines of force pass across from A% 1 Those Charges Fio. 860. — Method of determining direction of an elt-rtric on the two faces *' current. Volta's condensing electroscope. nearest each other ; but if the wire leading to A l is broken, and the plate A% is then removed, the negative charge on A1 will spread over the whole plate and may be detected and studied. In Volta's arrangement the plate Al was the top plate of a gold-leaf electroscope, and A., was a similar plate coated with a thin layer of shellac and carried by a •_rlass handle. Therefore in this apparatus, after the wire leading to Al is broken and A.2 is then removed, the nega- tive charge will spread over the plate and the, gold leaves, which will then diverge. If now a glass rod which has been rnhhed \\ith silk is brought near the electroscope, it will induce a positive charge on the leaves, which will in part neutralize their negative charges, and so they will collapse. It', on the other hand. J "J, < Fj, the gold leaves will be- come charged positively, and a charged glass rod will cause ELECTRODYNAMICS them to diverge still farther. In this manner, then, it may be determined whether F2 > Fr or F^< V^\ if the former is the case, and if a wire is made to join the two conductors, the direction of the current is from the one at potential V% to the one at potential Vl ; in the contrary case, the direction of the current is opposite to this. Detection of a Current. — When an electric current is flow- ing in a conductor, its temperature rises, as explained above, owing to the work done by the electrical forces against the molecular forces of the conductor. But this fact does not lead to a simple direct means of ob- serving a current, because with a feeble current the change in tem- perature is small. The magnetic action of a current offers, however, an extremely simple and direct method of detecting and even measuring a current. It was discovered by Oer- Apparatus of Oer- sted> a Danish physicist, in 1819-1820, that a wire carrying a current had a magnetic field around it. We shall take up this question more in detail in a later chapter ; but one or two facts may be stated here. sted for studying the action of an electric current upon a magnet. FIG. 361. If a magnetic needle is pivoted so as to be free to turn about a vertical axis, it will assume a north-and-south posi- tion, and now if a conductor carrying a current is placed parallel to it, but above it, the needle is deflected ; if the current is reversed, so is the deflection. Similarly, if the current is parallel to the needle, but below it, it is deflected ; but the direction Of the deflection is Opposite FlG. 862. -Section of a simple galvanoscope. PRODUCTION OF ELECTRIC CL'RREXT* to what it would be if the current were above the needle. Hence, it follows that if the conductor carrying the current is made in a loop lying in the magnetic meridian and inclos- ing the magnet, the deflection will be increased ; and if many loops are used, forming a flat coil, the deflection will be still greater. This constitutes a "gal- vanoscope." Another mode of increasing the deflection still more, and at the same time of avoiding, to a large extent, any disturbances of the magnet due FIG. 868. -An astatic combination to other actions than those of the current in the coils, is to attach rigidly to it another magnet of equal magnetic moment, but turned so that its axis is in an opposite direction. Thus, a north pole of one comes opposite to the south pole of the other. If, now, one of -<j magnets is inclosed in the coil and the other is either above or below it, the deflec- tive force of the current is in the same direction on both magnets ; but the action of any other magnetic field is almost entirely prevented. Such a combination of mag- nets as this is called an Fio. 864. —Section of • palvanoscope with MUttc needle. "astatic needle," because if tin -ir magnetic moments were exactly equal, and if their netic axes were exactly parallel, the system would not be under a directive force due to the earth, and would remain stationary in any position. Actually these conditions are not satisfied ; and the earth has an action, but it is extremely small. So, by bnnun,ILr another magnet in-ar the astatic needle, it may be made to take any position that 670 ELECTRODYNAMICS is desired, and the action of the earth may be neutralized as completely as is desired. By thus using a 'control magnet," then, the coils to carry the current may be kept in any position which is convenient, and the astatic needle may be made to lie in their plane, while the field of force due to the earth and the control magnet may be very' small. This last is shown by the period of the magnet becoming very long when it is set in FIG. 866. — Galvanoscope. I £ vibration, for T — ZTT^ — - (see page 611) ; and so, if R is small, T is large. The field of magnetic force near any current may be studied by the use of iron filings or of a small magnetic needle, as* was described on page 603. It is found that the lines of magnetic force form closed curves around the cur- rent; the directions of the current and the lines of force being connected by the right-handed-screw law. Thus, if LINES OF FORCE A ' CURRENT Fio. 866. — Diagram Illustrating connection between the direction of a current and that of the lines of magnetic force. AB is any portion of a conductor carrying a current from A to B, the lines of magnetic force near it are as shown ; or, if the total electric circuit is considered, the lines of force pass through it from one side, and return outside. Thus, a current and any one of its lines of magnetic force form two closed links threading each other, like two links of a chain. If a current, then, is passed through the coil of a PRODUCTION OF ELECTRIC CURRENTS 671 galvanoscope, the magnetic needle will be deflected; and if the current is reversed in direction, so is the deflection of the magnet. If the current in the coil is in the direction of the motion of the hands of a watch as one looks at the coil from one end, the magnetic force is directed away from the observer, so that a north pole is forced away from, and a south pole is forced toward the observer. By means of such an instrument one can determine, then, the direction of a current, and can roughly estimate its strength. Tangent Galvanometer. — If the coil of the galvanoscope is a circular one, that is, if the cylinder on which the wire is wound has a circular cross section, the intensity of the magnetic field at the centre of the coil may be proved (see page 711) to vary directly as the current strength and inversely as the radius of the cylinder referred to. Thus if i is the current strength and the radius, the intensity of the magnetic force is pro- portional to -; it also varies directly with the number of a turns of wire in the coil ; if this is n and if the turns of wire are so close together as practically to coincide, the intensity may be written /= c— , where c is a factor of proportionality. a The numerical value of c depends, of course, on the units chosen for the magnet ie and electric charges. If the coil is placed in tin- magnetic meridian and a current is passed around ic magnet (not an astatic one) sus- pended at its centre is under the action of two opposing couples, one due to the magnetic field of the earth, and the other to that of the current in the coil; and it comes to rest when these balance each other. If M is the magnetic moment of the ma-net. // the l,,,rj- ELECTItOD YNAMICS zontal component of the intensity of the earth's field, and N the angle that the magnet makes with the magnetic meridian when it comes to rest, the moment due to the earth's force is JIM sin N; and that due to the electric current is /M"cos N. Since these must bal- ance each other, or FIG. 368.— Tangent galvanometer. Therefore the strength of the current is meas- ured by the tangent of the angle of deflection ; that is, the strengths of two currents vary directly as the tangents of the angles of deflection. Such an instrument as this is called a " tangent galvanometer. " Electro-magnetic Unit Current. — The current strength is defined in terms of the quantities of charge which pass a cross section ; but actually these quantities cannot easily be measured directly. So, it is more convenient to define a " unit current " in terms of its magnetic properties, and then from this to deduce the value of a new unit quantity. Thus, a " unit current," or one of unit strength, is defined to be such a current that if flowing in a galvanometer coil of one turn whose radius is 1 cm., the intensity of the magnetic field at its centre equals 2 IT dynes. (The reasons for this choice of unit current will appear later.) Thus, using the same symbols as in the above formula, /= c — , this may be expressed by saying that, when i — 1, n = 1, and a = 1, PRODUCTION OF ELK'TUH' CURRENTS 673 /'= -2 TT : so this < It-tin it ion of a unit current is equivalent to putting c = - TT in the formula. This unit current is called th. ••( .(..S. electro-magnetic" unit, because its definition depends upon the intensity of a magnetic field ; that is, upon the force acting upon a unit magnetic charge. Then in a undent galvanometer the formula becomes .) _^ or, t = H-- tan N. Ha '2 -mi - - is called quantity it (r, i= - - tan N. IT and G may be measured, and the "galvanometer constant " ; TT N observed ; so the strength of a current may be measured. Tin- C. <i. S. electro-magnetic unit quantity of electricity is, then, the quantity carried past any cross sectio |
n of the conductor in one second by a unit electro-magnetic current. Thru- must, of course, be a constant relation between this quantity and tin* ('. (i. 8. electrostatic unit quantity as de- fined on page 640 ; and experiments prove that one C. G. S. electro-magnetic unit charge equals 3 x 1010 C. G. S. electro- static units of charge. (It should he noted that this ratio of th«- units equals the velocity of litfht.) The work ivqnhvil to carry a unit electro-magnetic quan- tity of charge from one point to another is the difference of potential between them exprened on the C. G. 8. electro- magnetic system. Kxperimeiits sho\\ that the numbers which would be required to BZpI68fl ordinary potential differences on this system are so large that they are inconvenient. Consequently other units arc used in practice. Thus a utial dilYcreiK t^on the C. G. S. electro-magnetic system of I'" "100,000, or 108, is ealled one -volt/' Measurement of Quantity of Current. — It is often necessary to n 'ie t.-tal <piant it y of current which llows in a short time, <>.//. \\ : 1 1 -<• harmed. In this case the current is not constant, and further the time of flow is SO AMEfl'8 PHYSIC'- • 7 i ELECTRODYNAMICS short that it is not possible to secure a permanent deflection of the galvanometer needle. If such a current is passed through a galvanometer, the needle will be acted on by an impulse and will have a certain " fling " ; that is, it will be deflected from its north and south position and will proceed to make a number of oscillations before coming to rest again in its former position. If the period of vibration of the needle is so long and the time of passage of the current so short that we may consider the current as over before the needle is deflected an appreciable amount, the maximum angle of deflection measures the quantity of current. The exact formula may be deduced without difficulty. If e is the quantity of the current on the C. G. S. electro-magnetic system; JV, the angle of fling from a north and south direction ; 6r, the galvanometer constant ; H, the horizontal component of the earth's magnetic force ; T, the period of vibration of the magnet, An instrument specially designed to measure quantities of current, as distinguished from current strengths, is called a " ballistic " galvanometer. Measurement of Electro-motive Force. — Since an electro- motive force is a difference of potential, it may be measured by any electrometer. (See page 658.) But, in general, other methods are adopted. One is to join the two points which are at different potentials to a condenser of known capacity, and then to discharge it through a ballistic galva- nometer. If E is the difference of ^potential and 0 the capacity of the condenser, the quantity of current measured will be CE. (The value of 0 on the electro-magnetic system must be used, if E is to be measured; but, if two electro- motive forces are to be compared, it is not necessary to know the value of (7.) PRODI < PlOJi "F l-'.l.KVTRIC CURRENTS 675 Another method of comparing differences of potentials depends upon the fact, which will be discussed more fully later, that in the case of a steady current its strength is directly proportional to the E. M. F. producing it; but, if the E. M. K. is applied at the ends of a long, fine wire, the current is small, while if the wire is short or thick, the current is large. In the former case there is said to be a great " re- sistance"; in the latter, a small one. Thus, if a P — » steady current is flowing through the conductor 0000 PQ, and the value of the R difference Of potential be- FIO.SW.— Diagram Illustrating a method of i tween two points A and B is desired, these points may be connected by wires to a gal- vanometer, #, through coils of wire, R, which are so long and so fine that they offer such an opposition to the passage of a current that practically none flows from A around through Q- to -B, and so no difference is made in the conditions at A and B. If, however, the galvanometer is sufficiently sensi- tive, it will measure this minute current; and its strength is directly proportional to the difference of potential between A and B. Other methods may be found described in labora- tory manuals. Steady Current. — If by any means, mechanical, chemical, thermal, etc., it is possible to maintain a constant difference of potential between two conductors, a steady current may be produced by connecting these two conductors by a wire or other conductor. There are at least four methods by which this constant diffen -nee in potential may be produced. 1 in electrical machine such as described on page 685 is turn.-.! at a uniform rate, it may be used to furnish a steady current. It' \\\ nt metals, sin-h as y.inr and copper, a re partl\ iniinei ><-d in soiur li.pfid conductor other than a fused metal, Midi as a solution of sulphuric acid in 676 EL ECTROD YNA MICK water, it is found that the rods are at different potentials. If a closed metallic circuit is made by joining several wires of different material in series, and if the junctions of the different wires are at different temperatures, a current is pro- duced in the circuit. Again, if a closed circuit of some wire is moved about in a magnetic field in such a manner that the field of force through the circuit varies, a current arises; and, if this change in the field continues at a uniform rate, the current is steady : this constitutes a " dynamo. " Primary Cells. — Experiments show that, when a solid con- ductor is immersed, partly or completely, in a liquid conductor other than a fused metal, there is a difference of potential between them, which is characteristic of the two conductors. So, if two solid conductors dip in the same liquid, they will be at different potentials ; and, if they are joined outside the liquid by a wire, a current will flow in it. This fact was first observed by Volta (1800), who used zinc, copper, and dilute sulphuric acid in this manner. This is said to be a "Voltaic cell." It is a question of experiment to determine which of the solid conductors has the higher potential. In the case of the voltaic cell, the copper rod is at a higher potential than the acid, and the acid is higher than the zinc ; so the current in the connecting wire outside is from the copper to the zinc. It is observed that, as the current continues to flow, the zinc gradually dissolves away and bubbles of hydrogen gas collect on the copper rod or break loose from it and rise to the surface. It is observed, further, that there is a current also through the dilute acid, and that its direction is from the zinc to the copper. Thus the current flows in a circuit; FIG. 870.— Voltaic cell. PRODUCTION OF ELECTRIC CUliliK\T.< 677 outside the liquid, from copper to zinc ; inside the liquid, from zinc to copper. Since the direction of a current is itl ways from a point of high potential to one of low, it is thus evident that at the boundary separating the zinc and the acid there must be some mechanism which raises the poten- tial: so that the points on the zinc must have the lowest potential in the whole circuit, and contiguous points in the dilute aeid must have the highest potential. This phenom- enon is evidently connected closely with the dissolving of the zinc in the acid. If pieces of zinc are placed in dilute sulphuric acid in a tumbler or beaker, it is noted that the zinc dissolves, that hydrogen gas is evolved, and that the temperature of the acid is raised. This proves that, when zinc dissolves, energy is liberate! 1 : in the simple chemical experiment this energy is spent in producing heat effects; in the voltaic cell it is spent iii raising the potential of points in the acid, and this maintains the current and so heats the conductors, etc. At the surface of the copper, where the current enters it from the acid, work is required to raise the potential of the plus charges from that of the acid to that of the copper, and to lower that of the negative charges which are going in the opposite direction. This difference of potential at the sur- face is due to the evolution there of the hydrogen gas. The mechanism of the current through the acid and at the /ine and copper rods will l»e discussed in the next chapter. The two solid conductors which dip in the liquid are called ••poles"; t he one which 18 at the higher potential is called the positive "ne, while the other is called the negative one. The latter is always dissolving as the current flows; so if it contains any metallic impurities, e.g. if the zinc has particles of iron in it. there will be local currents from the zinc to the acid, th.n to i1, iron, and thence to the zinc, etc. These currents have no external action ; and so should be pre- vented, it possible, because the zinc consumed in producing 678 them is \\;i>tr<l. This can be done in many cases by rubbing mercury over the xinc rod before it is immersed in the liquid, and thus making a surface of mercury amalgam with the metal, which is practically uniform. As the current flows in a voltaic cell, hydrogen bubbles collect over the copper pole, and thus hinder the action of the cell. Various devices have been invented in order to prevent this. The most successful is due to Daniell. He made a cell, which bears his name, consisting of a porous cup — such as unglazed porcelain — inside a larger vessel ; the cup contains a saturated aqueous solution of copper sul- phate, and the outer vessel, dilute sulphuric acid ; the zinc rod dips in the latter, the copper rod in the former. When the two rods are joined outside by a wire, the current flows from the copper to the zinc. As it flows, the zinc dissolves as before, but now copper is deposited out of the copper sul- phate solution on the copper rod. Consequently there is no change in the nature of the surface of the latter. This cell of Daniell is a typical "two-fluid cell." Other cells, both one and two fluid, can be made by using other metals |
than zinc and copper, and other liquids than sulphuric acid. They are called " primary cells " in distinc- tion to " secondary " ones, which will be described presently. Cells may be joined "in parallel" or "in series." Thus if Ol and Z1 are the posi- tive and negative poles of one cell, <72 and Z2 those of the second, etc., the cells are said to be in series if Ol and Zv 02 and Zy etc., are con- nected by wires ; while if Ov Cv <73, etc., and Zv Four cells joined tn parallel. 2' 3' Vt0m 87i. nected, the cells are said PRODUCTION OF F/./.r//;/r i 679 t«> he in parallel. If the cells are all of one kind, let E be tin- difference of potential between the two poles of each; then it n cells are joined in series, the difference in potential between C" and Zn is nE. Whereas, if they are joined in parallel, the only effect is to make what is practically one (dl \viili poles n times as large; this does not affect the difference of potential between the poles. A mechanical analogy of a simple voltaic cell is furnished by a pump or paddle wheel working in a horizontal tube connecting two tall vertical pipes containing some liquid, such as water. If the pump is open, the liquid will stand at the same level in the two vertical pipes; but, when the pump or wheel is set in action, the liquid will be forced through so as to stand higher on one side. A difference of pressure on the two sides of the pump or wheel is thus produced ; and, if sufficient, it will stop the action of the latter. If now a connecting tube between the upper portions of the pipes is opened, the liquid will flow from the one at the foot of \\hich the pressure is the higher over into the other, and a continuous current will be produced. This pij.j- in which tin- pressure against the pump or wheel is the greater corresponds to the o<>pp« -r rod in the voltaic c«-ll ; tin- other pip.- to tin- xinc; and the pump or wheel to the energy furnished by the dissolving zinc. taic cell. Fio. 872.-Model rep- resenting action of vol- Thermoelectricity. — If a closed circuit of linear conduct- ors, like wires, includes at least two different substances, tin-re is in general an electric current produced in the circuit if the junctions nf the.se substances are kept at different temperatures. Thus, if two \vires I and II make up a cir- cuit having junctions at .1 and B, there will be, in general, a current if the temperatures <.f A and //are not the same. The direction and strength ,,f the current depend Upon the t \vo substances and upon the difference in temperature. It 'iind by experiment that, beginning with a condition when A and //are at the same temperature, if that of A is 680 EL EC Tit OD YNA MICS kept unchanged and that of B is continuously increased, the current will be in a definite direction and will gradually increase, while if the temperature of B is decreased, the current will be in the opposite direction and will gradually increase ; as the temperature of B is made to differ more and more from that of A in one direction, — in certain cases when it is higher, in others when it is cuit made up of two con- lower, — there comes a point when the doctors i and //, having current begins to decrease, and finally junctions at A and B. ° J one at which the current ceases; while if the difference in temperature is increased still more, a current is produced, but it is in the opposite direction to that which it was before, and as the change in the tem- perature of B continues, this reverse current increases in strength. If tA is the temperature of A during the experi- ment, and if £/ is that of B at which the current ceases, FIG. 378. -A closed cir- experiments show that their mean A^'r is a constant quan- tity for any two substances : it is known as their " neutral temperature"; and tj is called the "temperature of inver- Zi sion," corresponding to tA. In order to explain these thermocurrents, as they are called, it is necessary to assume that at any cross section in the conductors where two different substances come in con- tact there is a difference of potential. If P and Q are two different substances meeting over a surface, the fundamental experi- ments of electrification show that, when they are separated, one is charged with plus, the other with minus electricity. This proves that when they are in contact there is some electric force — due to the difference in the elec- tric properties of the molecules — acting at the surface of con- FIG. 874. — Junction of two con- ductors P and Q. PRODI < won or I-:LK<-TIUC CURRENTS 681 t.ut, and resulting in a separation of the plus from the minus charges. Let us suppose that P is the substance which is charged positively; then the direction of the force producing this charge must be across the surface of contact from Q to P. As a result of the plus charge on P and the minus charge on Q, the potential of P is higher than that of Q ; so that if P ami Q are conductors, and if they are joined by some wire, a current would tend to flow, owing to this fact, from P through the wire to Q. This difference of potential at the surface of contact would be maintained by the molecular forces. Calling this difference of potential E, we may say that there is a " contact electro-motive force " E at the boundary. The proof of the existence of this E. M. F. across the surface of contact is afforded if P and Q are conductors, and if an electric current is forced by some source, such as a voltaic cell, across this surface, first in the direction from P to (), then in the opposite direction. It is found that in the former case the temperature of the junction rises; in the latter it falls. If the current i flows for an interval of time t from P to Qi the electricity is passing from high to low potential, and so the external electric forces do the work itE at the junction ; mid this energy appears in the form of heat effects. If, however, the current is in the opposite direction, the elec- tricity is having its potential raised at the junction, and so the work itE must be done on the electric forces at the expense of the energy of the molecules at the junction ; and there- fore its temperature falls. (Or, we may say that in the f on un- case work is done against the molecular forces which produce the electrical M-paraii«>n : \vhih- in the latter, these forces do work themselves in helping on the current.) These forces at tin- surface <•!' miitact of two substances are called "Pel- tier elect i-M-im.tive forces," having been first discovered by him. They can be measured by putting a junction in a rimeter "t water, ;m«l measuring the heat produced, the rui-rent, and the time. Direct experiments prove that they 682 ELECTROJ) Y.\A MICS vary in amount with the temperature. Thus, in the thermo- couple described above there are two such forces, at A and B\ and, if the temperatures at these points are different, these forces are unequal. But there are other similar forces in each conductor between A and B, if the temperatures of these points are different. For, consider either of these conductors, the two ends of which are at different temperatures ; if a section is taken across the wire at any intermediate point, the tempera- tures on its two sides differ slightly, and so the condition of the molecules which are in contact across this section is different on the two sides. Therefore, we might expect an electro-motive force at each point in the conductor. This was proved by Lord Kelvin — then Sir William Thomson — by the following experiment : let an electric current be forced through a wire of some definite material whose ends are kept at a higher temperature than its middle point, and let the temperatures be noted at two intermediate points, one in each half, which are such that their temperatures are 100o Oo 10Qo the same when no cur- rent flows; it is observed p J^ Q FIG. 875.— Diagram representing Thomson's that, when the Current is flowing, the tempera- ture at one of these points rises, while that at the other falls. Thus if the wire is PQR, let the current be from P to R, and the temperature of P and R be higher than that of Q\ and let A1 and Az be the two points whose temperatures are the same before the current begins. The molecular forces at Al producing the E. M. F. due to the temperature effect just described are either in the direction from Q toward P, i.e. from a cold point to a hot one, or from P toward Q. If the former is true, as the current flows from P to Q, the temperature at Al rises ; while if the latter is true, the temperature at A1 falls. Similarly, the temperature at Az either falls or rises ; but, if the temperature at Al A^ R PROi>r< y/o.v of t-:i.i-:< TR1C CURRENTS G83 . if the molecular forces are in the direction from a cold point to a hot one, the current at A1 is in a direction opposite to that of these forces, while at A^ the forces and the current are in the same direction, and so the temperature at A^ falls. These forces in a wire which is homogeneous except for differences in temperature are called "Thomson electro-motive forces." In a simple circuit made up of two w i res there are then these forces at each point of both. The electric current produced in a circuit made up of dif- ferent substances whose junctions are at different tempera- tures is due to the Thomson and Peltier electro-motive forces. These currents were discovered by Seebeck in 1821, but their explanation was not known for muny years. It is evident that, if a sensitive method is known for the detection of an electric current, a means is offered for detecting differences in temperature between two points ; for the junctions of a thermocouple may be placed at them. The sensitiveness of the instru- ment may be increased by joining in series several pairs of tli«- two conductors, as shown in the cut. If the alternate junctions are kept at one temperature, and the other junctions are kept at a different one, the current will be increased; and so a less |
difference in temperature mav l»e detected. Such an ins! ruim-nt is called a " thermopile." A cut of an actual in>truuicnt is shown. Fio. 876. — A thermopile. CHAPTER XLV MECHANISM OF THE CURRENT Electrolysis. — It is found by experiment that many liquids are conductors, while others are not. A metal in a liquid condition is a conductor, and its properties are exactly like those of the solid conductor. There are, however, certain liquid conductors such that, when a current is made to traverse them, there is an evolution of matter at the points where the current enters and leaves. The liquid must be held in some vessel and two metal rods or wires connected with some source of electric current — such as a series of cells — must dip into it. The conductor, or "elec- trode," at which the cur- rent enters the liquid is called the "anode"; that at which it leaves, the "cathode." Thus the potential of the former is higher than that of the latter; and the direction of the current in the liquid is from the anode to the cathode. The matter that is liberated at the anode and cathode may bubble off in the form of a gas, it may combine chem- ically with the metal rods themselves, or it may simply form a solid deposit on them. Liquid conductors which have this property are called "electrolytes"; and the process of conduction in them is called " electrolysis." A careful study of the nature of electrolytes has shown that in every case they are solutions, e.g. common salt or sulphuric acid in water; and 684 MECHANISM OF THE CURRENT 685 that the solutions are of the kind which exhibit an abnormal osmotic pressure, an abnormal depression of the freezing point, and an abnormal elevation of the boiling point. (See pages 2(H and -J76.) Faraday's Laws. — A careful study has been made of the character of the substances which are evolved at the anode and the cathode in different electrolytes. It is found that hydrogen and all metals are liberated at the cathode, while oxygen, chlorine, iodine, etc., are liberated at the anode. Further, the amounts of the substances evolved under dif- ferent conditions were systematically studied by Faraday. As a result of his investigation, he was able to describe all of his observations in two simple laws which bear his name. Before stating these, however, it is necessary to define a chemical term which was used much more commonly in former days than now, and yet which is convenient. Ex- p« i iments have established the fact that a molecule of any chemical compound consists of a certain number of smaller parts, called atoms, the atoms of any element being alike in all respects. Thus, a molecule of steam consists of two atoms of hydrogen and one of oxygen, so its symbol is HaO; a mole- cule of sulphuric acid may be expressed by the symbol H2SO4 ; one of copper sulphate by CuSO4 ; one of hydrochloric acid IIC1; etc. A molecule of hydrogen gas has the symbol II., : "iic of oxygen gas, Oa; etc. The kk molecular weight" iy definite compound has been already defined to be a number which is proportional to the weight of one of its molecules; and a method lias l>een described for the determi- nation of this (piantity in certain cases. Other methods arc kn«.\vn. Similarly, the •• atomic weight " of any element number proportional to the weight of one of its atoms. Thus, the molecular weight of oxygen is 82; so its atomic weight is lr, : . • It is seen from the above illustrations of the com] -.f molecules that in some cases one atom, in others two, of hydrogen are contained in a molecule. Thus, ill hydrogen gas and hydrochloric acid, one atom of hydrogen combines with an atom of hydrogen or one of chlorine respec- tively; in steam and in sulphuric acid, two atoms of hydro- gen combine with one of oxygen or with the " radical " (S()4); etc. The number of hydrogen atoms which is re- quired to form a stable molecule with the atom of a substance or with a certain "radical" (or group of atoms), is called the " valence " of that substance or of that radical. Thus, the valence of hydrogen and of chlorine is one; that of oxygen and of SO4 is two ; etc. Experiments show that if any molecule is regarded as made up of two parts, the valences of the two parts are the same. Thus, since a mole- cule of copper sulphate is CuSO4, the valence of copper is two, as is shown also by the fact that the saturated oxide of copper is CuO. (An atom may have a different valence in different compounds ; but only one of these is in general a stable mole- cule.) The ratio of the atomic weight of an element, or of the sum of the atomic weights of the atoms in a radical, to its valence is called its " chemical equivalent." We can now state Faraday's two laws : 1. The quantity or mass of a substance liberated from any electrolyte at either the anode or the cathode is directly proportional to the quantity of the current that passes. 2. The masses of different substances liberated at the anode and cathode in any electrolyte by the passage of the same quantity of current are directly proportional to their chemical equivalents. If the current flows through several electrolytes arranged in series, let ml and m^ be the masses of the substances liberated at the anode and the cathode in the first electrolyte, and cl and <?/ their chemical equivalents ; mv w2', <?2, and c2f be similar quantities for the second electrolyte, etc. Fara- day's first law states that any m varies directly as the quan- tity of current ; so, if the current is steady, and if i is its strength, m is directly proportional to the product of i by £, MXCHAX18M <>r THE CURRENT 687 tlu- interval of time taken to liberate law states that m^. //^': wa: mz' : etc. = Voltameters. - - The first law offers a convenient method for the mass m. The second : etc. tin- comparison of the strength of two different currents. An elec- trolyte is placed in series with the two currents in turn, and the (plan titles of matter liberated in < It-finite intervals of time at either anode or cathode are measured. If the strength of one current is called ir and if the mass it liber- - in an interval of time tl is mv raj is proportional to tfa. So, if the strength of the other current is called /2. and if it liberates a mass wa of the same substance in time tv 7H2 is proportional to t.,<\, : »»r ml:mz = tlil:t^r Hence /,:/, = 5j . ?b. /, t% i.il instruments have been designed for ihjs ]nirix>ae of coniparin.i,' cunvnt -lli>: tli. -van- call.-.l * voltaui.-t. : () of the commonest form- i> -ho\vn in tin- rut. It is rall«Ml a "water volta- meter": an. I in it the electrolyte is dilute Hul j.l i uric acid, and the anode and cathode are sheets of platinum inserted at the lower ends of HM- t\\ • {» long U tube. The substances li are oxygen gas at the anode, and hydrogen gas at the catlx.d.-: tlu>se collect in tl.. U|.J..-i : ,,, ,!„, ilistrum.-i.l ,,ay be measured. (Tl,.- mid. II.- ti.U-. as Fiu. 873. -A water volUm«t«r. 688 EL KCTR 01) YXA MICS in the cut, is open to the air at the top, and the quantity of liquid in it may be varied so as to make the pressure on either gas equal to that of the atmosphere, when its volume is measured. From a knowledge of its pressure, volume, and temperature, and of its density as given in tables its mass may be calculated.) In another form of instrument the electrolyte is a solution of copper sul- phate, and the anode and cathode are both sheets of copper ; the quantity of copper deposited on the cathode is deter- mined by weighing it before and after the current passes. This is called a "copper voltameter." In the standard instrument the electrolyte is a solution of silver nitrate in water, of a definite concentration; the anode is a plate of silver and the cathode which receives a deposit of silver is the platinum bowl which holds the silver nitrate; this bowl is weighed before and after the current passes, and suitable precautions are taken to prevent any mechanical currents in the liquid, for it is found that they, for purely chemical reasons, affect slightly the quantity deposited. FIG. 879. — A simple form of silver voltameter. Electro-chemical Equivalent. — By means of Faraday's second law we can calculate the relative amounts of differ- ent substances which are liberated by the same current in the same time. The quantity, i.e. the mass, of any substance which is set free at either anode or cathode as a unit C. G. S. electro-magnetic quantity of electricity passes is called the u electro-chemical equivalent" of that substance. It is a matter of experiment to determine its value for any one substance ; but, being known for this, its value for any other is given at once by the second law. Careful experiments show that the electro-chemical equivalent of any substance apparently varies with the kind of voltameter used ; but the variations are undoubtedly due to secondary reactions or causes. If the voltameter is so constructed as to avoid MECIIAyiSM OF THE CURRENT 689 these, the elect ro-eheniical equivalent of silver is found to be 0.011175 g. The chemical equivalent of silver is 107.93; so calling the electro-chemical equivalent of any other substance M and its chemical equivalent <?, we have the relation <».« ill 175: w = 107.93 :c, _ 0.011175 xc= c Thus, for hydrogen, in = 0.00010354; for copper, m = 107.93 9658* 0.003211:.".!; for zinc, m = 0.0033857 ; etc. Since m is the quantity of any substance liberated by a unit electro-magnetic quantity of electricity, the quantity liberated by 9658 such units is a number of grams equal to c, the chemical equivalent. Ions. — The explanation of electrolysis which was advanced 1 » y Faraday was that in any electrolyte there are present cer- tain charged particles, some with +, others with — charges, and that these particles are driven by the electric force in one direction or the other. Positively charged particles will move in the direction of the force, that is toward the cathode; while those negatively charged will move in the o |
pposite di reet ion, toward the anode. These charged particles Fara- day called " ions "; and those which move toward the cathode he called "cations," while those which move toward the anode he called "anions." Thus all cations are positively charged; all anions, negatively. The electric current in the electrolyte consists, then, from this point of view, of the passage in opposite directions of these two sets of charged hodies. When they reach the electrodes, they give np their charts and in some manner cause the liberation of uncharged Allies or ordinary matter. Hydrogen and all metals are liberated at the cathode; therefore we must consider a hy- drogen ion or any metallic ion as being positively charged. Similarly, we must consider an oxygen or a ehlorine ion as being negatively charged. \MI *'«. i Hvsics — 44 690 ELECTRODYNAMICS Since the electrolyte itself is not electrically charged, any volume of it must contain as much positive electricity on its cations as negative on its anions. The current consists of the passage across any cross section of the electrolyte of these ions ; if the current strength is i, and if ^ is the positive charge carried on the cations, and z'2 is the negative charge carried on the anions, i = z\ + i2. In the main body of the electrolyte the positively and negatively charged ions balance each other ; but in the immediate neighborhood of the cath- ode, positively charged ions carrying a charge ^ come up in a unit of time, and negatively charged ions carrying a charge iz leave in the same time ; so this space gains in this time cations carrying charges (i^ + i%) which are not balanced by anions ; and these give up their charges to the cathode. Similarly, at the anode, anions carrying a charge i2 come up, and cations carrying a charge i^ leave, thus causing a con- centration, as it were, of anions carrying a charge (^ -h ^'2) which are not balanced by cations. In other words, the current of strength i enters the electrolyte owing to the fact that anions carrying a charge i are liberated at the anode ; and it leaves the electrolyte owing to the fact that cations carrying a charge i are liberated at the cathode. This fact may also be expressed by saying that the same quantity of electricity — not regarding its sign — is carried on that number of ions of any substance, whose mass equals its electro-chemical equivalent. This may be described differently : When a unit quantity of electricity passes, tj + «2 = 1 ; if ml and m2 are the electro-chemical equivalents of the two sets of ions, m1 and m2 grams are liberated at the two elec- trodes in this interval of time. The liberation of the m1 grams at the cathode is due, as has been shown, in part to the bringing up of the cations carrying the charge il and in part to the withdrawal in the opposite direction of the anions carrying the charge ia ; so the effect is the same as if there were no anions, but only cations, carrying a unit charge. In Ml-:< HAM-.M OF ////,' Cl'RRENT 601 other words, a mass of cations equal to //^ carries a unit elect n '-magnetic charge ; and similarly a mass of anions equal to 7H.2 carries a unit cliarge. 'I'll us, in the case of hydrogen, the electro-chemical equiva- lent equals §3*53; and therefore this number of grams of hydrogen ions carries a unit charge, or one gram of hydrogen ions carries a charge equal to 9658. Similarly, a number of grams of ions of any substance equal to its chemical equivalent carries a charge equal to 9658. (The ratio, then. of the charge carried on a hydrogen ion to its mass equals ^g^, or, approximately, 1 x 10"4.) Faraday showed that both of his experimental laws could be explained if it were assumed that the ions of any one substance were all alike, and that the charge on any ion was proportional to the valence of the substance. For, if these assumptions are true, it is evident that the quantity of elec- tricity carried through the liquid must vary directly as the mass liberated at either electrode ; this is the first law. Further, if the same quantity is being carried by two dif- ferent sets of ions, the masses of these substances liberated, if the charges carried by all the ions are equal, are propor- tional to their atomic weights; whereas, if each ion of one set carries twice the charge carried by each ion of the other set, — that is, if the valence of the former set is twice that of the latter, — only one half as many of the ions of the former are involved in the current as of the latter, and so the ratio lie mass of the former to that of the latter is equal to that of their chemical equivalents; this is the second law. On this assumption, the charge carried on an ion whose valence is one is the smallest charge involved in electrolysis. It is called an "atom of electricity." Nature of Ions. - The question as to the nature of the ions is a most important one. As was said above, all electrolytes are solutions which show abnormal osmotic pressures, depres- ^ of the free/in^ point, etO.J and it is shown in ; 692 ELECTROD YNAM1CS on Physical Chemistry that these various abnormal phe- nomena can all be explained if it is assumed that in these solutions a certain proportion of the dissolved molecules are dissociated into simpler parts. It is natural to expect that, if a molecule is broken up into parts, they should be electric- ally charged, so that equal amounts of positive and negative electricity are produced. If this is the case, it is seen at once that the charge on any atom or radical is proportional to its valence. Thus, if a molecule of hydrochloric acid, HC1, breaks up into two parts, H and Cl, and if one is charged positively, the other will have an equal amount of negative electricity ; and the valences of hydrogen and chlorine are the same. Similarly, if a molecule of sulphuric acid, H2SO4, dissociates into three parts, H, H, and (SO4), the two hydro- gen atoms will be charged alike, and therefore the radical (SO4) will have an opposite charge equal numerically to twice that on a hydrogen atom ; and the valence of (SO4) is two. We assume, then, that when an electrolytic solution is made, a certain proportion of the dissolved molecules are dissociated into simpler parts, and that these parts are elec- trically charged, some positively, some negatively, so that the total charge is zero. (This condition of dissociation is not to be thought of as a static one, but as dynamic ; mole- cules are constantly dissociating, and others are being formed by combinations of the parts, which are moving about in the solution ; but at any temperature and concentration a certain definite proportion of the molecules are in a state of dissocia- tion.) These charged fragments of molecules form the ions when two electrodes at different potentials are lowered into the solution ; the positively charged ions move toward the cathode during their intervals of existence, before they com- bine with other ions and form electrically neutral molecules ; the negatively charged ones move toward the anode. The fact should be emphasized that the ions are produced in the act of solution, not by any action of the electric current : MECHANISM OF THE CURRENT 693 the current merely liberates the matter at the electrodes. It should also be emphasized that an ion is an electrically charged atom or radical, and is not a molecule ; and that the properties of matter as we observe them, e.g. gases, liquids, . are the properties of molecules or groups of molecules. Tli us, there is no connection between the general properties of a hydrogen ion and those of a hydrogen molecule. Again, as an ion moves through a solution, it is extremely probable that it carries with it a certain number of molecules, and that the number associated with a negative ion is not the same as that associated with a positive ion. If this is true, the effec- tive mass of an ion is much greater than its actual mass, considered merely as a fragment of a molecule. The student should consult Jones, The Modern Theory of Solution, New York, 1899, for the original memoirs of Van't Hoff, Arrhenius, and others. The question as to whether an ion is positive or negative is settled by observing whether it is a cation or an anion. Thus an ion of hydrogen or of any metal is positive ; while one of oxygen, or chlorine, etc., is negative. Again, since the same masses of any substance are liberated by the same quantity of electricity, regardless of the nature of the elec- trolyte, e.g. when hydrogen is liberated from dilute sulphuric acid, or nitric acid, or hydrochloric acid, etc., it is proved that an ion of any one substance always has the same charge in an electrolyte, no matter to what molecule it owes its in. Thus a hydrogen ion in a liquid always has a defi- nite plus charge ; etc. \\V shall now consider in detail one or t wo cases of elec- trolysis. If sulphuric acid, HaSO4, is dissolved in water, let us consider the ions as being II. II where the first two are charged with equal amounts of positive electricity, which we may call + «, and the last has a charge — 2 e. Under the action of the electrical force the hydrogen ions move in the direction of the cathode, they combine with SO4 894 ions, other molecules dissociate, etc.; but, as the current flows, hydrogen ions continuously come up to the cathode, give up their charges, combine with other hydrogen atoms to form molecules of hydrogen gas which is liberated. The (SO4) ions in a similar manner migrate toward the anode ; but, since a molecule of (SO4) radicals cannot exist under present conditions of temperature, pressure, etc., when these ions reach the anode and give up their charges, there is a reaction with the molecules of the water near the anode, which takes place according to the following formula : S04 + H20 = H2S04 + O. Consequently for each (SO4) radical an oxygen atom appears, and these oxygen atoms form molecules of oxygen gas which may bubble off at the surface of the electrolyte or may ac |
t chemically upon the anode and oxidize it. Again, let the electrolyte be a solution of copper sulphate, CuSO4, in water; and let both the electrodes be copper plates. A molecule of copper sulphate dissociates into two ions, Cu and (SO4); the former is charged positively, the latter negatively. When the copper ions reach the cathode, they form molecules and are deposited on it. When the (SO4) ions reach the anode, they react upon the copper mole- cules of the plate in such a manner that the copper goes into solution. Since metal ions are charged positively, the copper dissolves in the form of positive ions ; so the current is carried off the anode by them. These + ions serve, then, to balance the — (SO4) ions which are being continually brought up to the anode. This obviously offers a method for "copper plating" an object. Its surface must be so prepared that it is a conductor and that copper will adhere to it; and it then must be used as a cathode in an electrolytic bath of copper sulphate, the anode being a plate of copper. Similarly, in an aqueous solution of silver nitrate, AgNO3, the ions are Ag and (NO3); the former is the cation, the OF V7//-; < ri;i;i-:.\T 695 latter tin- union. If ;i plate of silver is the anode, it dis- solves, and silver is deposited on the cathode. This offers a method of silver plating. In tin; last two illustrations it is seen that the mechanism by which tin- current enters the electrolyte from the anode consists in the copper or the silver plates dissolving; this is done by the positively charged ions of copper or of silver leaving the plates and entering the liquid. In the water voltameter, where the anode and cathode are platinum plates, the case is not quite so simple. The negative ions of oxygen are formed at the anode by the reaction of SO4 upon the water molecules ; and in some manner positive charges pass from the anode to certain of these ions, first neutralizing their negative charges and then giving them positive ones ; and, after this takes place, a positive oxygen ion combines with a negative one and forms a molecule of oxygen gas, whieh bubbles off at the surface. At the cathode, the mech- anism is similar. Thus, in the case of copper sulphate, the positive copper ions reach the cathode ; under the electric force negative charges pass from this upon certain of the copper ions, making them negative ; then a negative copper 01 M hinrs with a positive one to form a copper molecule which is deposited on the cathode. Polarization. — If two platinum electrodes dip in a solution of sulphuric arid, and if a \.-ry small elect ro-inotive force is applied to these electrodes, a current will flow, but will i cease. This is owing to the fact that under the aetion of t he electrical force the positively charged hydrogen ions collect at thr rathodr, and t hr negatively charged anioiis at the anode : |Q that, if theiv is not sullicieiit force to make the ssary charges pass from the electrodes to the ions and then to form the molecules, these charged particles will lower the potential at the anode and raise it at the cathode until the. 1 1 iv electrical force in t IK; electrolyte, and the current stops. As t he applied elect n • mot ive force 696 ELECTRODYNAMICS is gradually increased, a value is reached which will cause the evolution of the gases at the electrodes ; and the current will now continue to flow. The same description applies in general to any case of electrolysis ; a definite E. M. F. must be applied before electrolysis begins; but its value is dif- ferent for different electrolytes. This may be calculated, however, from a knowledge of the heats of combination of the compounds which are separated by the electrolysis and of their electro-chemical equivalents. Thus, experiments prove that, when 18 g. of water are formed by the combination of 2 g. of hydrogen and 16 g. of oxygen, 68,800 calories of heat energy are evolved; and, therefore, when 18 g. of water are broken up into 2 g. of hydrogen and 16 g. of oxygen, an amount of work equal to the mechanical equivalent of 68,800 calories must be done, i.e. 68,800 x 4.2 x 107 ergs. As a quantity of currents equal to e is passed through the electrolyte consisting of H2SO4, the quantity of hydrogen evolved is me, where m is the electro-chemical equivalent of hydrogen ; and an "equivalent" amount of oxygen is liberated at the anode. Since it requires 68,800 x 4.2 x 107 ergs to liberate 2 g. of hydrogen, that required to liberate me g. is me 68>80° * 4>2 X 1QT. If E is the E. M. F. applied to the electrodes, which just causes the electrolysis, the work required to pass a quantity of electricity e between the electrodes is Ee. Therefore, since this work is spent in liberating the hydrogen and oxygen, — neglecting the work done in heating the electrolyte, and assuming that there is no other source of energy, — we have the equation „ me 68,800 x 4.2 x 107 — -' or E = m 68,800 x 2.1 x 107. For hydrogen, m = 1.035 x 10~4, and therefore E = 688 x 1.036 x 2.1 x 10* = 1.5 x 108 = 1.5 volts. The E. M. F. of a Daniell cell is approxi- mately 1.1 volts; so at least two Daniell cells are required to decompose water. Calculation of the E. M. F. of a Primary Cell. — We may consider the mechanism of a voltaic or of a Daniell cell from this standpoint of ions. In the former, the cause of the current is the solution of the zinc in the acid, that is, the MECHANISM OF THE CURRENT 697 parsing off of positive zinc ions into the liquid. The chem- ical action is the solution of the zinc in the acid and the evolution of hydrogen at the copper pole ; and experiments have proved that when 65.4 g. of zinc are dissolved in dilute sulphuric acid, 38,066 calories are evolved. The electro- chemical equivalent of zinc is 0.00338. So when a quantity of current e passes off the zinc rod into the acid, the mass of zinc dissolved is 0.00338 e ; and the energy liberated is 38.066 x2x 10' If E is the difference of potential between the copper and the zinc electrodes, it follows that ~ ,-, _ 38,066 x 4.2 x 107 x 0.00338 or E = 38,066x4.2x107x0.00338 = ^ x 10- = 0.83 volt* 65.4 In this calculation we neglect the loss of energy of heating tin- liquid, and we assume that the only source of energy is that furnished by the solution of the zinc. In a similar manner, when a current is produced by a Daniell cell, /inc dissolves at the zinc plate and copper is deposited at the copper electrode. When 63.6 g. of copper dissolve in sulphuric acid, 12,500 calories are evolved ; and the electro-chemical equivalent of copper is 0.00329. So the E. M. F. of the cell is found by calculation to be 1.1 volts, making the same assumptions as before. Storage Cell. — In certain cases of electrolysis the anode and eith'ide are so modified l>y the passage of the current, and the < >nsequent liberation of matter at them, that they be used afterward t«. form a coll for the production of a current. Therefore, if the battery of cells or the dynamo which was producing the current through the electrolyte is removed, and if the electrodes are joined by a wire, a cur- rent will flow through it. This action does not continue 698 EL ECTI! 0 1) YNA MICS indefinitely, for in producing a current, those modifications which were the result of the electrolysis are reversed, and the electrodes return to an inactive condition. If the bat- tery of cells or the dynamo is again used to send a cur- rent through the electrolyte, the process may be repeated. Such a combination of elec- trodes and electrolyte is called a "storage cell" or a "secondary cell"; and when it is in a condition to produce a current itself, it is said to be " charged " — but it is evident that this expression has no connec- tion with what has been called a " charge " in pre- vious chapters. FIG. 880. -The ordinary form of storage cell. The commonest form Qf storage cell has dilute sulphuric acid as the electrolyte, and as the anode and cathode two lead grids whose interstices are filled with a paste of lead sulphate. When such a cell is charged, it has an E. M. F. at first of about 2.1 volts; but this falls to about 1.8 volts as the current flows. Conduction of Electricity through Gases. — In speaking in a previous chapter of the passage of sparks through air or any gas, occasion was taken to state that the discharge con- sists in the disruption of molecules into simpler parts, and that the path of a spark is an excellent electric conductor. We are thus led to believe that a gas becomes a conductor owing to the presence in it of small parts of molecules, which we may again call ions, although there is no reason for believing that these ions are the same as those in evi- dence in the electrolysis of a liquid. The charged particles v/.s.u OF nn: < t HHENT 699 may In- the same, hut the in -•>< iciat ed with thiMii are diileivMi ( >ee page 693). All the known facts in regard to condiu -lion through a gas are in support of this idea of tin- ionic nature of the proce The method of determining whether a gas is a conductor or not is, of course, to immerse in it two electrodes, at a small distance apart, and to observe whether a current flows when a difference of potential is produced between these plates by some cell or combination of cells. It is found that a pure dry gas is an extremely poor conductor ; but it may be made conducting by many means. A few will be mentioned. By passing a spark through any portion of the gas, all other neighboring portions have their conductivity increased. In many cases of complex gases, a sufficient rise in temperature makes them conducting. If the ultra-violet waves from a source of light pass through a gas, it becomes a conductor. There are many solids, e.g. alkaline earths and metals, which when illuminated with ultra-violet light make the gas near them conducting. This is called " photo- electric" action. Further, if metals (or carbon) are made very hot, they make the surrounding gas conducting. If the cathode ra |
ys <»r the X-rays (see below) traverse a gas, it i> made cm id noting. Certain bodies, known as "radio- active " substances, have the power of emitting radiations which make the i,ras through which they pass a conductor; these will be discussed more fully below. In all these cases the tid tn l>e •• iolii/ed." Spark and Arc Discharge. — If two electrodes immersed in a gas at not t ; a distance apart are raised to a Milli- cient difference <>f potential, a spark will pass between them, i. The character of the dUchar^e and the utial difference required depend upon the nature and the condition <»f the ^;,s: its pressure, its temperature, its purity, etc. If the two electrode^ are lu-oiiLrlit sulVicieiit ly Close together and ti iidiic.tOP, 700 ELECTRODYNAMICS so that a large current passes, the discharge is called an "arc," as is illustrated by the ordinary arc lights in the streets. In both the ordinary spark and the arc the elec- trodes are gradually vaporized, and their vapors, as well as the gas itself, are rendered luminous. The phenomena of the discharge are so varied that some special treatise on the subject should be consulted. The best is J. J. Thomson, Conduction of Electricity through G-ases, London, 1903. We shall describe here only one or two cases of special interest. Vacuum Tube Discharge. — If the gas is inclosed in a glass bulb into which two metallic electrodes enter, and if the pressure is gradually lowered by means of an air pump, the FIG. 381. —Vacuum discharge tube. character of the discharge changes in a most marked man- ner. When the pressure is about that of 1 mm. of mer- cury, it is observed that at the cathode there is a velvety light covering it entirely or in part, which is separated by a dark space from a luminous region, and that this is separated by a second dark space from a luminous striated column extending to the anode, the end of which — if it is a wire — has a bright spot of light. The first dark space near the cathode is called the " Crookes dark space " ; the second one, the " Faraday dark space " ; the region separating them, the " negative glow "; and the striated portion near the anode, the "positive column." As the bulb is more and more exhausted, the Faraday dark space extends farther down '01 or I III-: ' URRENT toward the anode, and another phenomenon becomes most prominent. There is a radiation of something from the cathode, proceeding in straight lines perpendicular to its surface, quite independ- ent of the position of the anode. This radia- tion produces a faint luminescence of the traces of gas left in the bulb ; and so the path of the rays through the bulb may be seen. They are called the "cathode rays." Where they strike the walls of the hull), it is made lumin- ous and its temperature a. 6. Fio. 882. — Two forms of vacuum tube discharge: a, moderate vacuum ; b, high vacuum. rises ; the color of the light which is thus produced depends upon the material of the bulb, but ordinary glass emits a greenish yellow light. (If certain other solids are intro- duced in the bulb in the path of the rays, e.g. coral, they emit characteristic colors.) The fact that the rays proceed in straight lines is proved by introducing in the tube solid bodies which are opaque to the rays; for it is observed that they cast sharp shadows on the walls of the tube. The radiation passes directly through metallic films, if sufficiently thin. The path of the rays in the tube is deflected by a strong electric field, provided the vacuum is p< •! feet enough ; and the deflection is in such a direction as to l.-ad one to believe that the radiation OOnifotl of n. -Datively _M-d particles. In fact, all experiments lead to this con- •ItiiT.l t.y r:ith'»lr rny*. 702 ELECTRODYNAMICS elusion. If the rays are made to enter a hollow cylinder which is connected to an electrometer, it is seen that the cylinder is receiving a negative charge ; and the fact that, when the rays strike a solid its temperature is raised, is ex- plained by assuming that the rays consist of material parti- cles. Again, as will be shown in a few pages, a charged particle in rapid motion has the same action on a magnet as does an electric current ; and, since a wire carrying a current may be made to move under the action of a magnet (which is simply the reverse of the fact that a current can move a magnet), so a particle in rapid motion may have its direction altered if it is charged ; and experiments show that the cath- ode rays may be deflected by a magnet in exactly the man- ner which one would expect if they were negatively charged particles in rapid motion. The velocity of these rays may be measured in many ways ; and it is found to depend upon several conditions : difference of potential between the elec- trodes, pressure of the gas, etc. ; its value is not far from one tenth of the velocity of light. The masses of these particles and their electrical charges may also be measured with a fair degree of accuracy ; and it is believed that the charge of any particle is the same as that on a hydrogen ion in ordinary electrolysis, while its mass is approximately one thousandth of that of a hydrogen atom. So far as experiments can prove, these particles which constitute the cathode rays are the same no matter what gas is put in the tube. If the cathode is in the form of a metal plate with many small openings in it, it is observed that in addition to the cathode rays which are emitted from one side there are rays proceeding in the opposite direction apparently through the holes in the cathode. These are called " canal rays," and have been proved to consist of positively charged particles, moving much more slowly than the cathode rays. Their charges are probably the same as those of the latter rays ; but their masses are comparable with those of an atom. MECHANISM oF THK (Tltl:i-:\T 703 Tin- gas which is traversed by cither the cathmle or the canal rays is ionized; that is, becomes a conductor. This was proved for the former rays by Lenard, who constructed a glass bulb in such a manner that a portion of the wall which was struck by the rays had an opening in it which \\as covered by a thin layer of aluminium. Surrounding this bulb was another which could be exhausted if desired. So, as the cathode rays passed through the aluminium win- dow and entered the outer bulb, their action could be studied. Pto. 884.-AnX-«ytube. The aluminium itself emits cathode r,(\s, as a result of being struck by the interior cathode rays, in addition to t: mittiiiLT some ; and the total radiation from the \vindo\\ is called "the Lenard r. This includes also* t\ pe of radi- ation cut irel\ distinct from cathode rays. first sho\\ n d that whenever cathode rays strike lid BQrfaoe, • radiation is produced which lias most in- -tiir_: properties, and to which he gavr the name X-rays. Thesr raya pro eed in -n-ai^ht lines from their sniirce; they arc not reflected, ivt'ra. • ij thej arc not 704 ELECTRODYNAMICS affected by an electric or magnetic field ; they cannot be polarized ; they affect a photographic plate ; they cause many bodies to become luminous ; they pass with ease through many bodies opaque to light, e.g. wood, aluminium, the human flesh, etc. ; they are absorbed by other bodies much more than is light, e.g. glass ; they ionize a gas. It is believed that they are transverse pulses in the ether, in dis- tinction to being trains of waves ; and all their properties may be explained on this assumption. If we consider what we would expect to happen when a moving charge is suddenly brought to rest, as the cathode rays are by the solid which they strike, we can understand how these pulses are pro- duced, and how irregular they are. Rontgen's original papers and Stokes' memoirs giving an explanation of the X-rays are reprinted in Barker, Rontgen Rays, Scientific Memoir Series, New York, 1899. Radioactive Bodies. — Certain substances of high atomic weight, namely uranium, thorium, radium, and one or two others, possess, in common with their compounds, such as thorium nitrate, etc., the most remarkable property of emit- ting spontaneously a radiation which can ionize a gas ; they are said to be "radioactive." This radiation is complex, consisting of what are called " a rays," which are positively charged particles analogous to canal rays, and "/3 rays," which are negatively charged particles analogous to cathode rays. Radium also emits rays which are like X-rays in many respects; they are called "7 rays." This radiation is accompanied by changes inside the mole- cules of the substances which emit it ; and in some cases the products of the molecular changes are gaseous. Thus, if thorium nitrate is dissolved in water and ammonia is added, a precipitate is formed, which may be separated by filtration. The precipitate is not radioactive at first but gradually becomes so, in fact returning to the same condition as the original thorium nitrate. The filtrate, on the other hand, is 3/AY7M-V/.S.V <>r mi-: CUKRKM 705 mo>t radioactive, but loses this property in time; as it does this, it L,rives off a ^a>emis emanation, which is radioactive. This emanation is at lirst uncharged, but by losing its nega- tive charges it becomes positively charged and may be attracted to any negatively charged body. It undergoes further changes; and during each, radiations are emitted. The emanation of radium finally decomposes into helium gas. During these processes, further, heat energy is evolved, and the temperature is raised several degrees Centigrade. Electrons. — When a gas is ionized, either by the action of X-rays or by the rapid motion of minute positively or nega- tively charged particles through it, all the negative ions are found to be alike in all respects and to be the same as the eathode rays. It is therefore believed that, as such charged particles or as pulses pass through a gas, they break off from its atoms these negative ions. A theory of the constitution of an atom has |
been based on this idea. An atom is thought to contain within it a great many minute negatively charged particles which are making rapid revolutions — not unlike the constitution of the solar system, the other portions of the atom making up the positive charge. Then ioni/ation would consist in causing one or more of these negative par- ticles to leave the atom. These particles while inside the at oiii — and when outside also, provided they have no other mat. -rial particles elin^in^ to them — are called "electrons." Their vibrations inside the atom irive rise to waves in the ether. So tar as is known a moving piece of uncharged matter does not affect the ether; but, it' charged. it does; and. if it> motion has a* • //. waves are produced. It is known from theoretical considerations that when a charged particle U in motion, its kinetic energy is greater than it would he if it were not charged; and, therefore, an electric charge in motion by itself — quite apart from matter would have kinetic energy; that is, would have : The (jiiestion thm arisi .1 not the inertia of matter really AMKft'8 PI1T8IC8 — 46 706 ELECTRODYNAMICS due to the motion of electric charges in its minute parts? In other words, is not a moving charge the fundamental fact in nature ? This question is fully discussed, and a most in- teresting description of the general properties of electrons is given, in a series of papers in the London Electrician during 1902-1903, by Sir Oliver Lodge. Conduction in a Solid. — It has been proved in what has gone before that conduction in an electrolyte and in a gas consists in the actual motion of charged particles, called ions. In the case of a solid conductor the question as to the mechanism of the conduction of a current is not so simple, owing primarily to the fact that the particles of a solid have so little freedom of motion and can only vibrate. But there is every reason for believing that in a solid also the process of conduction is by means of ions. The existence of free electrons moving about inside the solid conductor from atom to atom may be proved by many experiments ; and the evidence in favor of this explanation of conduction is accumulating continually. Convection Currents. — It was Faraday who first conceived the idea that the essential feature of an electric current was the rapid motion of an electric charge ; but the first to prove by direct experiment that such a charge in motion had the same magnetic action as an ordinary current produced by a voltaic cell was the late Professor Rowland. He charged a circular metallic disc and caused it to rotate rapidly on an axle perpendicular to its faces; he observed that when a magnetic needle was brought near the disc, it was deflected exactly as if electric currents were flowing in circles in the disc. He was able to prove that, within the range of veloc- ities used, a charge e moving with a velocity v is equivalent to a current whose strength is ev. It is probable that this is true, even if v is very great, much greater than it is possible to attain by any mechanical means. A current due to a moving charge is called a "convection current." CHAPTER XLVI MAGNETIC ACTION OF A CURRENT General Description. — In a previous chapter a general description of the magnetic field due to an electric current was given ; and it was seen that a conductor carrying a cur- rent is surrounded by a field of magnetic force, such that the lines of force form closed curves around the current. The relation between the direction of the current and that of the lines of force is given by the right-handed-screw law. The magnetic field, then, due to a circuit carrying a current is the same as if a great many minute magnets, of the same length and strength, are taken and placed side by side so that their north poles are all turned one way and their south poles in the opposite direetion, thus forming what is called a "magnetic shell." having the same contour as is made by the conductor earn inur the current. For. lines of force would proceed out from the north poles of the slu-11 and all return tn the south poles. We < an thus speak <>f the M north face " (.!' a eiivuit irivnt and of its "south face." Again, if a wire — or other eoiid net «.r — is wound in the form of a helix, and if a current is 708 ELEL'TRODYSAMICS passed through it, thus forming a " solenoid," the magnetic field enters one end and emerges from the other exactly as if it were a bar magnet. Electro-magnets. — If, then, a bar of iron or of any mag- netic substance is inserted in a solenoid, it is magnetized because each little molecular magnet turns and places itself along a line of force. A bar of iron wrapped with a helix of insulated wire is called an "electro-magnet." The bar is FIG. 886. — A powerful electro-magnet arranged to show magnetic or diamagnetic action on suspended sphere. usually made in the shape of a horseshoe, or of the general form shown in the cut. These are used in a countless number of instruments, such as call bells, telegraph instru- ments, etc. The first electro-magnet was made by Sturgeon (1825) ; but the idea of wrapping several layers of wire, like thread on a spool, is due to Joseph Henry. By this means the intensity of a magnet may be greatly increased. Electro-magnetic Forces. — Since a solenoid is equivalent to a bar magnet, it will, if suspended free to move, turn and MA'.M-TK X OF A CURRENT plarr itM-lt in the magnetic meridian; and, if another solen- oid or a liar magnet is brought near it, it will be seen that 7'".' like i H.Irs irjH'l and unlike ones attract. These motions, pro- duced owing to the magnetic field of a current, are said to be diit to •• electro-magnetic forces" ; and we can describe them in another manner which is perhaps more definite. Since the south end of a solenoid is the one which tubes of force enter, and since it is the one which is attracted by the north pole of a magnet or a solenoid from which tubes of force proceed, it is evident that by the act of approaching one another, more tubes of force enter the south end of the solenoid than before. Other cases of attraction and repulsion may be con- sidered in a similar manner ; and it is easily seen that they may all be described by saying that motions of conductors • any ing currents take place in such a manner that as many tubes of force as possible emerge through their north ends or faces. This is equivalent to saying that the electro-magnetic forces are in such a direction as to produce these motions. If a loosely wound spiral spring is sus- pended in a vertical position with its lower end just dipping in a cup of mercury, and an electric current is passed through it, the separate turns of the coil will attract each other, because by coming closer together more tubes of force pass through them, instead of escaping out from th»- sides; as th«- >]>ring thus contracts, the electrical connection at the bottom i- lu-okt-n. and so the force of attraction vanish*-* : tin- >j>rinij tli* ii drops, connection is again made, the ..t Hows, etc. The action is increased «'rting an iron rod inside the coil. Plo. 8W._Kli0tro.IIIIIfBrtte tllf-e. UOD of parallel current*. These phenomena of electro-magnetic forces were discov- 1 by Ampere, and lie invented many most beautiful experi- 710 ELECTRODYNAMICS 999.9 - 9999 merits to illustrate them. They may be found described in many text-books. He also proposed a theory of magnetism based upon his observations. He advanced the hypothesis that in each molecule of a magnetic substance there is an electric current, flowing in a fixed chan- nel; and therefore if a bar of sucli a substance is brought into a magnetic field, — due either to a magnet or to a solenoid, — the molecules will all turn so as to include as many tubes of magnetic force as possible. The bar will be saturated magnet. Ampere's theory when the molecules have so arranged themselves that their currents are parallel to each other and to the ends of the bar ; and under these conditions the lines of force due to these currents will emerge at one end and return into the other, exactly like a solenoid. If two parallel wires or rods are placed in a horizontal plane and are joined by a fixed wire BO, containing a cell, and by a movable wire PP', which can roll on the wires, FIG. 888. — South pole of * ELECTRO MAGNETIC FORCE Fio. 890. — Relation between directions of current, magnetic Fio. 889. — Electro-magnetic force : magnetic field is upward fleltl> and electro-magnetic force, through circuit. Each is perpendicular to the other two. this movable wire will be set in motion if there is a mag- netic field through the space between the parallel wires, because by so moving a change is made in the number of tubes of force which pass through the circuit (PP1 OB). If MA<.\I-:TI<- ACT1OB or .1 i-rniiKNT 711 the current is in the direction shown in the cut, the upper face is tin- north one; and, if the magnetic field of force is in tin- direction shown, tin- cross wire PP' will move trd the right, so that the circuit incloses more tubes coming out from its north face. This law of force is given by the diagram which describes the connection between t la- directions of the magnetic force, the electric current, and tlu- electro-magnetic force. Magnetic Force Due to a Current. — Since the magnetic lines of force due to a current form closed curves around it, it would require a certain amount of work to carry a unit north pole around such a line of force in a direction opposite to that of the line itself; and it is evident that this work must vary directly as the strength of the current. It may be shown by experimental methods that the amount of work required to carry a unit pole around any closed curve encir- cling the current is the same for all paths. Thus, calling this work IT. we may write W= d, where c is a factor of proportionality depending on the system of units adopted for the measurement of the current. The C. G. S. e |
lectro- magnetic system is based on the definition of such a unit current as will make this factor equal to 4 TT. (The reason for this choice depends upon the connection between a cur- ivut and a magnetic shell, and need not be explained here. It may 1..- found in any advanced text-honk.) If the unit pole is carried around the circuit /// times, the work done is evidently ITT////. One consequent- of this definition of a unit current is that the intensity of the magnetic field at the centre of a circular coil of radius a, of n turns, carrying a current of strength /'. — . (See page 678.) If the condun m of \\ helix, the magnetic fOTOe in-ide il. . !i it. may he deduced at once. Tin- work done in carrying a unit pole around a closed path tf/W \\n in the cut, where ab is parallel 712 ELECTRODYNAMICS to the axis of the helix and inside it, be and da are perpen- dicular to this axis and cd is outside the helix, equals the product of the intensity of the magnetic field inside and the length of the path ab, be- cause the lines of force are along the axis, and so no work is required to tra- verse the portions da and be, and the force outside is so small that it may be neglected if the helix is long. So, calling the inten- sity of the field inside R and the distance ab x, the work is Rx. But, if there are N turns of the helix per unit length, there are xN turns in the length x ; and since each of these carries the current t, the work done in threading them by a unit pole is, in accordance with the above definition of a unit current, 4 irN x i. So, Rx = 4 TrN x i, or, R = 4 irNi ; a most important formula. . — Magnetic force inside a long solenoid. If there is a rod of iron, of permeability ^, filling the solenoid, the number of tubes of induction per unit area passing through the iron is proportional to pR. (See page 615.) Since p for iron is large, this means that the number of tubes of induction passing through the solenoid is greatly increased by inserting in it a rod of iron. These additional tubes are due to the magnetization of the iron by the current. Since each of these tubes of induction passes N times through a circuit carrying the current i in a unit distance, it is evident that the magnetic action of the solenoid is propor- tional to that of a single turn of wire carrying a current equal to N*i. Energy of a Current. — The fact that, when a current is flowing in a conductor, forces may be experienced in the surrounding medium, proves that there is a certain amount of energy in this medium due to the current. This energy is MAGNETIC ACTION OF A iTRRENT 713 not in the form of a strain — no sparks are observed in the medium, etc. So it is natural to think of the energy as being kinetic- in its nature; and this idea will become more evident in a later chapter. What will be shown is this : as a current is first started, e.g. by joining the two poles of a primary cell, the i urrent does not rise to its full strength instantly, for part of the energy furnished by the source of the current is spent in producing those motions in the surrounding medium which constitute the magnetic field, and it is not until these motions are established that all the energy of the source of the current goes into forcing the current through the con- ductor, and so heating it. Similarly, if the source of the current is suddenly removed, the current does not instantly cease, because the energy in the medium disappears gradu- ally, being spent in maintaining the current for a short time. Compare the case of a railway train starting from rest; it does not In its full speed instantly because the energy furnished by the loco- motive is used in producing kinetic energy ; but, after the desired speed in reached, the only work done is against the friction of the wheels, the resistance of the air, etc. Similarly, when the train is to be stopped, the power furnished l»y the locomotive is shut off, but motion continues until the kinetic energy of the train is used up in ov« -n •oiuin^ friction. The analogy with an electric current is not particularly good, because in the latter caae the kinetic energy is not in the conductor, but in tin- medium, while the work done in producing the heat is spent in the conductor. From what was said above in regard to solenoids, it is evident that th<5 energy of their currents is proportional to vhere N is the number of turns per unit length. There are many electrical instruments and machines whose artiMiis depend ap<m electro-magnetic forces: and a few will The Electric Bell. — This consists «,f a gong Q- ; an elect n> magnet / .iluatiii^ clapper of spring brass which has a small knob If with \\ hich to st like the gong, and a strip of iron ihe polo of the electro-magnet; a stiff piece 714 ELECTRODYNAMICS of brass A> which is attached to the clapper, and presses against a lixed metal stud O. One end of the wire around the electro-magnet is joined to the clapper, while the other FIG. 392. — Electric bell and push button. end is connected to a binding post at D. When the bell is in use, D and 0 are joined to some primary cell B ; a contact key P being introduced in the circuit. When the key is pressed so as to make contact, the current flows through the Fi<;. :!!»:;. Tin; relay. electro-magnet, attracting the clapper and thus ringing the bell ; but as the clapper moves toward the magnet, contact is broken at (7, and the current ceases ; then, owing to its MAGNETIC A(T1<>.\ oF .! rri;i:i-:NT 715 elasticity, the clapper vibrates back, making contact at (7, and the current again flows; etc. Relay. — A relay consists of an electro-magnet in front of which is an iron plate called the armature, carried by a metal rod pivoted at its base. The wire around the electro-magnet may be connected to a primary cell or a battery of cells at a distance, with a key in circuit. So, when the key is pressed, a current will flow around the magnet. Even if the current is extremely feeble, the armature will be attracted ; and by means of suitable contact points a sec- ond cell may be closed through any circuit; and thus any elect n> magnetic effect may be produced, . Vi:i|.l. k.'\ .'111.1 -oini.liT. such us rin'_miL,r a bell. etc. NVheii the key is broken, the current ceases, and the armature is drawn Lack from the electro-raagiK ' i»iral spring attached to its rear. It* a second eh-- \\itlianannatuivisintroduced in the circuit <>f the second » -ell, the sound made by the 716 ELECTRODYNAMICS armature as it clicks against the electro-magnet may be clearly heard. This is the principle of the ordinary telegraph system, different letters being distinguished by different combinations of " dots and dashes " ; that is, short and long intervals of time between consecutive clicks of the " sounder." Duplex Telegraphy. — In the " Duplex " system of telegra- phy it is possible to receive and send messages from a station at the same time. The instrument consists essentially of a receiving instrument E, such as an electro-magnet or a galvanoscope, around which are wound two coils of wire in opposite directions ; so, if equal currents are passed through both coils, no effect is pro- duced, while, if a current passes through one coil alone, there is an effect. One of these coils is con- nected through the "line wire " to the distant sta- tion, while the other is joined through several coils of wire of adjust- able lengths, R, to the earth. The arrangement of the cell and the key K is as shown in the cut. The coils R are so adjusted that when the key K is pressed and makes connection with the cell jB, equal currents pass around the receiving instrument E, and there is no effect ; but a current passes over the line wire to the distant station. When a current is received over the line wire, it passes to the earth, entirely regardless of the position of the key K, thus affecting the instrument. (If the key K is pressed down when a current is being received from the distant station, we may regard the current in the line wire as FIG. 895. — Diagram of one form of duplex telegraph instrument. if.-i<..Y /•:/•/<• ACTION OF A rr/,'/;/v\vr 717 neutralized. Init part of the current from U passes around E through R to the earth. Telephone. — The ordinary Bell telephone receiver consists steel magnet M, around one end of which is wound a coil of wire, and in front of which is a thin iron plate railed the dia- phragm. This is at- tra.-t ed toward the magnet, but is kept from motion as a whole by the frame. If a current is passed through the coil, it will either strengthen or weaken the steel magnet, drpniding upon its direction; and so the diaphragm is either attracted or repelled. Thus, if the current fluctuates, the diaphragm vibrates. , ,, K i... 396.— The Bell telephone. Microphone or Transmitter. — This consists of a thin metal diaphragm whose edges are held but whieh can vibrate like a drumhead ; against its centre presses a carbon " button," which is held firmly by suitable metal supports. A cell is connected to the carbon button and to its supports. There is poor electrical contact between these; and it varies in its conducting power as the pressure of the diaphragm against the carbon button varies. If the pressure is increased, more current flows; if it is decreased, less current. So, if the diaphragm vi- brates, the current fluctuates in st rength. It is thus evidrnt that, if a telephone I1]1 is included in the mi« roplion. FIO. 8»7. — nimrnun of BUk« vibrations of tin- mi.TMpli. -m- diaphragm produce corresponding \ M. rations of the telephone diaphragm. In this manner. the MT produced b\ the human voice will cause \ MS of 718 EL ECTUOD YXAMICS the telephone diaphragm, which will in turn send out sound waves ; and thus sounds may be said to be transmitted. D'Arsonval Galvanometer. — This consists of a permanent steel horseshoe magnet, between whose poles a coil 0 is supported by means of a vertical wire. The wire in this coil is continu- ous from A to B, two |
fixed binding screws. When no current is passing, the coil is held so that its plane is parallel to the line joining the two poles ; but if a current is transmitted through the coil by means of A and B, it will turn so as to include as many of the tubes of force of the magnet as possible. It will be brought to rest by the torsion of the wire ; and so its deflections meas- ure the current strength. PIG. 898. — D'Arsonval galvanometer. Practical Instruments. — In most practical work, such as measuring the electric currents of telegraph systems, lighting systems, dynamos, etc., instruments are used which are porta- ble. They are sometimes called "practical instruments." The principle used in them all is to have a permanent steel horseshoe magnet, between whose poles is supported on pivots a coil of wire through which the current to be measured is passed. This coil turns so as to include as many tubes of force as possible ; but, as it turns, it winds up a flat coiled spring, and so is finally brought to rest. The angle of deflection is measured by a pointer. ACTION OF -l CUBE 719 Radio- micrometer. — This is an instrument invented l»y Pn»fexx,,r P,,.\«, I'm- the detect inn and measure- ment of radiation in the ether. It consists of a thermocouple and a loop of wire, used according bhe principle of the coil in tlu« D'Arsonval galvanometer. A loop of copper wire ends in tine strips of bismuth and antimony, A and B^ which are soldered together. This loop is then suspended by a fibre between the poles of a permanent magnet, with its plane parallel to the line joining them. The junction of the two metals is blackened, and is exposed to the radiation ; as it absorbs energy, its temperature rises, a current flows in the loop; this is then deflected so as to FIG. 899. — Weston's am: t'X). — Boys' ra/llo-nilrroinft.-!-: A and include as many t uhesof ;' OS possible, and it finally to TCSt wllCtt this R are two different raetalu fonnlnR a th-nn , elect ro-ma^llct ic force is l»al- '"' 111 PI aneed by the toi-Mon of the fibre. This deflection evidently measures the intensit\ of the radiation al»s..rlied |,v the l.lackeiieil junction. Electric Motor. This oonaiiita of aD elect: poles are turned to fa< c each other, and of an "armat which is a shuttle-shaped piece of iron or an iron rin which are wound coils «,f irOO. 'I P« eninii with metal strips, insulated from . a. h other, mi the 720 ELECTItOD YNA MICS the armature ; and on these " commutator bars," as they are called, rest two metallic rods or "brushes," which are joined to some source of a current such as a dynamo. It is easily seen how it is possible to make the connection of the coils with the bars in such a manner that the current passes through one coil of the armature, which is in such a position that it does not include as many of the tubes of force due to the FIG. 4oi. -An electric motor. electro-magnet as it would if the armature turned on its axis ; therefore, the armature will turn ; and, as it does so, another coil on the armature comes into the position occupied by the previous coil ; then the current will pass through this ; etc. In this manner continuous rotation of the arma- ture may be secured ; and by means of its shaft, work of various kinds may be done ; e.g. street cars may be moved. When the motor is doing work, the energy is furnished by the source of the current. If the electro-motive force of this source is E and if the current is i, then the energy it furnishes in a time t is Eit. Part of this goes into Fl°- 401 «— Method of windin* coil8 heating the conductor, and the rest into the motor. This last may be written eit, where e is called the "back E. M. F." of the motor. So the amount of energy that goes into heat effects is (E — e)it. of wire on a " drum " armature. CHAPTER XLVII LAWS OF STEADY CURRENTS Steady Current. — In the foregoing chapters the various properties of electric currents, viz., heating, magnetic, < trolytic, etc., have been discussed and illustrated ; and several methods for the production of currents have been described. A current is called u steady " if these properties remain con- stant, e.<j. if a constant deflection of a galvanometer needle is produced, if heat energy is developed at a constant rate, if matter is liberated in an electrolyte at a constant rate, etc. ; and experiments prove that, if a current satisfies one of these conditions, it satisfies all. A " variable " current is one that is not steady. In order to produce a steady current one may use a source of constant K. M. F., sin -h as a I Smell's cell, or a thermocouple whose junctions are maintained at constant temperatures, or a dynamo — as will be described in the next chapter. Uniformity of Current. — One of the most important properties of a steady current is that its strength is uni- form throughout the circuit ; that is. if the circuit includes conductor* of diflVrent material, of different sizes, etc., the strength of the current is the same in them all. This may !>•• shown by proving that the magnetic or the hea1 • ii of the current is the same for all portions of the lit. A L: i in, if the current were not the same at all points, there would be accumulations of charges at certain points; and, as th. 'sr innv.e>c<l, they could be detected; but such is not the case. mlailv. if at an\ point of the circuit, it branches SO as i 721 722 EL ECTItOD YNAM1CS to form two or more parallel conductors, the strength of the current in the single conductor must equal the sum of the strengths of the currents in the branches. This may be expressed in a formula, K,W, x. „„ is the current in any FIG. 402. — A divided circuit. J conductor at a branch point, the direction of the current being called positive if it is toward the point, the summation of all the currents at that point is zero, or, in symbols, 24 = 0. • • °- Ohm's Law. — We have seen that a current flows between two points of a conductor only if there is a difference in potential between them ; so that we may in a way regard the E. M. F. as the cause of the current, and it is not un- natural, judging from analogy with the flow of heat in a bar owing to difference in temperature between two points, to advance the hypothesis that the current strength in a con- ductor varies directly as the E. M. F. between two points, provided the current is steady. (We are considering the case where there is no cell or other source of E. M. F. introduced in the conductor between the two points.) That is, if A and B are any two points in a circuit in which is flow- ing a steady current of strength i, and if E is the E. M. F. between FIG. 408. — Diagram to illustrate A i T» .LI i A i • • 1 1 Ohm's law. A and B, the hypothesis is that E = Ri, where R is a constant depending upon the nature of the conductor between A and B, but not upon the values of E or i. This hypothesis has been found to be true, so far as experiments can decide ; E has been varied by intro- ducing more cells or a dynamo, and the resulting current or STXADT m;i;i-:yT8 723 has been measured. It is called Ohm's law, having been proposed by Georg Ohm in the year 1826. Resistance. — It is evident from the formula that if R is lar^e. / i> >mall, provided E remains constant; while, if // small, i is u'reat. For tliis reason R is called the "resistant. " of the conductor between A and B. This law can also be E 1 H H written i = —- ; and, if for — the symbol 0 is substituted, the formula becomes i = CE. For obvious reasons O is called the u conductanee " of the conductor between A and B. If the conductor between A and B is a uniform wire, it is evident that the K. M. F. between A and B is exactly twice what it is between A and a point halfway to B. Therefore, since the current is uniform, the value of R for the conductor l>et ween .1 and Jt must be twice that for half the length. So, in general. the value of R for any portion of a uniform con- ductor of constant cross section varies directly as the length of this portion. It, while a constant E. M. F. is maintained between the two points A and B, a second conductor is introduced be- tween them, identical with the first one, each will carry a current i = — , and so the current is doubled or the total E H resistance is halved. The same would be true if, instead of using two conductors, one of twice the cross section were in trod need. So, in general, the resistance of a conductor \aries inversely as its cross section. Direct experiments show that if the same E. M 1 applied at the ends of conductors of the same length and M section, but of different matt rials, the resulting cur- rent is different. This and the two previous statements may be expressed in a formula. p a where R is the i i uniform conductor of length / ~- t i 724 ELECTRO!) YNA AIICS and of constant cross section a, and c is a constant for a con- ductor of any one material, but differs for different ones. This quantity c is called the " specific resistance " of a sub- stance, or its "resistivity." Similarly, the conductance C = — = --, and may be written C = k- , where k = -. This R c I I c constant k is called the " conductivity " of a substance. Illustrations of Ohm's Law. — 1. Con- ductors in series. Let the circuit consist of several conductors in series, and let the resistances of the portions A^AV A2 A^ ^.3^.4, be Rv R2, Rs ; further, let the potentials at the points A^ A2, A%, etc., be Vv Vv V& etc. Then applying Ohm's law to the separate sections, FIG. 404. — Conductors in series. Hence, I = The total resistance between A1 and A± is by definition i = 1 "7 — - ; and it is seen that its value is R1 + R2 + Ry In general, then, the total resistance of a number of conductors in series equals the sum of the resistances of the separate parts. (The fact that R varies as I, the length of a conductor, is a special case of this.) 2. Conductors in parallel. — Let the circuit branch at any point A into two or more conductors which meet again at B; let the resi |
stance of these branches be Rv R2, R& etc. ; and let the currents FIG. 405.— Conductors in parallel. LAWS «/• STEADY friiliBNTS flowing in each be t\. i.2. iy etc. The total current is / = /! -f /2 4- i3 -I- ••• ; and the total resistance between A and y _ YR B is by definition — — . Applying Ohm's law to the various branches, we have *1 D D ' 8 D ; etc. and therefore, calling the total resistance R R RZ R This may be expressed more simply in terms of conductances, for 0 = — ; hence, C= C + Ca+ C + — • or. in a branched circuit the total conductance equals the sum of the conductances of the branches. (The fact that R varies inversely as the cross section of a conductor is a special case of this.) It should be noted, further, that the ratio of the currents in any two branches equals the inverse ratio of the resistances of these branches. Thus, 1 . 1 3. Wheatstone bridge. — This is a particular arrangement of •onductors ; four form a < iivuit ABCD, and two connect the diagonal points A and <\ and B an<l />. This network of con- ductors is used in many experi- mental methods ; l.ut only one will be descrihe.1 here. In this a cell is introduced in one of the diagonal branches, say AC. and 726 ELECTRODYNAMICS a galvanoscope in the other. If the arrangement is such that A is joined to the positive pole of the cell, the potential of A is higher than that of (7; and the potentials of B and D are both less than that of A and greater than that of C. So it must be possible to find a point D in the branch ADC whose potential equals that of any given point B in the branch ABC. If this is the case in the actual arrangement, no current will flow across from B to D, and the current flowing from A to B will equal that from B to C ; and that flowing from A to D will equal that from D to C. Call the poten- tials, at A, B, C, and D, VA, VB, V& and VD (it should be noted that VB = F^>); the resistances of AB, BO, AD, and DO, RV R2, Ry, and R± ; and the currents in AB and BO, FIG. 407. — Arrangement of Wheatstone's bridge for the comparison of resistances. -F, e'j ; and in AD and DO, ^'2. Then But F»= Vn\ hence 4? = or This formula evidently offers a method for the comparison of resistances. For, suppose AB and BOa,re two conductors the ratio of whose resistances is desired. They can be joined in series, and A and C can be joined by a uniform wire ; then, making the diagonal connections and introducing the galvano- scope and the cell, the end D of the wire connected to the LA M'.s OF STEADY < URHEb tS former may be mo\ed aloiiLj tin- uniform wire until there is DO deflection of Uiegalvanoacope needle. Then Ffl= J^aml the bridge is said to be - balanced." As just proved, when ^ 2 Pro. 408. —A Wheatstone " wire bridge." 7? 7? 7? this condition is secured, = - hut —2 equals the rat in of the lengths of the portions of the wire AD and DC; and, -f2 -"4 -**4 as these can be measured, the ratio — 1 is known. Similarly, 72 even if the conductor ADC\x not a uniform wire, but if the it /;. As will ratio ' '• is known, that of Jl{ to 7?2 may be determined. in the next of mer- (•iii-y atO° C., of a uniform cross section of a length !<>«;.:; ,-m.. and hayiiiLT a mass .if 1 I. I.VJ1 <j.. has a n-sistanee which be explained a column section, ilh-d an ••ohm " : .sniisin^ a "wire bridge," thai in which A IK ' is a uniform \\ in-, ami introducing the colinnn of mcrciii-y in 1 he " arm " . I />'. i \\ i re forming t he "arm " />< ' >c made, b) altci-iii'^ its length, to 1; "f 1 ohm, or of 'I olims, etc. So, combining these and l:iiL,r in an «.1.\ i,,iis maniH-r. a scries of coils can be made whose n I. J. l« ..... . -"'000, etc., ohms. Tln-s.- c.iils ma\ : I'aii^'-d in a c. >n \ en i.-n! that any on. ,-. .nibinat K>M of I hem ma\ be iis.-d. The 728 ELECTRODYNAMICS ends of each coil are joined to large brass blocks which are insulated otherwise from each other, as shown in the cut ; so that there is a continuous circuit from the first block to the last through the coils. These blocks may, however, be directly connected by the insertion of brass "plugs"; and when any one plug is in place, the corresponding resistance coil is "short circuited"; that is, its resistance is so much greater than that of the plug that any current through the box will pass directly through the plug. (It may be assumed for ordinary purposes that the resistance of the blocks and FIG. 409. —Resistance box, showing method of winding coils. plugs is zero.) These resistance coils are wound, as shown in the cut, in such a manner that the current flowing in one portion is immediately next a current flowing in the opposite direction ; so that the coil has no magnetic action. (The magnetic shell to which it is equivalent has an extremely small area. See page 707.) Definition of the " Ohm," the " Ampere," and the " Volt" - Methods have been described in previous chapters for the measurement of current and E. M. F. on the C. G. S. electro- magnetic system, and so the resistance of any conductor may be determined, simply using Ohm's law E=iR. (Other methods will be given shortly.) It is found that if the C. G. S. electro-magnetic system of units is used in express- ing the values of current and E. M. F. the numbers obtained for the resistances of ordinary conductors of moderate lengths LAWS OF STEADY CURRENTS are enormous ; and so in ordinary cases a unit of resistance 109 times that on the C. G. S. electro-magnetic unit is used. This unit is called an "ohm." As the practical unit of K. M. F. a volt is used (see page 673), which equals 108 times that ('. <i. S. electro-magnetic unit. Therefore, if the value • >f the K. M. 1 . applied at the ends of a certain conductor is £l volts, JE=E1 108; and, if the resistance of this con- ductor is Rl ohms, R= Rl 109. So the current strength t is E If1 — = — '-£. Therefore, if the numerical value of the current K 10 /fcj is in l>e deduced from Ohm's law, using ohms and volts in which to express resistances and electro-motive forces, a unit current must be defined whose value is one tenth that of a C.G. S. electro-magnetic unit current. Such a current is called an -ampere." For if, in the above experiment, the cur- rent strength is i, ampere^ i = -1 = — * — - ; or il = 1. The V 1 V <piant ity nf current carried by a current of tj ampere flow- ing fort seconds is called (tj*) "coulombs"; and the capacity of a condenser, which when charged with ,> coulombs is V 10 Rl 10 /i'j volts, is said to be -^ "farads." A capacity of one millionth '1 is called a "micro-farad." The definitions just given of the ohm and the volt are imt. strictly speaking, correct. As may be easily understood, the amplest way of defining a unit oi nee is to select some standard conductor and call its resistance the unit : hut. <iner there are grent advantages in using the 0. G. 8. sys- «»r (|iiantities which may he expressed in t. •: it by ; lain number of factors 10, the h«-st manner of defining a unit is to select some condm tor whose resistance, when de- t'lmiiM'd as accurately as possible in termi of the (/. G. S. system, is simph --\ pressed in terms ..t n. and then to adopt the resistance of this conductor as the unit. Thus, the ohm i d. -lined to he t ipial to the resistance of a column of mereui\ 730 KlJX'TltoDY \AMirs at 0° C., of uniform cross section, of length 106.3 cm. and having the mass 14.4521 g. (This column, then, has a cross section of almost exactly 1 sq. mm., accepting the usual value for the density of mercury at 0° C.) The resistance of this column of mercury is equal to 109 C. (i. S. electro-magnetic units, to within the limits of accuracy of our present experimental methods. The volt is defined to be the E. M. F. which, steadily applied to a conductor whose resistance is 1 ohm, will pro- duce a current of 1 ampere. It is therefore practically equiv- alent to 108 C. G. S. electro-magnetic units. The E. M. F. of a certain cell, known as the " Clark cell," which can be made in a definite manner, is found by careful experiments to be 1.4322 volts at 15° C. The E. M. F. of the " cadmium cell," which is another standard cell, is found to be 1.0186 at 20° C. (The E. M. F. of the latter cell changes with the temperature much less than that of the former.) Heating Effect. — It was shown on page 664 that the heat energy developed in a conductor carrying a current of strength i in a period of time t was Hit, where E is the dif- ference of potential at the ends of the conductor considered. As was also noted, the number expressing this quantity -of heat is in ergs, if the C. G. S. electro-magnetic system is used. If the current is steady, this quantity may be expressed in other ways, for E — iR. So, writing W '= Eit, we have W= izRt = —~. It is seen, then, that the heating effect R is independent of the direction of the current, because tin- square of the current enters the formula, and it has the samr value for either a plus or a minus sign. If the current is z\ amperes and the resistance is R1 ohms, i = ^ , and R = R1 109; so W= i?Rj 107 ergs. But 107 ergs equal 1 joule, and the power of a machine which does an amount of work of 1 joule per second is said to be 1 watt. LA II - ; AI>Y CURRENTS 731 I currents are measured in ampeivs and re.xistanees in ohms, tin- power of the current is i?Rl watts. Measurement of Resistance Absolutely. — By means of a Wheatstone bridge one may determine the ratio of the resist- ances of two conductors, but it does not furnish a method tor the measurement of a resistance directly. The formula just deduced for the heating effect of a current does, however, suggest a method. If a coil of wire is immersed in a calorimeter containing water, the heat produced in a given time by passing a current through it may be measured in calories, and since the mechanical equivalent of heat is known, the value in ergs may be deduced. The current strength may be measured by a galvanometer and, since thus TF, t, and t are known, the value of R may be determi |
ned ; for Ttr R = — . (There are other methods whicli are more accurate.) Temperature Effect. — Experiments show that the resistance of a given eoml urt<>r varies with its temperature. This is what might be expected, be- no.— < mwwuring the hmtiaf effect of • current. •• it was shown that the resista ne.- < •!' any uniform con- • I net or of length / and of < ross section a could be expressed // = r , where c was a constant, characteristic of the con- a doctor. Hut. if the temperature of the conductor is altered, the motion and the distance apart of its molecules are ted: and BO ii is no longer the same snl»staueo. It is found that U the temperature i> inen-asrd the resistance of all solid conductors — with one or two excepti 732 ELECTRODYNAMICS increases, while that of most liquid conductors decreases. (The effect in this last case is complicated by the fact that the extent of the dissociation produced in the act of solution varies with the temperature.) As the temperature of a pure solid is lowered toward absolute zero, its resistance almost vanishes. This change in resistance of a conductor with change in temperature offers at once a method of making a "resistance thermometer " ; and, in fact, a " platinum thermometer " con- sisting of a coil of platinum wire whose resistance can be measured is at the present time the most satisfactory ther- mometer in use for accurate work. Similarly, the same phenomenon is made use of in the " bolometer," an instru- ment for the detection and the measurement of radiation in the form of ether waves. A strip of platinum is covered with lampblack, so that it absorbs as completely as possible all radiation that falls upon it, and is made to form one arm of a Wheatstone bridge. As ether waves are incident upon it, its resistance changes, and the amount of the change measures the intensity of the radiation. CHAPTER XLVIII INDUCED CURRENTS Tin: discovery by Oersted of the fact that an electric current produced a magnetic field, and the subsequent dis- covery of methods for making a bar of iron a magnet by means of a current, led many investigators to seek for means « »t' producing an electric current by means of a magnet. The method of doing this was discovered independently by Joseph Hi-nryin America and Michael Faraday in England about 1831. Experiments of Henry. — Henry's experiments were the • •arlier. H« observed that, if a circuit in which there was a UUUUUUi Fio. 411. -A solenoid, Ultutntlog Henry'* fl«t «p«rtm«ot rv «.f rells was lu«.k m at any point, there was a f spark; and, further, if the l>n-ak was made by means of the 788 734 EL ECTR ()1)Y . \A.M /r.s hands, so that the circuit was completed by the arms and body, a shock was felt. He noticed, too, that both the effects were increased by increasing the length of the conductor and by coiling it up into a helix. There is thus an " extra- current " on breaking a circuit, in addition to the one due to the battery; and Henry's experiments prove that this cur- rent varies as the magnetic field of the original current ; for, if the conductor forms a helix, the magnetic field is much greater than if the conductor forms simply an approximately circular circuit. A few years later Henry observed that if a wire were wound around the soft iron armature of a horseshoe electro-magnet and if the current were suddenly broken, or if the armature were suddenly removed from the magnet, a shock would be felt, if the two ends of the wire were held in the hands ; or, if these ends were joined to a galvanometer, a sudden deflection of the needle would be produced, but the needle would return to its original posi- tion. The same effects are produced if LJI V FIG. 412.— Diagram rep- resenting Henry's second the current is again made or if the arma- ture, when separated from the magnet, is brought close to the magnet, but the current in the galva- nometer is in the opposite direction. The quantity of the current in the galvanometer, or the shock received by the arm, varies with the number of turns of wire on the arma- ture ; and the shock varies with the suddenness of the motion of the armature ; the current also varies with the material of the conductor, while the shock does not. It is evident that these "induced" currents, as they are called, are due to the change in the number of tubes of magnetic induc- tion which pass through the coil of wire wound on the armature. INDUCED CUi;i;i 73f> Experiments of Faraday. — Faraday's experiments were somewhat different, lie had two separate coils of \\irc wound on the same iron ring, one coil being joined to a cell, the other to a galvanometer ; and he observed that, if he broke the current or made it again, there was a sudden fling of the needle, but that the current was only a transient one. Here, again, the induced current is due evidently to the change in the number of tubes of magnetic induction through the circuit which is GAIVANOMFTCT joined to the galva- nometer. Faraday then showed by a series of most brilliant experi- ments that if the num- ber of tubes of magnetic induction inclosed by any closed conducting circuit is varied in any manner, e.g. by bring- ing up or removing a magnet or another cir- cuit carrying a current, there is an induced cur- ivnt, whose strength varies directly as the change in the number of tubes of magnetic induction and as the r.ite of this change, and also depends upon the material of the cir- cuit. If iron is inside the circuit, it is magnetized bv the current ; and thus the induction is changed. (It \\a>, in Fio. 418. - Faraday '» double coll in his first experiment. iiUr to this Study of induced currents that Farada\ Was h-d to his Conception of tubes of induction and to the id.-. i of these tubes being continuous through a magnet. See page 61 M.in\ years later Faraday rediaoovered tin- phenomena ,,f the e\t ra current on I,: i^'ed his apparatus as sho\\ n in Fi^. II I, where K i^ a cell, C is a helix 736 ELECTRODYNAMICS or electro-magnet, and A and B are the two ends of a broken wire in parallel with the helix. He observed that, if A and £ are held in the hands and the electrical current is broken at E^ a shock is felt. Similarly, if A and B are joined to a gal- vanometer, there is a sudden fling of the needle when the circuit is broken. Just before the current is broken, there is a magnetic field through the helix ; but, when the circuit is broken at E, there is still a closed circuit around the helix and through ^ 5 'dnd the magnetic field in thi» ^W de- creases, since the cell is out of circuit, and so there is no E. M. F. to maintain the current. Owing to the change in the number of tubes of induction in this cir- cuit there is the extra current. Law of Induced E. M. F. — All of the facts discovered by Henry and Faraday in regard to the strength of induced currents may be expressed by saying that, when the number of tubes of magnetic induction inclosed by a closed conduct- ing circuit is varied, there is an induced E. M. F. in this circuit whose value is proportional directly to the change in this number, and inversely to the time taken for the change. If there are n turns of the wire, as in a helix or coil, the tubes pass through each, and the induced E. M. F. is n times as great as if there was but one turn. Thus, calling the change in the number of tubes of magnetic induction A^V, and the time taken for this change A£, the induced E. M. F. during this change equals c-r—, where c is a factor of pro- portionality. It may be proved by methods of the infinitesi- mal calculus that using the C. G. S. electro-magnetic system to express the E. M. F. and the C. G. S. definition of a unit pole as the source of a unit magnetic tube, this factor c has A7V the numerical value 1. Thus, we may write E = -—-. INDUCED CURRENTS 737 If A* is the resistance of tin- circuit, tlic strength of the induced current is ', «>r i = — — — . Hence the quantity of /i R A£ the < -in-rent in time A£, or i&t, is •--•• The current strength varies as the change is made, but the total induced quantity, or the summation of ?Af during the entire change, equals the suiuinatioii of ~JT\ that is, it equals the total change in -ZV divided by A*. It must be remembered that if the wire is coiled up, this quantity N varies directly as the number of turns. Thus the induced I-;. M. I-', varies with the sudden- ness of the change, while the induced quantity does not; the latter depends upon the resistance of the circuit, and upon the total change in the number of tubes; i.e. upon the field of magnetic force, and upon the area and number of turns of the coils. These facts are shown by Henry's and Faraday's experiments; because the shock received by one's arms is conditioned by the induced E. M. F., while the fling of the Lcalvanometer needle measures the induced quantity of the current. It must he particularly noted that there is an induced cur- rent in the ciivuit only so long as there is a change in the number of tubes of magnetic induction through the circuit. The direction of this current was investigated by both Henry and Faraday. Their conclusions may he expressed by saying that, if the change in tin- magnetic field through the eireuit is an ////•/•• <(*•' in the number of tulx's of induction. the induced current is in such a direction as by its Own inai,'- netir lield to decrease the number : or, if the change in the lield is a decrease in the number «.f t ul».-s. the induced current h a direction as to incre<i*> tin- number. In general. then, the induced em-rent produced by any change in the magnetic field throu-h i h a direction as to tend to neutralize this change. ( If this \\.-re n..t true, an increase in AMBS'8 FIIT«ICfl — 47 738 ELECTRODYNAMICS the magnetic field would induce a current in such a direction as to increase the field still more ; this second increase would produce a second induced current in the same direction, etc. ; so conditions would be unstable.) If there is already a c |
ur- rent flowing in the circuit, the induced current is superim- posed upon it, either increasing or decreasing it. Special Cases. — A few simple cases will be considered ; if a current is flowing in a circuit, and if a bar magnet is made to approach it or to recede from it, the direction of the induced current may be at once predicted. If the north pole of the magnet is nearest the south face of the circuit, some tubes due to the magnet pass out of the north face of the circuit. So, if the magnet is brought nearer the current, more tubes will pass through it, and the induced current will be in such a direction as to oppose this change; i.e. it will be in a direction opposite to that of the original current. Thus the current in the circuit is decreased as long as the magnet is approaching. This means, expressed in other lan- FIG. 415. — Diagram to illustrate induced guage, that WOrk is required to move the magnet, and since this is done by the current, only part of the energy of the cell is available for forcing the current around the circuit. Con- versely, if the magnet is withdrawn, the field of force through the circuit is decreased, and the induced current is in the same direction as is the original current. This means that work is being done by whatever agency moves the magnet ; and this work appears as an increased current. The case when the bar magnet is turned with its south face toward the south face of the circuit may be treated in a similar manner. Earth Inductor. — If a coil of wire is arranged so as to turn on an axis parallel to its plane faces, it may be so placed < r ///;/:. \ ro 739 that this axis is vertical; and then, if the face of the coil is perpendicular t<> the magnetic meridian, it will include a tit-Id of magnetic force due to the earth. If the coil is turned into the magnetic meridian, t licit- is no field through the coil : and, if it is turned 90° farther, the original field of force will pass through it, hut in the opposite direc- tion with reference to the coil. So it is just the saun- as if the coil had remained ionary and the field of force had changed from N to 0 to -N. The total change, then, is 2 N. If A is the area of the face of the coil, if there are n turns of wire in the coil, and if // is the horizontal component of the earth's magnetic Held, N= nAH. So, if the terminals of this coil are joined to a hallistic ^;i\ \ anometer, the (juantity of cur- rent measured when the coil just described is suddenly turned through 180°, from a position perpendicular to the R magnetic meridian, equals " — , where R is the resistance •in- circuit. Similarly, if the coil is so turned that its axis of revolution is in the magnetic meridian and its faces an- hori/nntal. X=nAV* where V is the vertical component of the earth's magnetic force; and if the coil is turned on the axis through 180°, the (piantity of the induced current is -•If V '—= — Therefore the ratio of these two quantities equa 1 which is the tangent of the angle of dip. (Seepage 619.) Such an in^t i inimt ix .-ailed an "earih inductor." It was invented by the great German physicist, Weber. 740 ELECT ROD YN AM n .s Induction Coil. — A case of special interest is one studied by both Henry and Faraday : a coil of wire is wound on a spool or cylinder of such a size that it will slip inside another Fm. 417. — Methods for production of induced currents. cylinder, on which is wound another coil. Call the first coil J., the second B. If the terminals of A are joined to a cell, and those of B are connected to a ballistic galvanometer, no 711 current will flow in the hitter until tin- current in .4 is varied; but. it' this is <h>nc. a current is induced. If there is a rod of soft iron inside A. the induced currents are greatly increased when the current is made or broken; because when the cur- rent is flowing, this rod is magnetized, and so the magnetic field is increased. If a magnet is brought near this iron rod or is taken away, there are also induced currents. Similarly, if the terminals of B are held in the hand, a shock is felt when the current in A is made or broken ; and this shock is increased by inserting a piece of iron in A. The shock on breaking the circuit is greater than on making, because the time taken for the change in the magnetic field is less in the former ease, and so the induced E. M. F. is greater. The induced nuantitii is the same in both cases. If a thin copper (or conducting ) cylindrical tube is interposed between 1 /A it prevents almost completely the shock felt at the terminals of B, but does not affect the quantity of current as shown by a ballistic galvanometer. The reason is that, as the current in .1 is changed, electro-motive forces are in- duced both in B and in the copper tube; and so, as the (in rent in the tube changes, it also induces an E. M. K. in J9; these two induced electro-motiye forces are in opposite directions and so the resultant effect is small. If the current in A is large, and there are enough turns of wire on B, an E. M. !•'. may be induced in B. when the current in .1 i- broken, snllicicnt t o spark across considerable distances in case the terminals of B are separated. This ui.il. rdinarj ••induction c».il," as shown in Fig. 418. I [| a mechanical arrangement for automatically break- md making the circuit in A. \\hose principle is evident. The iron core of >iidi a coil is alwa\s made of iron wires insulated from each other and is not a solid rod, because as the current in .1 d. induced currents would be pro- duced in i iron rod in circles around the axis of the rod, and thoe arc piv\ciitc«l by the division of the rod into 742 ELECT R OD YXA MICK wires. (These currents produced in a solid core are called "eddy," or Foucault currents.) A condenser is always in- troduced in the battery circuit in parallel with the " primary " coil A. One of its chief functions is to prevent sparking FIG. 418. — Diagram of induction coil. at the points where the circuit is broken ; it does this by diverting the extra current in the primary from the two points where the circuit is broken into the two plates of the condenser. (In other words, to produce a spark, a definite potential difference is re- quired, depending upon the distance ; and the differ- ence of potential of the two plates of the condenser does not rise sufficiently high to allow a spark to pass, provided the capacity is great; for Vl—V^ = ^> See page 652.) Thus, if the extra current on breaking the primary circuit is prevented, the change in the field through the " secondary " circuit B is very sudden, and the induced E. M. F. is intense. FKJ. 418 «. — Induction coll. . , -, , ., i\i>i < /•:/> CUBES* 743 When the current in the primary is sixain made, the change in the magnetic field is comparatively slow, and so the induced E. M. F. in the secondary is not great Self and Mutual Induction. — 1. Self-induction. If a cur- rent is Mowing in a circuit, it has a field of force of its own : if / is the current strength, the number of tubes of magnetic induction which thread this current is proportional to it and may he written LL where A is a constant for the given con- ductor and for a given medium surrounding it. L is called the "coefficient of self-induction" or the "inductance." It is evident that L -varies directly as /A, the permeability of the medium ; for the magnetic induction equals p times the mag- netic force, and the latter depends simply upon the current and the shape of the conducting circuit. (Thus the effect of introducing an iron rod into the circuit is explained. In the case of iron it must be remembered that /u, is not a constant, lor it depends upon the intensity of magnetization. So L is not a constant unless the medium is kept the same.) 1 Hither, L must increase as the area of the circuit in- creases, because the circuit will include more tubes. In tin- case of a solenoid which has N turns per unit length, the magnetic force inside has been shown to be 4 irNi ; therefore, it I is the length of the solenoid, each tube of force passes through the current Nl times; and, it .1 is the area of the cross section of the solenoid, L = 4 7rjV-/M it' the medium is air, and equal> 4 Tr^N^lA in general. Therefore, if the em-rent is varied in any \\a\...;/. by alter- ing the K. M. V. of the cell, there will be an induced I-;. M. V. whose value equals the rate of cha ii-' ..l A/: and the greater /. i, so moch the greater is the indaoed K..M.K. The induced quantity of « urrent equals the total change in the number of tubes of induction divided l>\ the resistance of the circuit. If the applied K. M. V. is 1 1 1 ; i 1 . so as to tend to increase the current, and thu^ Increase the li«-M . l he induced '•nrr. -MI must he in the opposite direction ; and as a c<- 744 ELECTRODYNAMICS quence the current does not rise instantly to the value corre- sponding to the applied E. M. F. Similarly, if the applied K. M. F. is decreased, the induced current is in the direction of the original current ; and so this does not decrease instantly to its final steady value. These facts may be expressed in a formula by writing the induced E. M. K. = , where the minus sign means that if AJVis positive, the induced E. M. F. is negative, while if AiV is negative, it is positive. Particular cases of these changes are -when the circuit is suddenly broken and when it is suddenly made, e.g. by removing one of the electrodes from the cell and by then plunging it in. In the former case, the current does not instantly fall to zero ; there is the extra current, as shown by the spark, etc., as observed by Henry and Faraday. In the latter case, the current does not rise instantly to its fixed value. The time taken for these changes evidently varies directly as L ; so that L measures what may be called the 44 inertia of the current." When the circuit is broken, the energy of the magnetic field is no longer maintained by the cell, and it returns into the conductor, contin |
uing the current until all the energy is consumed in heating the conductor. Then the current ceases. Similarly, when the circuit is closed, part of the energy fur- nished by the cell is spent in producing the magnetic field, and only a portion of it is available for producing the cur- rent in the conductor. It is not until the magnetic field is established, then, that all the energy supplied by the cell goes into maintaining the current. As a consequence it takes time to produce a steady current. These intervals of time required for a current to come practically to rest when a circuit is broken, or to be produced, are, as a rule, extremely short, a few milliontlis of a second; but if the circuit has ti large value of L, e.g. if it is in the form of a D rr/;/;/-;.vrs 71-". solenoid inclosing a rod of iron, the time may be as great second. The energy of the current, i.e. of the magnetic field due to it, is thus in the surrounding medium. 2. Mutual induction. — If a circuit carrying a current is near another circuit also carrying a current, some of the tubes of force due to each current will pass through the other riivuit. Thus if the currents in the two circuits are tj and /2. the number of tubes due to the first current which pass through the circuit carrying the second one may be written Mfa. (It must be noted that if the second circuit has n turns, the tubes pass through its current n times.) Simi- larly, the number of tubes due to the second current that pass through the first may be written Mj,v It may be proved by the infinitesimal calculus that Afj = Mv and that this quantity is a constant for the two circuits, depending upon their shape, size, number of turns, and relative positions, and also upon the permeability of the surrounding medium ; it :lled the coefficient of "mutual induction," or the "mu- tual inductance." (The unit of induction is called the •• Henry." It is the induction which a coil has if it is of such a size and shape that a variation in it of a current at the rate of one ampere per second produces an induced E. M. 1 . of one volt.) Tin- value of M may be calculated provi<l»l </// th,- till,-* due to one current pass thron<//t (ill the coils of the other l> This condition may be secured practically in two wa ' 1 ' I' IM. ing one helix inclose the other and making them nearly as possible of the same length and < TOSS section, and then 111! n><ll|rlll'_r ,, lnllLr ,.,„! ,,(' jm,, ; iniliiig the t\v<> helices on a rin^ of iron. ;is shown in the cut : for iron lias such a great |>erme;il»ilit v ,((mp;uvd with air .... ... Fie. 419.— Dteffnun of ttmntibnntr that all the til! ' ir. illy May , colU wound on . 746 ELECTRODYNAMICS in the iron and do not escape into the air. The former piece of apparatus constitutes an " induction coil " and has already been described ; the latter is called a " transformer " and is used for many practical com- mercial purposes. FIG. 419 a. — Section of a com- In the case of an induction coil, the intensity of the magnetic field at a point inside the primary far from the ends owing to a current iv is 4 TrNfa, where -ZV^ is the number of turns per unit length. If there are N2 turns per unit length in the secondary, there are N2l turns in a length I ; and if A is the area of the cross section of the primary, 4 irfjbNlN,2lAil tubes of induction thread these coils ; so for a length I of the induction coil — near its middle part — M = 4 jr^^N^A. It is thus seen how the induced E. M. F. in the secondary is increased by introducing an iron core — so as to have p large, and also by having N^ and N^ large. mercial transformer. Transformer. — In the case of the transformer the case is somewhat different, owing to the fact that it is assumed that none of the tubes escape into the air, and an exact calculation can be made of the effect of all the coils, not simply of those near the middle. If n^ is the total number of turns in the primary, and n2 that in the secondary, the coefficient of self-induction of the primary is proportional to nf ; and the coefficient of mutual induction of the two coils is proportional to n$iv So, if the current in the primary suddenly ceases, or if it is reversed in direction, the E. M. F. induced in the primary is proportional to nf, and that induced in the sec- ondary is proportional to n^ny As will be shown in a few pages, it is possible to construct a machine that produces an E. M. F. that is rapidly reversed in a continuous man- ner ; this is called an " alternating" E. M. F. A particular case is one that may be written E=El cos pt, where El is INDUCED CURRENTS 717 a constant. In tliis cast- tin- K. M. F. obeys a "sine cur rising t<. a inaxiniuni value Er <l< :;,1 then being- reversed to — -#r etc. It such ail K. M. F. is apj to tin- primary of a transformer, it will produce a similar induced. E. M. F. in the secondary, which may be written 't—N). The phenomena are all then periodic, with a period -TT/P, or a frequency p/Ztr. From what has been shown above E1: E^=n^: 91^1., = i^ : n.,. Thus, if /^ = 100 14, L\ — WQE2. Therefore bv means of a transformer an alter- nating current with a large E. M. F. may be made to produce another alternating current with a small I-".. M. F. This plan is used in lighting houses with lamps rendered incandescent by means of alternating currents; the street current has a large E. M. F., but by means of a transformer the cunvnt produced in the house lias a small E. M. F., which is not dangerous to life or property. iee the energy supplied by the primary current is pro- portional to E^ and that spent in maintaining the current in the secondary is proportional to E^iv it follows that, if we neglect any losses, E1il = E,iiv or the ratio of the current equals tin- inverse ratio of' t he electro-mot i ve forces. rge electromotive forces are used in the street circuits because if such is the case, the conductors may !>•• m;i<le of smaller sized wires than would be possible if tli-- I . M. r irere small. ] iurtor must be able to furnish a large amount of «Mi«Ti:y. !.*•. AVV must be large; but the rtioiuil to /,-/,'.' where R is the resistance; in or.l.-r to have thi^ ^inall. i, inns! IM- as small as possible, and ,, \\lii. -h is made large. Dynamos. —The simplest case of a so-called dyn that of the M Gramme-ring H type. It consists of tu t and the anna! ure. The magnet is one so mail.- that th-- north i li p«.l«'s < «• me opposite each other, shown in tl-o ent. In praotioe it is always an elect i-o-nia-'iift which is ina^nd .in clcri :'ont. The armature ronsisteof a soft ir«»n rin- wl made up 748 ELECT ROD YNAMWS of insulated iron wires bent into circles, and around which is wound a continuous copper wire carefully insulated from PIG. 420. — Gramme-ring dynamo. the iron. The armature is rigidly fastened to a shaft per- pendicular to its plane, and the shaft is placed perpendicular to the magnetic field of force between the poles of the mag- net. If the shaft is revolved, cur- rents will of course be induced in the coils of wire wound around the ring, because the field of magnetic force through them is constantly changing as the ring rotates. On the shaft of the armature is fastened FIG. 420 a. — Simplest form of commutator. what is called the " commutator," which consists of metal strips or "bars" along the shaft, each insu- lated from its neighbors, and resting across these bars are two so-called " brushes," which are metal strips, one on one side 1NDI '< i:i> of the commutator, the other diametrically opposite, so ar- raiiLTfd a> to touch opposite bars of the commutator at the saint- instant. These brushes are held stationary as the < imitator revolves, and they are joined by a conductor through which a current is desired. Each of the bars of the commu- tator is connected by a wire to different points of the wiiv which is wound around the iron ring. Consequently, as tin- armature is turned by means of the shaft, the brushes are always joined to those portions of the wire around the iron i iiiL,r which occupy in turn the same positions in the magnetic field. The lines of force from the north pole of the magnet pass into the iron ring, and around through the ring to the side opposite the south pole, where they pass out and cross the air gap to the south pole. They do not pass across the ring, but are divided, as it were, into two sections, which crowd through at the top and bottom of the ring. Conse- Fro. 481.— LlM* of pole piece* of dynamo when there U DO oarrrat quently, as the armature revolves, turns of wire which are hori/ontal have no field of force through them; but as they reach the top or bottom of their path, and so are placed vertical, there is a strong field through tin-in. din are, therefore, induced in thnn, and th.ir dir.-ctions are easily deduced. If the magnetic poles are as shown in tin- cut, and the annatn 'iirnrd in t lie direction indi- cated, it is seen that tin- rum-nts in all tin- mils on tlu- ascending half of tin- rin- are «! 1. \\hiU- those in the T half arc also d»<\n\\ard. So, if tin- l» rushes are - Hooted with tfaota ; mnutator \vhii-h are 750 ELECTRODYNAMICS joined to the top and bottom coils of the armature, a current will be produced which will flow from one brush around to the other through the external wire, thus reaching the top coil, where it will divide into two branches which meet at the bottom coil; it then flows back to the other brush, etc. This process is perfectly continuous, and a steady current will be produced. In practice, the brushes are not connected with the coils at top and bottom, but with those a little farther in advance, in the direction of rotation. This is because, as the current flows in the armature coils, it produces a magnetic field which so influences the field due to the magnets that the coil where the current tends to branch is no longer exactly at the top, but is a little in advance. If, instead of driving the shaft of this armature by means of sonic exter |
nal power, and so producing a current, a current is sent through i 1m armature from some other dynamo or battery, the armature will revolve and the shaft can be used to furnish power. This is, of course, the prin- ciple of the motor, as already men- tioned. In other types of dynamos, the armature consists of coils wound on a shuttle-shaped piece of iron, but the principle is the same as that of the ring dynamo just described. In order to avoid induced currents in the iron body of the armature, it is divided transversely into a great number of thin plates which are insu- \ MAIN CIRCUIT FIG. 422. - Diagram of a scries dynamo. lated f r°m each °fcher> but . clamped together so as to form a solid frame. The current to magnetize the field magnet is furnished in general by the dynamo itself, although in some cases a current is provided from a < ri;i; T.-.1 :;vte source. In SOIIM- forms «•! inarhines the total mnviit flows around tin- iiia-^u.-t i/in- r«.il>. \\hilt- in others the current is divided and only part is so used. 428. — Alternating uno. The current produced by tin- <l\n;mt.»s just described is dir.-ct ; Init l>y siinph- rliuni^-s tin- in.icliin.^ may be SO arraii^'-d as to i»i-n<lncr altrniatin^ rurn-ntg. A common Jtemating <lvnaim>" is slmwii in Q that :•! «>t" li,i\ iir_r only two pnlf j.ir.-.-s. : al an. i ar<>un<i .1 circh- ami ><• oetixed |] ..•:i rt - Dtafmn of winding In alUrnriaf mnwi in th.-n \ ^ S iture UK! iii opposite 7.~>l> ELECTRODYNAMICS directions; so, as the armature is revolved, the induced electro-motive forces are in the same direction in them all, but they are reversed in direction as the coils pass from one magnetic field into the next. If there are n pole pieces, and if the armature makes ra revolutions per minute, the E. M. F. will be reversed nm times each minute. On the armature shaft there are two conducting rings, which are insulated from each other, and to which are joined the terminals of the wire wound on the armature. On them rest the two brushes joined to the external circuit ; so an alternating E. M. F. is applied to it. The pole pieces of the dynamo are magnetized by a separate direct-current dynamo. Oscillatory Discharge of a Condenser. — It has been stated in the description of the discharge of a condenser that under certain conditions it is oscillatory. The reasons for this may now be discussed more fully. When the condenser is charged, the surrounding medium has a certain amount of electrostatic energy. If its two plates are joined by a conductor, a cur- rent will flow in it, and thus, if the resistance is small, so that the energy is not in the main spent in heating the con- ductor, a considerable amount of it will be consumed in pro- ducing a magnetic field. When all the electrostatic energy is thus exhausted, the electro-magnetic energy will flow back into the conductor, continuing the current in the same direc- tion, and thus charging the condenser plates again, but in the opposite manner to their original charges. Finally, all the energy which has not gone into heating the conductor or to producing waves in the ether is again in the form of electrostatic energy ; and the process is repeated in the oppo- site direction ; etc. After a certain number of oscillations the energy is all exhausted, and everything comes to rest. It is evident that the greater the self-induction of the con- ductor joining the condenser plates, so much the more energy goes into the magnetic field for a given current. IMil'CED CUR If I A useful mechanical analogy is furnished by the vibrations spiral spring carrying a heavy body. When the sj st stretched, the energy is entirely poten- tial ; then when the body passes through its position of equilibrium, the energy is entirely kinetic; as the body moves np, the spring '•d, the strain being op| what it was originally, and the energ again potential; etc. Further, the greater the mass of the moving body, so much the greater is the kinetic energy for a given velocity. As the vibrations continue, the amplitudes decrease gradually. owin^r t" l< of energy by friction, and finally the motion ceases. The period of the motion is evi- dently increased if the mass is increased, and is decreased if the stiffness of the sprii increased. In the electrical oscillation, then, electro- static energy corresponds to potential energy. and electro-magnetic to kinetic. The potential Fio. 4»4.-Vibrmtlnir •plrd .print. charged condenser is and for a given charge is greater if C is small ; so - corre- C spends to the stiffness of the spring. The self-induction of the elertrical conduct.. r corresponds to the ii; the vibrating matter. By means of the infinitesimal calculus it maybe shown that if the resistance of the conduct' small, the period of oscillation of tl irge is given the formula T l-rrVEG; so, if L is large, the period is great; and. t f is large, the period is small, other things _T equal. — 48 AMKft'8 754 ELECTROD YNAMICS Electrical Waves along Wires. — This mechanical analogy is illustrated still further by the phenomena of electrical waves along wires. If the potential of one end of a long conductor is first raised, then lowered, raised again, etc., or, in other words, if it varies periodically, it is found that the potential at all points along the conductor is not the same at any one instant, and that there is a train of waves of elec- trical potential passing down it. This is the process by which messages are transmitted across submarine cables, and by which telephone messages are sent. The velocity of these waves can be measured, and the wave length also. Experi- ence and theory both agree in showing that the velocity of short waves along wires is the same as the velocity of light in the free ether, viz., 3 x 1010 cm. per second. These waves may be compared with ordinary mechanical waves along a stretched string or rope. The self-induction of a unit length of the conductor corresponds to the mass per unit length of the string ; the reciprocal of the capacity per unit length of the conductor, to the stiffness of the string ; the resistance of the conductor, to the internal friction of the string. Waves of all lengths decrease in amplitude as they advance along the conductor ; but the long waves decrease least ; so as a complex train of waves advances, it becomes more and more simple, because its shorter components vanish. If the self-induction of the conductor is increased sufficiently, not alone is the attenuation of all waves decreased, but it is the same for waves of all wave lengths ; thus there is no distor- tion of the waves. In Pupin's system of constructing tele- phone lines, this condition is secured by inserting in the line, at regular intervals of every few miles, a helix whose self- induction is large. By this means it is possible to telephone with distinctness over intervals of a thousand miles or more. Electrical Waves in the Ether. — Electrical oscillations pro- duce waves in the surrounding ether, as has been already i.\in < i:i> < n;i;i-:\ i s stated. These waves are identical in their properties with those which produce the sensation of light; and there are many ways by which they may be detected. They will pro- duce oscillations in other conductors : and these may be made evident by sparks or by the heating effects. Again, if a n umber of small metallic particles are put loosely together, hey offer a great resistance to a current : l»nt, when these long ether waves fall upon them, they cohere in such a manner as to have comparatively Mnall resistance. It the particles are jarred slightly, after the v as, their resistance again increases. Such an apparatus is .ailed a "coherer." It is evident that by introducing wires int- two ends of a coherer, and putting it in circuit with a battery and a Lralvanoscope, or a telegraph sounder, one can observe the passage of these " electric waves." There are many m« >i ••• methods, descriptions of which may be found in advanced text-books. The papers of Henry and Faraday, on the subject of induced currents, have been reprinted in the Seiei. liemoin - Vols. Xi and Xll, New York, CHAPTER XLIX OTHER ELECTRICAL PHENOMENA THERE are several phenomena, showing the connection between electricity and light, which should be mentioned. Faraday Effect. — The fact discovered by Faraday that, if a beam of plane polarized light is passed through a strong magnetic field, parallel to the lines of force, the plane of po- larization is rotated has already been discussed. (See page 564.) The direction of rotation follows the right-handed- screw law ; so that, if the field of force is inside a solenoid, the direction of rotation is that of the current flowing in the coils. This rotation leads one to believe that associated with a line of magnetic force there is a rotational motion in the ether ; so that any vibration in the ether at right angles to the line of force will have its direction changed. The energy of the magnetic field of force should then be considered as due to this motion ; and it is thus seen why it is kinetic. Hall Effect. — This rotational action of a magnetic 'field is shown also in what is called the " Hall effect," a phenome- non discovered in 1879 by E. H. Hall, now of Harvard Uni- versity. If an electric current flows through a thin metallic film, it will so distribute itself that corresponding to any point on one edge of the film there is another on the op- posite edge which has the same potential. Let A and B, in the cut, be two corresponding points ; if they are joined by a wire which includes a galvanometer, no current will flow in it. If now this film is placed between the poles of a mag- net, so that a magnetic field is produced perpendicular to the current sheet, a current will flow from A to B through the 766 KLECTRICAL 757 galvanometer, showing that they are no longer at the same potential. This proves that the lines of flow of the m. Fio. 488. — Lines of flow In a thin metal Htrip. |
Dotted lines are lines of constant potential have been rotated by the magnetic field. \\\ moving one terminal of the wire slightly along the edge of the film, another point may be found fur which there is again no current. Before the magnetic field is produced, the lines of tlo the current are as shown in the cut ; and lines of constant ;itial, at right angles to these, are r A also shown by dotted lines. After tin- field is produ these lines are all rotated, as shown. The direction of the rotation is different in different films, and is reversed if FM. 416. - Dtacrmn ahowlor Hall field or the cu s reversed. •i may be described in a different way. Before Mtroduced into the magnetic field, the lin. ,,f the electric force may be represented by ~FQ\ after the fiel produced it U Lfiven by FQV if the r< the diree indicated. This is equivalent to saying that the effect of magnetic field is to cause a tran*vtr*< I M I-'., ' 758 ELE( 'TROD r\A MICS Certain metal films, notably bismuth, have their resistance apparently increased when placed perpendicular to a magnetic field. This may be described by saying that the field causes a longitudinal E. M. F. opposite to that existing before; because, Ji as shown by Ohm's law, i = ^, a decrease in E will produce a decrease in i, exactly as if R were increased. This apparent increase in resistance may be explained if it is remembered that a current consists of the motion of charged particles ; and such a particle in motion will have its direc- ». 427. - Transverse tion of motion changed by a magnetic field, exactly as cathode rays are affected by a magnet. Therefore, owing to this deflection of the moving particles, the strength of the current lengthwise of the film is decreased. (It should be noted also that the flow of heat through a body is affected by a magnetic field in a manner exactly similar to this just described for a flow of electricity ; and the reason may be explained as a conse- quence of the existence of the Thomson electro-motive forces.) Zeeman Effect. — A most interesting phenomenon was discovered by Zeeman in the year 1897, which has an im- portant bearing upon theories of light. He observed that if a gas which was rendered luminous in any way was placed in a magnetic field, the spectrum " lines " were all affected ; any single line was changed into two, three, or even more lines. These effects may be explained only by assuming that the ether waves emitted by the gas are due to the vibrations of charged particles. These are the electrons inside the atoms. By measuring the effect of a given magnetic field upon any particular radiation, it is possible to calculate the ratio of the charge of an electron to its mass ; and it is found -•/•;/: >;/,/•:< //;/r. i/, PSSNOM that this value is the same as for the negative particle- by radio-active substances and as for the cathode m\ It is t lius seen how intimately connected an- tin phenomena icctrical charges and radiations in the ether. Fur sin««- a moving charge has an inertia of its own, quite apart from that of the material particle carrying it, it is not impossihle that all the phenomena of matter may be explained as due to the motions of electrons. The original memoirs of Faraday, Kerr. and Xeeman are given in Lewis, The Effects of a Mayi" / >ldon Radiation^ Scientific Memoirs Sc;;.-. N. ,\ fork, I'.'OO. Historical Sketch of Electricity The first to use tin; word "electrieity '' or "electrical" was Gilbert in his treatise on Magnetism published in tin- year 1600. Many electrical phenomena, however, were known to the ancients. Thales, of Miletus, who lived about 600 B.C., is reported to have described the power which was produced in amber by friction and whieh enables it to attract bits of straw and other li-ht bodies. The earliest description, however, of this property which is extant is that given by Theophrastus, 321 B.C.; and from the writings of totle, Pliny, and others we learn that the ancient natu- >t8 were aware of the electrical phenomena in the shocks of the torpedo. Gilbert was the first to make a sharp ion between magnetic and electrical forces, and from dmost every natural philosopher perform- • nU in this m,,st interest in^r lidd of sricnce. The tied bodies n>u <'ach butfd to V<»i G about 1670), It was he also who invented the fir- mac bine, which was aftcr\\aid imj. roved 1 B i others. the po\v* os to conduct libuted to Stephen <.i.t\. \\h.. di.d in 1736. The fact that I 760 ELECTRODYNAMICS and negative, was discovered by Du Fay in 1733 ; and the idea that in every process of electrification equal quantities of opposite kinds are produced is due to Symnier, 1759. The phenomena of electrostatic induction and of charging by induction were first investigated by Canton, 1753, and .<Epinus, 1759. We owe many important ideas to Benjamin Franklin. Among other things, he established the identity of atmospheric electricity as manifested in lightning, etc., with electricity as obtained by ordinary means. Many im- portant phenomena were discovered by Cavendish in the last few years of the eighteenth century, but unfortunately his researches were not published for nearly one hundred years. Cavendish was the first to prove the law of the inverse square for electrical forces ; to study the capacity of condensers ; to discover the effect of introducing dielectrics other than air; to propose a law for an electric current, which is practically equivalent to Ohm's law ; and to meas- ure roughly the electrical resistances of many conductors. , The most important work done within recent years has been that of Michael Faraday, who was the first to recognize the importance of the surrounding medium in all questions dealing with electrical forces. The varied phenomena dealing with the properties of electric currents and the names of those scientists associated with their discovery are given in the preceding pages. BOOKS OF REFERENCE PERKINS. Outlines of Electricity and Magnetism. New York. 1896. An elementary treatise, in which the phenomena are all explained in terms of tubes of induction, either electrostatic or magnetic. WATSON. A Text-book of Physics. London. 1899. The sections on Electricity are particularly good. J.J.THOMSON. Elements of Electricity and Magnetism. London 1895. This is a text-book which treats the subject from a more mathemati- cal standpoint than most other elementary books; but it may be consulted with advantage by any student. INDEX Aberration, chromatic, 4S9 ; of light, 568 ; spher Attenuation of vibrations and waves, 828, 846, leal, 451. 465,48* Abeotata tamperacwe. M8, M7. Abv.lut.- zero. -.'Jo. :;.•;. A».>..rpti.,n. t,.-ly. MS, DBT; of .-ru-ivy. -.1.7 ; Mirf;i<'c. •_".'•.', ."•»•>. At.-or].;r,.- pOwW, -"-".'. Acceleration, angular, 55; composition and resolution. 41. 44 ; due to the earth, "g," 4, •r, 89. 545. Action and reaction, prim-:; Adiabatic curves, 806. Adiabatic expansion, 194, 387. Airy, rainbow, M? ; stellar aberration, 569. Alternating dynamo, 746. Ampere, electro-magnetic experiments, 709. 729. Ampere's theory of magnetism. 71". Amplitude of harm, ;•; waves, 842. Analog.-. sJtdrifl VM 1 ML -••iuii.-'ii Mnr,- v,.I •MMl&W, 8M, 8M, nt 7M; bBMtotlM an.l r..?.itior,. .',7. '.O. 11J. Analysis of musical notes. 896. Analyzer in polarized Hght, 560. Andrews' IsothennatoTor OO% 277. ,.f ,!,M, •;..:,. m\ "f «::,.,!,•,,.. Mfj ^.\ ; ,,| minimum deviation. 460, 506; orr.-r!,,-tlofl. ..,,'.,. I ;] ; ,,f ,,•-,„ •:.,,,. .;7.(. l ||| i.o'jn/i'ii: v. • Angstrom unit for wave lengths, 687. AMteMMfcMHM, MaSiSri Md N!M A. -I,-. .X, l^i«:2!»B?!k Arehlmedel' prtndple In Hydrostatics, 164. \r, IMBI MK Mt, Attracted disc electrometer, 659. Attraction and repulsion, electrical, 627, 642; magnetic. 596, 617. Atwood's machine, 71. Audibility of air waves, 886. Average or mean values, 29. Avogadro's hypothesis, 2uo. .Ml. : a crystal, 542.; of lens, 461 ; of rotation, 85, 52, 94. Back E. M. K. in motors, 720. HB::I:HT wheel ofwatoh, Itt. Ballistic galvanometer, 674. Barometer, 176 ; correction of, 177. Base ball, curves of, 169. 718. as of, 861. BtL Blaxal crystal-. 268; effect of dissolved sub- stances on, 275 ; effect of pressure on, 268. BafaMter, -".'». 782. Boon m, nmn ••..::„..„„•.,•. ,..-:.. Boyte's Ir.w, 194, 200. Boys, rmdlomlcrometer, 184, 719. Mr:,.-,'. M., , -t,, ,,.!,..!. .,!.,-!. r. Hi BradleyVmetiiod of determining veJodty of !.•« at friction, 190. ••: •i it, MT, •"! . ilorto, Ml, . Ml CUortaMtar, 151. .. 719. ktBsM \t, ;v:; Atomic WtljM, Mt MB "f radlatkw, 801 m Capillan «»»H - hi • i i 9 - : 762 INDEX Cathode rays, 701. Cation. 68t. ( austic, by reflection, 451 ; by refraction, 465. Cavendish, gravitation experiment, 183 ; lu\v <>f electric force, 64U ; measurement of capacity, 660. Cells, primary, 676; secondary, 698. Centigrade scale of temperature, 222. Centimetre. '-'I. Centre, of gravity, 99, 134; of mass, 61, 80; of pressure, 174. Centrifugal force, 72. Charge, electrical, 638 ; magnetic, 607. Charles' law, 240. Chemical equivalent, 686 ; electro-, 688. Chladni's figures, 861. Chromatic aberration, 489. Circle, uniform motion in a, 40, 47. Clark cell, standard of E. M. F., 730. Clinical thermometer, 225. Clock, pendulum, 186. Clouds, formation of, 267. Coefficient of expansion, 280, 234, 287. Collimator, 505. Colloids, 141. Color, 586. Color blindness, 592. Color sensation, 592. Colors, absorption, 587 ; complementary, 524, 587 ; connection between wave number and, 424 ; mixtures of, 587 ; of thin plates, 528 ; polarization, 557 ; surface, 588. Combination, chemical, heat of, 285 ; of lenses, 487 ; of notes, 412. Combustion, 285. Commutator of dynamo and motor, 748. Compensated pendulum, 232. Complex pendulum, 322. Complex vibrations, 320. Complex waves, 347, 754. Composition, of a uniform acceleration and a uniform velocity, 45 ; of displacements, 36 ; of forces, 60, 75 ; of harmonic vibrations, 318, 822; of moments, 91 ; of velocities, 87, 53. Compound microscope, 504. Compound pend |
ulum, 185. Compressed glass, double refraction in, 542. Compressibility, of a gas, 192 ; of a liquid, 171 ; of a solid, 150. Concave grating, 537. Concave mirror, 447. Condensation of vapors, 279. Condensers, electric, 650 ; capacity of, 652, 660 ; discharge of, 654, 752 ; energy of, 656 ; sec- ondary discharge of, 662. Conduction, electric, 668 ; of heat, 227, 287. Conservation, of electricity, 638; of energy. 110, 115, 218, 810; of linear momentum, 78; of mass, 65. Conservative forces, 107. Consonance, 414. Contact, difference of potential, 681 ; electrifi- cation, 625. Continuity of matter, 282. Convection, electric currents, 706 ; of heat, 227, 286. Converging lenses, 470, 475. Convex mirror, 452. Cooling, Newton's law of, 297. Coplanar forces, 96. Cords, vibrations of stretched, 851, 400 ; vocal, 407. Coulomb (unit of electricity), 729. Coulomb's law of electrostatic force, 640. Coulomb's law of magnetic force, 608. Couple, 100 ; theruio-, 294, G7'J. Critical angle, l.">r,. Critical temperature, state, etc., 278. Crookcs, fourth state of matter, 208. Crookes, radiometer, 204. Cross hairs of telescope, 506. Cryohydrates, '^t>2. Cryopnorus, 270. Crystalloids, 141. Crystals, biaxal and uniaxal, 542; expansion of, 281. Current, electric, 663. d'Alembert, laws of mechanics, 188. Dalton's law of mixtures of gases, 190, 195. Damping of vibrations, 828. Daniell's cell, CMN d'Arsonval galvanometer, 718. da Vinci, lever, 123, 187. Davy's experiment on the nature of heat, 309. Davy's safety lamp, 289. Declination, magnetic, 618, 622. Density, 146 ; of a gas, 192 ; of a liquid, 147, 176; of a solid, 147, 165; of water, 14G, 14s. Depolarization of light, 560. Depression of the freezing point, 261. Descartes, laws of mechanics, 138 ; theory of rainbow, 517. Deviation, angle of, 459 ; minimum, 460. Dew, 267. Dew point, 268. Dewar flask, 228 ; liquefaction of gases, 280. Diamagnetic bodies, 595. Diaphragms, in optical instruments, 507. Diatonic scale, 416. 615. Dielectric constant, 6*1, 661. Dielectrics, 640. Differential notes, 412. Diffraction, around an edge, 887; through a small opening, 890. Diffraction, grating, 530. Diffuse reflection. 429. Diffusion, 140, 202. Diopter, 488. Dip, magnetic, 618, 620, 622, 739. Direct vision spectroscope, 514. Discord, 414. Dispersion, 423, 491, 508 ; anomalous, 514. Dispersive power, 512. Displacement, angular, 52 ; linear, 86. Dissociation, 201, 284 ; electrolytic, 692 ; heat of, 285. Distortion of waves, 848. Diverging lens, 480. Divided circuits, laws of, 722, 724. Dollond, achromatic lens, 493. Doppler's principle, 845, 898, 584. Double refraction, 541. Ductility, 17. Dulong and Petit's law of specific heats, 256. Duplex telegraphy, 716. Dutch telescope, 502. Dynamics, principles of, 59. Dynamo, 747 ; alternating, 750. Dynamometer, friction, 120. Dyne, 67. Ear, the human, 898. Earth, density of, 134 ; magnetism of, 619. Earth inductor, 789. Extra runvnt on breaking and making, 784, ot. of gases, 191; of liquid.-. r:..4-ucity. i:;y. Mi-; coefficients of, 144. "conductance. 668. . .induction through gases, 698. Electric current, 868; energy of, 664, • m:uriletiC enWt -f, M* ; |,,,.:iMin-m,-I,t --I, 668, 671;stea.! mitof, 672, 729. ic field, energy of, 649. Electric furnace, 667. Electric potential, «42 ; measurement of, 666. <• resistance, I *1 instrument.*, practical, 719. Electrification; 625; by induction, 688; energy Electricity, distribution of, 646 ; electro-mag- 678; electrostatic unit*, 640; poattta :,!„! negative. B7, 888; prurtir:,: units, 728 ; quantity of, 688, 674. ..f. mi ••>«--. •.»'. Electro-chemical i->.-tr...iii:ur:.,-!. Electro-magnetic force, 710. Deetro-nagBetk Induction, r.-t. |-:!,,-tr.,.,,,:»iri,.-ti,- unit,. .-,:::. 7'>. equivalent, 668. ne. •-magnetic waves along wires, 754. Beetro-Mfnetk wave* In -...-.••• .:. 60ft n:,Ttr...M...tiv,-f..r.-,.. 8tt; teAMtd. M; irenea! ..f. 806. <•:;. \--. •.,-.. .;M ; ..r. ,-,:,;: .t.ii,.l:ir,t -r. 780; Thomson, .',>•.•. . Beotrotafta, <-i DMtrofyta, 684. vters 7.0. •rdinary ray. 542. liuiiian, &90. Eyepiece, 494. t!itTin.. in,-t«T scale, 288. Falling »HKlies. law of. 44. Farad. I Faraday, diauiagnetism, 624; lee-pail experi- nifiit. JWtf ; indiii-i-d currents. 785; Influence lium in electro-magnetic phenomena. 595, 624, 656, 760 ; laws of electrolysla, 185; rotation of plane of polariration, 668, T66; tubes of induction, 615. Field, electric, 629 ; magnetic, 602. rind |, . n,t, „!:»:!„ M.I..I ,,,T,r. —.-;;. Flieau, measurement of velocity of light, 570, 572. Flt-xur. », sensitive, 192. bodies, 179. Flowing, in irregular tubes, 168; In uniform tubes, 168. Fluid pressure, 158. Fluids, properties of, 157. Focal length, 476. Focal lines, 485, 451, 466, 467. K.M-I. .-..ni'iiMt.'. 44-. .il. 484,486. Force effects, 67. Footpound. 11.' K.. ..-.-,! rfbratldM, tte Forces, 12, 60, 67 ; composition and resolution live, 107; measurement of. -••; |.aralU-l.«ram of, 75; Mtawll -i. 76; unit of, .',7. flit ilt, measurement of velocity of light, -•99. Fourier's theorem. 821. fo«rtt rt»»a of matter, IT. -.'i«. Fraunhofer. crating-., etc., 089. Fraunbofi-r llni-s. 511.587. Energy, 10ft. 1 18 ; conservation of. 1 10. 115, SI 8, nf\ totsnd, US, MB; IdMtic. io6, 114; potential, 109. 114. 117; of electric currents. 7*4; ofel,Ttr . ft .•t», «16. «19, 810; of magnetic in. rorve of radiation. 294. u«; steam. 971. ; • f • Uwied body. 97. 1" Kqnlpot. f a rigid M. SIT. . (topMMlot a£M] 1 * ... ' ••! . -4 .1 t'rnal. 189. 188. 2O2. indamental and i-artial vlt a,8»,897 '« A"/'. .liV,'":.i.-. ' '"'• "r lit; avtarMl, !!•>. to rohnw M; a«J < M; 19; rnVet of matter opon, US, of motion 44 46. 187; Btksrwmrw, •.-.-. t.-. TM r.vhua--. hwwt • .f, ••.-:. 'xolute. 9». »«; BiifNUTnt. «M : Gal .. 186 ; pltrh of soanda. 897 : tilnBgai. metar, «6; v^odty of Hgnt, 191; 674; tangent, 871. 764 INDEX of, 287, 246 ; in motion, properties of, 167, 191; internal work in. -Jl.". : kinetic theory of, 197; law for a, 7T= AM/7*. 1>W : lique- faction of, '279 ; specific heat ..f. •.•:••_' : " stand- ard conditions " of, 289 ; " velocity of sound/' 881 Gas constant //„. i'lo. •jii. Gas thermometer, 2'28, 80S. Gay-Lussac's law for a pas, 240. Geissler-Toepler air pump, 211. Gilbert, William, />« Jfagnete, 624, 759. Glaciers, motion of, 259. Glow lamp, 666. Graham, diffusion, etc., 140. Gram (unit of mass), 66. Gramme armature, 747. Graphical methods, 160, 196. Gratings, concave, 587 ; plane, 580 ; resolving Gravitation due to earth, 101, 130; universal, power of, 585. 129. Gravitational waves on liquids, 172, 839. Gravity, centre of, 101, 134; value at different latitudes, 181. Gregory's telescope, 498. Griffith's, mechanical equivalent of heat, 304. Ground ice, 259. Guard ring electrometer, 659. Guericke, von, 167, 759. Hall effect, the, 756. Harmonic motion, rotation, 55 ; translation, 48. Harmonic vibrations, composition of, 318, 322.' Harmonics, 321, 397. Harmony and discord, 414. Heat, flow of, 305; mechanical equivalent of. 226, 804 ; of evaporation, 269 ; of fusion, 260 ; of solution, 283 ; sources of, 215 ; specific, 250 ; transfer of, 227, 286. Heat effects, 216, 303. Heat energy, 219, 226. Heating effect of electric currents, 665, 730. Helmholtz, analysis of sounds, 396 ; explanation Henry, electrical waves, 656 ; induced electric of harmony, 414. currents, 738. Henry, the, unit of inductance, 745. Herschel, fluorescence, 578 ; infra-red radiation, 310. measurement of, 613. HerschePs telescope, 497. Homocentric pencils, 435. Homogeneous waves, 428. Hooke's law, 145. Horizontal intensity of earth's magnetic field, Horn blower, steam engine, 275. Horns, 858, 404. Horse power, 115. Humidity of air, 268. Huygens, clock, 186, 138 ; experiment in double refraction, 547 ; eyepiece, 494 ; fixed points of temperature, 226; impact experiments, 156; principle, 866; reversible pendulum, 136; theory of reflection and refraction, 307, 871 ; variation in "g," 181 ; wave surface for Iceland spar, 548. Hydraulic ram, 206. Hydrostatic press, 163. Ice, lowering of melting point of, by pressure, 258. Ice calorimeter, 250, 252. 789. Iceland spar, 541. linage.-, real and virtual, 435. Impact, 7*. l.V>. Impulse, C.'.t, 106. Incandescent electric lamp, 666. Incidence, angle and plane of, 869, 481. Inclination, earth's magnetism, 618, 620, 622, Inclined plane, 42, 119, 126. Independence o!' forces, principle of, 60. Index of refraction, 482, 455, 457 ; measure- ment of. 4;.C., 4»SO, 465, 506. Indicator diagram, nil, 274. Induced electric currents, 738. Induction, electro-magnetic, 707 ; electrostatic, Induction coil. 741. Inertia, 18, 14, 15; moment of, 88; principle 631, 645 ; magnetic, 597, 614. of, 59. Infra-red radiation, 293, 424. Insulators, 626. Intensity, of electric field, 641 ; of light, 487 ; of magnetic field, 609 ; of magnetization, 610 ; of sound, 397 ; of waves, 814, 828, 330. Interference of light waves, 374, 424, 519 ; of sound waves, 374 ; of waves on surface of liquid, 374 ; over long paths, 527. Internal energy, 218, •-'»:;. Internal work done when a gas expands, 248, 248, 253. Interval, musical, 416. Inverse square, law of, 180, 608, 640. Inversion, thermoelectric, 680. lonization of gases, 699. Ions, 689, 692. Isoclinal lines, 621, 628. Isogonal lines, 620, 622. Isothermals, 195, 267, 276. Jar, Leyden, 653. Jets of liquid, 189. Joly, steam calorimeter, 252. Joule, determination of mechanical equivalent, 303, 804, 810 ; internal work in a gas, '243. Joule, the unit of energy, 112. Jupiter, occultation of satellites, 421, 5G6, 574. Kaleidoscope, 446. Kathode (see Cathode). Kelvin, absolute scale of temperature, 307; electrometers, 658; expansion of gases through porous plugs, 243 ; thermoelectric- ity, 682. Kepler's laws, 132. Kerr effect, 565. Kinetic theory, of evaporation, 143, 264; of gases, 197, 241, 254; of matter, 142; of vis- KirChhofFs law of radiation and absorption, 301, cosity, 148, 202. 579. Knot, 37. Koenip, manometric flame, 858. Kundt, measurement of velocity |
of sound, 864. M& Lamp, arc, 665; incandescent, 666; Nernst, Langley, bolometer, 294, 732. Laplace, velocity of sound, 888. Latent heat, 809. Leibnitz, laws of mechanics, 188. Lenard rays, 708. Length, units of, 21. INDEX Lenses, 461, 467 ; achromatic, 498 ; combination ^7; converging, 475; dlvergii . Lever. Leydenjar, 658; oscillatory discharge of, 654, mes, vibrations of, 861. Mersenne, vibrations of stretched cords, 414. utensltv <>f, 437; interference T17. 429, 482, 465 ; standard - velocity ,.f. 4J1. :>66; wave theory of. Limiting value of a ratio, 28. ..1 for liquefaction of gases, S80. sph.-ri.-u! aberration, 4;-i. 1 Microscope, compound, 004 ; .vex, 408; 445 ; rotating, 444 ; measuring spedflc .m.ix ,,f. l-l; Uould elast elasticity of. 171 : expansion »f, £tt ; in mo- in revolution. Ussalous' figures, 8M. I Local action in primary cells, 677. Loops and nodes, 849. Luminescent Luminosity, 489. Lummer-Brodbum photometer, 448. Machine Magic lantern, 501. Magnetic effect of electric currents, 66- Magnetic elements of earth, 619. Magnetic Held, 602, 609; attraction and n-i.ul- sTon In. 617 ; energy of, 616, 71 slty of, 609 ; measurement of. 618. U»*M*i ten, Iwratiil; :.:,.- .,r. .-..;. Magnetic Induction, 597, 614. 780. llHMtteaMldlM -f.-arth. -'.I-. Magnetic moment, 610 ; measurement of, 618. Marnetie needle, 096 ; astatic, 669 ; vibrations - of elasticity. 144. ' M..:,-riibr Moknte, P.-. Moment, of a force, 86; of inertia, 88. Momentum, angular, 98 ; conservation of linear, 78; linear. 69. Monochromatic illuminator. 511. Moon, motion of, iai. Motion t, 56 ; laws of, 09. :ar sense, 12. 14. Musical instruments, 400. M.i.i.-a! notation, -un. Maaiml MtM/MA, 740. lamp. 666. . .•..tiiltt.riiim, 104. MaVnJtfc permeability, 609. Newoomb, velocity of lit), M:uM,,n, ,...;,.. Urft, ML agnetic rotation of the plane of polarltatlon, ''' ' Mar.-M, -'.t.-nnv IN in ii la for tho volocltv of a lonjrltuiiinal wave, .:i . ::.-.:!,:,•.. w-. M . JM ,,f ,,..,rv. In a circle, 47 ; telescope, 494, ' radiometer. 294. uXSSS&ff. Majfnlfvlng |H,w,.r. 4.'^. *sj Maloa, polarlxation of light by reflection. 001. Ume. 857. •>n of. 60; meannretnent rt6; and weight. 64. Matorial ' C.; enntiniiltv s. .:,...-.! Mon, mi ' 1 . 286. metar, 220. objtetlve Mean fre* path of rookcoU. SOI. »> ad vantage of machine*. 1«. M-.-h.^l.-i! ...;•'• - ' ' .t ,'l Mrm,,k.,,.,nt (M r« ..... p..) •:. :» ,ui bubble*, 18«, 244. . of. 7 7RI. 72». - ,• ..• . •••. trte oarrvni*. ",ti. 1-7 190. 766 INDEX Opacity, 427. Optic axis of crystals, 542. optically active bodies, 563. Ordinary and extraordinary rays, 542. Orir:in pipes, 355, 401, 402. Oscillations, electrical, 816, 654, 752. Osmosis, 141, 179. Osmotic pressure, 179. Overtones, 897. Parabolic mirror, 408, 454. Parallelogram of forces, 75. Partial vibrations, 322. Particles, reflection by fine, 480, 555, 589. Pascal's law, 1U-J. Path, of molecules, 202. Peltier K.M.F., 681. Pencils, astigmatic and homocentric, 485. Pendulum, compensated, 282; complex, 322; compound, 185 ; magnetic, 610 ; reversible, 136; simple, 74, 91. Penumbra, 426. Period of vibration, 48, 817 ; measurement of, 322. (jnograph, 396, 407. Permeability, magnetic, 609. Phase of simple harmonic motion, 49. — logiston, 309. Phosphorescence, 299, 578, 588. Photo-electric action, 699. Photographic lens, 500. Photometer, 442. Photometry, 437. Physical quantities, 10 et seq. Physics, 7, 9 ; divisions of, 10. Piezometer, 150. Pigment colors, 587. Pile of plates, 554. Pin-hole images, 426, 495. Pitch, of a screw, 127 ; of sounds, 397 ; stand- ard, 418. Plane of incidence, 431 ; of polarization, 554. Plasticity, 17. Plate, refraction through, 458, 466. Plates, colors of thin, 523 ; vibrations of metal, 360. Platinum thermometer, 223. Points, effect of, on electric charge, 633. Polarization, angle of, 552 ; by double refrac- tion, 547; by reflection, 551, 554; colors due to, 557 ; plane of, 554 ; rotation of, 563, 750. Polarized waves, circularly and elliptically, 818, 560, 562; interference of, 556; plane, 313, 555. Polarizer, 560. Porosity, 18. Potential, electric, 642. Potential energy, 109, 117; and force, 114. Pound, «'•('•. Power, 115. Practical system of electrical units, 728. Pressure, 158; atmospheric, 166, 176; centre of, 174 ; unit of, 165 ; in a bubble or drop, 185 ; in a gas, 158, 198. Pressure in a gas, measurement of, 178, 196. Pressure in liquids, 158; due to cohesion, 162; due to gravity, 163; due to surface-tension, 185. Prevost's theory of exchanges, 297. Primary colors," fi'.i''. Primary electric cell, 676; energy of, 697. Principal axes of rotation, 94. Principal section of a crystal, 548. Prism, angle, edge, face, 458, 466; resolving power of, 510. Projectile, 45. Projection lantern, 501. Projection of a line or area, 28. Pulley, 128. Pulses, 342, 348, 394, 409, 425, 704. Pump, air, -joy ; water, 208. Quadrant electrometer, 658. Quality of sound, 895. Quantity of heat, 226, 809. Quarter- wave-plate, 561. Radian, 24. Radiant energy, 290. Radiation, 290 ; and absorption, 298, 576. Radioactive bodies, 7'H. Radiometer, Crookes', 204, 294. Rainbow, 516. Ram, hydraulic, 206. Ramsden eyepiece, 494. Ratio of specific heatsof a gas, 253, 254, 337. Ray of light, 881, 425. Reaumur temperature scale, 223. Rectilinear motion, 43, 70. Rectilinear propagation of light, 381, 425. Reed pipe, 403. Reflection, 383, 367 ; angle of, 369, 431 ; of , ' 8b < . ether waves, 867, 429, 430. Reflection of light, concave surface, 447 ; con- . vex surfacju.452 ; diffuse, 429; fine particles, regular, 430 ; total, 428, 456. Reflection of sound, 367, 408; diffuse, •Ir- regular. 430 ; total, 428, 456. Refraction, angle of, 373, 432; double, 4f><> ; in- dex of, 432, 455, 457, 460, 506. Refraction of light, concave surface, 467 ; con- vex surface, It'i!) ; plane surface, 371, 432, 455, 462 ; plate, 458, 466 ; prism, 458, 466. Refraction of sound, 408. Relay, 715. Residual electric charges, 662. Resistance, electric, 723, 726, 781 ; standards of, 729. Resolution offerees, 75. Resolving power, of grating, 535; of lens, 483 ; of prism, 510. Resonance, 292, 328. Resonator, Helmholtz, 396. Restitution, coefficient of, 144. Resultant, non-parallel forces, 96; parallel forces, 98. Reversible engine, 300. Reynolds and Moorby, mechanical equivalent of heat, 304. Right-handed-screw law. 58, 671. Rigidity, 1 ; coefficient of, 152. Ripples, 190, 840. Rods, vibrations of, 858. Rotner, determination of the velocity of light. 421, 566. polarization, 563, 756. Rontgen rays, 703. Rotation, motion of, 35, 52, 86 ; of the plane of Rotor, M. Rowland, convection currents, 706; dividing engine, .r):','.i ; gratings, 538, 540; measure ment of the mechanical equivalent of beat, 304. Rumford, experiment on the nature of beat, num and i lined pla ' plane, nulloy*. 76, 196. .:.».! lOlM lU lUwp«lM,ftL mometara, 994. rtk Sauveur. beau in sound, 418. Scattering of light by fine particles, 480, 666, Schuster and Gannon, mechanical equivalent MatfEttoft, iftrihr, 4-.-J. Beraw, i -.••;. Second, mean solar, 99. Bseoadan Mtttry, M Secular change, magnetic, 690; of MO in ther- mometars, 994. Self-Induction, 748. -h.-arir.ir strait, ...-. ,-.,:;,. :il. •.;..:,. r,-29. motion, 48; composition of, irve, 819. barometer, 177. 404. Size and shane, properties of, 16. or, St hart, MB. ' -.1. :,,i.|..V.M. 7-7. • '':••»' »f. i I'.1. SolkliflcaUon. chance of volume chance of volume on, 968. -j..i. I7B, Ml; h.-at ,,f. -.•<{. ....,.! •., MMTV ..f. 684: fr,-.-/ir,»r i-ii.t • . '. f, 04i fr,-.-/ir,»! of, 961 ; vapor preMore of, 976. Sounds. 898; analysis of. 896: limit of andl- Mhtv. §f8; loudnees. 897; |.itrh. 897; .|.i»l- tty. 896 ; .yn thesis of, 897. Sound waves, reflection and refraction of, 408; Sparks, electric, 646. 699. ' r. 147. B t M • • • Btarma, BM^MUB, 9J8. Straight line, motion In a, 40, 48, 70. >tr:»ii,. 146: -:.,t;:,. ,-.,•,-.• Strength of a couple, 100. Strew, 144. Stria- in vacuum tubes, 700. Stringed Instrument*. 400. tetefi, s.;... 'v oi b u mm mtmt IM. :w.; Sublimation, 282. Summational notes. 419. Son. heat of, 948 ; spectrum, 611, 687, 6m M.rJ-i.-.-ut-.ri.ti,,.,. -'•.', H& Surfan- t.-ri-.i..ii. 1-.' Synthesis of musical notes, 897. vibrations of, 851, 400. Tangent galvanometer, 671. T,-l,rru,,h. 71.V Telegraphy without connecting wtrea, 666. Telescope, 496, 601 ; resolving power of, 486. :tture, 290; change in, as heat eflect, 247 ; of inversion. 660 ; on kinetic theory oi gases, 197, 947. Tempered scale, equally, 417. T.-rn-trial BMMMdSA. 61& Thermal unit/996. principles o< 88ft, fco: air. 298: ciinkal, 996; tm and minimum, 996 . X{. '.-.••-• . • • •-;. ThMMOl Y. M K . 8M M nits of, 92. rlment, 167. TorrieelH's law of efflux, 181 498,466. Itonlm i.vj T-tnl r, Il,,-ti,,n. ToonMte*, M:- Transformer, 746. • i. mkrtta . BOH • oC •. '•:. TrariMoltter. 717. i • i| I r:ii,-|..ir.-ti.-x . JJT 814 Mb , . 614 ; mrmtfaff. 586 H,,,.,-tr,nii. .".'I. I.'-!. BI6, .'-I ; .t.,,.ri,Il..-: '- i;itn, xl,,U,thjrht,4$4. Speed, anmilar. 68 ; Hnew. 87. \ M-,x:,i,rx.t,!v M.' rr,,.v sjatln BMMll I MlNOoVsM, '.'i-., 4-AA . . * ( OiU 1 IMffMiML W { '' '' S ••' '•••» \.n,t.-, \\,-, •-'. . .,-. • (ML Stefkn'* law ..f radlatioo, 996. 768 INDEX Variation, magnetic, r>l-. -. •.'.">; composition :iinl resolution, •_'('>. Velocity. angular, M ; composition and resolu- tion i>f. :!7. M ; linear, 86; of ether \va\c.-. 4-_'l. 7M ; of M.unil waves, 837, 864; of water waves, 889. Vi-rtical circle, motion in a, 112. Vertical force, magnetic, 618. Vibration, electric, 316, -654; harmonic, 48, :U7 : of bells, 861 ; of columns of gas, 868, 400; of pUtMADtT membranes, 860: of rods, ••.' : of stretched cords, longitudinal,851, :;.>. KM) ; stationary, 849. Virtual images, 485. " Viscosity, 17, 189, 168, 202. Voice, the human, 407. Volt (unit of E. M. F.) 678, 780. Volta, condensing electroscope, 667; electro- phorus, 686. Voltaic cell, 676 ; source of energy of, 697. Voltameter, 687. Volume, change in, as heat effect, 229 ; work required |
to produce, 159. Vowels, characteristics of, 407. "Wallis. impact experiments, 156. Water, density of, 148, 286 ; equivalent, 251 ; expansion of, 236 ; waves, 172, 389. Waterman, calorimeter, 251. Watt (unit of power), 115. Watt, James, steam engine, 275. Wave front, 330, 541. Wave length, 344 ; of light, measurement of, 424. Wave motion, 811, 392. Wave normal, 544. Wave number. :?•!•_'. Wave surface in uniaxal crystals, 548. Waves, composition of, 847; elastic, 812; elec- tric, t;r>,r>, 7M ; intensity of. o!4, 330 ; on sur- face of liquids, :<:;<) ; primary and secondary, :;<;•; : velocity of. :'.:',((. Weber, law of radiation, 296. Weight, 12, 14, 101, 184; relation botud, mass and, 64. Wheatstone's bridge, 725. Whispering gallery, 409. White light, decomposition of, 423, 4- 577. Wiener, stationary light waves, 4-20, 529. Wind instruments, 401. Wireless telegraphy, 656. Wollaston, cryophorus, 271. Wood, It. W., anomalous dispersion, 516; sound waves, 410 ; treatment of caustics, 451. Work, 105; measure of, 111; necessary to change volume of a fluid, 159. Wren, impact experiments, 156. X-rays, 704. Yard, 22. Yellow spot in eye, 591. Young-IIelmholtz theory of colors, 592. Young's interference experiments, 374, 521. Young's modulus, 154. Young's principle, 874. Zeeinan effect, 758. Zero, absolute, 240, 807. |
emember that you can only use this equation when acceleration is constant, it is not true otherwise. We also have an equation that defines average velocity and is true in all cases, v = v0 + v 2 . (2.9) Let’s solve this equation for ∆x, and use the new equation for v, v = ∆x t . ∆x = vt, ∆x = ∆x = v0 + v 2 t 1 2 (v0 + v)t, (2.10) (2.11) (2.12) to get an expression for displacement in terms of initial and final velocities. We can get an even more useful relationship by eliminating the final velocity. If we use Eq. (2.8) to substitute for the final velocity, ∆x = 1 2 (v0 + (v0 + at))t ∆x = v0t + 1 2 at2. (2.13) This expression is often handy because it does not contain the final velocity. In many cases, we have information about the start of motion, but we rarely have information about the end of it (that’s usually what we are trying to predict). We can get another useful relationship by solving Eq. (2.8) for time, t = v − v0 a , (2.14) and substituting into Eq. (2.12), v − v0 a ∆x = ∆x = (v0 + v) 1 2 v2 − v2 0 2a v2 = v2 0 + 2a∆x. This equation is very handy if information about time is not given in the problem; it still allows us to calculate the final velocity without knowing how long the object was accelerating. 19 Example: Car Chase A car traveling at a constant speed of 24.0 m/s passes a trooped hidden behind a billboard. One second after the speeding car passes the billboard, the trooper sets off in chase with a constant acceleration of 3.00 m/s2. (a) How long does it take the trooper to overtake the speeding car? (b) How fast is the trooper going at that time? Solution: We first need to be clear about how we are measuring time and position in this problem. Let’s measure them both relative to the trooper’s motion. That is, x = 0 at the billboard where the trooper starts moving and t = 0 when the trooper starts moving. This means that the speeding car passed the trooper at t = −1 s. (a) Now we need to think about what the problem is asking us to find. It asks for a time (“how long”) at which the trooper overtakes the speeding car. At the exact moment that the trooper overtakes the car, their positions on our axis will be equal, i.e. xt = xc. So, we want to find the time at which the positions are equal. We have just derived an equation which gives position as a function of time for constant acceleration, ∆x = v0t + 1 2 at2. First check that we can use this equation; are both objects undergoing constant acceleration? Yes, the problem says the trooper has a constant acceleration and the car has constant velocity (a constant acceleration of 0). So let’s figure out the position of the trooper. The trooper’s initial position is x0t = 0, his initial velocity is v0t = 0, and his acceleration is a = 3.00 m/s2, so we have xt = 1 2 at2. Now we do the same for the car. It’s initial position is x0c = 24.0 m (it had a 1 s head start), it’s initial velocity is v0c = 24.0 m/s, and it’s acceleration is ac = 0, so we have xc − x0c = v0ct xc = x0c + v0ct. To find the time at which positions are the same, we set the two expressions equal and solve for time, xt = xc at2 = x0c + v0ct 1 2 at2 − v0ct − x0c = 0. 1 2 You get a quadratic expression for t, so you will need to use the quadratic formula (it’s in your textbook), t = t = v0c ± 24.0 m/s ± 2 a (−x0c) 0c − 4 1 v2 2 a 2 1 (24.0 m/s)2 − 4 1 2 1 2 · 3.00 m/s2 2 · 3.00 m/s2 (−24.0 m) t = 16.9 s. There is also a negative root, but since we know that the trooper could not overtake the car before it passed him, this root is not physically meaningful. 20 (b) The trooper’s speed at that time is a fairly straightforward problem. We have an equation that gives us final velocity if we know the acceleration, time and initial velocity (and we know all those now). v = v0 + at v = (0) + 3.00 m/s2(16.9 s) v = 50.7 m/s. As mentioned earlier, constant acceleration is particularly useful because objects moving due to the gravitational pull of an object will move with a constant acceleration if air resistance is neglected. Objects moving under the influence of gravity and without air resistance are said to be in free fall. Note that this does not mean that the object necessarily started from rest (v0 = 0). Objects can be moving upward, like when you throw a ball, and still be considered free-falling. The magnitude of the free-fall acceleration is denoted by g and has a value of 9.80 m/s2 on Earth (although it varies slightly depending on latitude). The direction of g is always towards the large object creating the gravitational pull. Because g is a constant, we can use all of the equations we just derived for constant acceleration. Example: Rookie Throw A ball is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The ball just misses the edge of the roof on its way down. Determine (a) the time needed for the ball to reach its maximum height, (b) the maximum height, (c) the time needed for the ball to return to the height from which it was thrown and the velocity of the ball at that instant, (d) the time needed for the ball to reach the ground. Neglect air drag. Solution: You can refer to the figure in your textbook (Fig. 2.20) to get a visual idea of what’s going on in the problem. We are expressly given an initial velocity and an initial height in the problem. There are a few other pieces of information that are not expressly stated in the problem. The ball is in free fall, so we know the acceleration, g = 9.80 m/s2, and we know that the velocity of the ball at it’s maximum height will be zero. Let’s set up our coordinate system so that y = 0 corresponds to the top of the building; this means that the bottom of the building is at -50.0 m. (a) The first part of the problem is concerned with the upward motion of the ball. We are given an initial velocity, we know acceleration and a final velocity, and we want time. We have an equation that contains all these quantities, v = v0 + at v − v0 a t = t = 0 − 20.0 m/s −9.80 m/s2 t = 2.04 s. 21 (b) Now we want to know the maximum height reached by the ball. We have all the information from before and we now have the time at which we reach this maximum height, so we can use y = y0 + v0t + 1 2 at2 ymax = (0) + (20.0 m/s)(2.04 s) + ymax = 20.4 m. 1 2 (−9.80 m/s2)(2.04 s)2 (c) Now the ball begins to move downward and we want to know how long it takes to get back to it’s initial height. We can use the same distance equation, but now use the fact that yf = 0 to solve for time. y = y0 + v0t + 1 2 at2 v0t + t(v0 + 1 2 1 2 at2 = 0 at) = 0. We will have two roots, t = 0 because the ball starts at that height, and the one we really want, v0 + 1 2 at = 0 t = − t = − 2v0 a 2(20.0 m/s) −9.80 m/s2 t = 4.08 s. Note that this is twice the time it takes to get to the maximum height. The velocity of the ball at that time is v = v0 + at v = 20.0 m/s + (−9.80 m/s2)(4.08 s) v = −20.0 m/s. (d) Now our final position is yf = −50.0 m, but this is essentially the same problem we just solved. y = y0 + v0t + 1 2 at2 1 2 at2 + v0t − yf = 0. This leads to a quadratic equation, so it’s a little more effort to solve, t = t = v0 ± 20.0 m/s ± 2 a (−yf ) 0 − 4 1 v2 2 a 2 1 (20.0 m/s).80 m/s2 2 · 9.80 m/s2 (−(−50.0 m)) t = 5.83 s. 22 Chapter 3 Vectors and Two-Dimensional Motion We live in a three-dimensional world and the objects around us move in that three-dimensional space. While some simple examples of motion can be described in one dimension, extend what we’ve learned about motion to more than one dimension will allow us to study many more systems. 3.1 Vector properties Displacement, velocity and acceleration are all vector quantities. That is, they have a magnitude and a direction. In one dimension, there were two options for the direction, left or right, and we could represent the direction of displacement, velocity, or acceleration simply by using a negative or positive sign. When we move to two dimensions, there are many more options for the direction of the vector and we will need to be more formal with the mathematics of vectors. Vectors are represented graphically as arrows with the length of the arrow representing the magnitude of the vector and the direction of the arrow giving the direction of the vector. This graphical representation might help you understand the basic arithmetic of vectors. Mathematically, we denote a vector, A, with an arrow over the variable name. If we use the variable name without the arrow, this means we are referring to the magnitude of the vector. Let’s set up a coordinate system at the starting end of the vector (see Fig. 3.1). Then the vector points to a specific location in that space. We know that we can give the coordinates of that location using either Cartesian, (x, y), or polar, (r, θ) coordinates. The polar coordinates of this point are quite straightforward; r is the length (magnitude) of the vector and θ is the angle between the vector and the x-axis. We can also determine the x and y coordinates by finding the projections of the vector on the x- and y-axes. The projection of a vector A along the x-axis is called the x-component and is represented by Ax. The projection of A along the y-axis is called the y-component and is represented by Ay. We can find the x and y components by converting the polar coordinates to rectangular coordinates, Ax = A cos θ Ay = A sin θ. (3.1) Note that we can go from the vector’s x- and y-components back to the polar representation using the Pythagorean theorem and the definition of the tangent: x + A2 A2 = A2 y Ay Ax . tan θ = 23 (3.2) Figure 3.1: The projections of a vector on the x- and y-axes. Projections give the x- and y components of a vector. Equality of vectors Two vectors are equal only if they have the same magnitude and direction (two arrows are the same only if they are the same length and point in the same direction). This means that you can move a vector around in space as long as you don’t change the length or direction of the ve |
ctor. Adding vectors Just as when you are adding scalar quantities, you must ensure that the vectors you are trying to add have the same units. Vectors can be added either geometrically (graphically) or algebraically. To graphically add two vectors, A and B, draw the vectors (using the same scale for both) head to tail on a piece of paper. The resultant vector R = A + B is the vector drawn from the unmatched tail to the unmatched head (see Fig. 3.2). You can lay out many vectors head to tail to find the result of them all. While graphical addition of vectors is useful for visualizing the addition process. You will most often be adding the vectors algebraically. To add vectors algebraically, resolve the vectors into their x- and y- components. All the x-components are added to get the x-component of the resultant vector. All the y-components are added to get the y-component of the resultant vector. Never add the the x-components to y-components. You can get the magnitude and direction of the resultant by converting from the Cartesian representation (x and y components) using the equations presented earlier. Negative of a vector The negative of a vector, A, is defined as the vector that gives zero when added to A. If you think about the graphical addition of vectors, this means that the negative of A must have the same magnitude as A, but opposite direction (180◦ difference). Subtracting vectors Subtraction is simply the addition of a negative quantity. Since we have now defined the negative of a vector, we can figure out how to subtract. Remember that the negative of a vector points in the opposite direction, so for subtraction we flip the direction of the vector to be subtracted, but otherwise use the same graphical method as for addition. Graphically flipping a vector corresponds to algebraically changing the sign of both the x- and y- components of the vector, so we can use the same algebraic method for subtracting vectors as we do for adding vectors. 24 AAxAyθ Figure 3.2: Graphical addition of vectors. To graphically add vectors, the vectors are drawn head to tail. The resultant vector is the vector that connects the two “loose ends”, drawn from tail to head. Example: Take a Hike A hiker begins a trip by first walking 25.0 km 45.0◦ south of east from her base camp. On the second day, she walks 40.0 km in a direction 60.0◦ north of east, at which point she discovers a forest ranger’s tower. (a) Determine the components of the hiker’s displacements in the first and second days. (b) Determine the components of the hiker’s total displacement for the trip. (c) Find the magnitude and direction of the displacement from base camp. Solution: (a) Let’s set the origin of our coordinate system at the camp and have the x-axis pointing east and the y-axis pointing north. On the first day, the hiker’s displacement, let’s call it A has a magnitude of 25.0 km with a direction θ = −45◦ — it’s negative because she is moving south of the x-axis. We can use Eqs. 3.1 to find the x- and y-components, Ax = A cos θ = (25 km) cos(−45◦) = 17.7 km Ay = A sin θ = (25 km) sin(−45◦) = −17.7 km. On the second day, her displacement, let’s call it B, has a magnitude of 40.0 km with a direction θ = 60.0◦ — positive this time because it is north of east. The x- and y-components of this displacement are Bx = B cos θ = (40 km) cos(60◦) = 20.0 km By = B sin θ = (40 km) sin(60◦) = −34.6 km. (b) The total displacement is the vector sum of A and B. We’ve just found the components of A and B, so we can find the components of the total displacement, Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km Ry = Ay + By = −17.7 km + 34.6 km = 16.9 km. 25 (c) We just found the components of the total displacement, all we need to do is convert them to a magnitude and direction. The magnitude is found using the Pythagorean theorem, x + R2 y R2 = R2 R = (37.7 km)2 + (16.9 km)2 R = 41.3 km. The direction is found using the tangent function, tan θ = tan θ = Ry Rx 16.9 km 37.7 km 16.9 km 37.7 km θ = tan−1 θ = 24.1◦. Don’t forget to check that the angle is in the correct quadrant. components are both positive, so we are in the first quadrant and the angle is correct. In this case, the x- and y- 3.1.1 Displacement, velocity and acceleration in two dimensions In one dimension, we defined the displacement as the difference between the initial and final positions of an object. The position in that case was determined by a single coordinate. We now want to define displacement in two dimensions where the position of the object is given by two coordinates. Let’s call the initial position of the object ri and the final position of the object rf , where r is a position vector that goes from the origin to the position of the object. The displacement is defined as the vector difference between the initial and final position vectors, ∆r = rf − ri. (3.3) With this generalized definition of displacement, we can also generalize the average velocity and average acceleration of an object, vav = aav = ∆r ∆t ∆v ∆t . Finally, we can also generalize the instantaneous velocity and instantaneous acceleration v = lim ∆t→0 a = lim ∆t→0 ∆r ∆t ∆v ∆t . (3.4) (3.5) (3.6) (3.7) If you look carefully at these definitions, you will see that x-component of acceleration is determined by the x-component of velocity which is determined by the x-component of the displacement. The same is true for the y-components of these quantities. This means that the horizontal and vertical components can be treated independently of each other. 26 Example: The swimmer The current in a river is 1.0 m/s. A woman swims across the river to the opposite bank and back. She can swim 2.0 m/s in still water and the river is 300 m wide. She swims perpendicular to the current so she ends up downstream from where she started. Find the time for the round trip Solution: Since the woman swims perpendicular to the current let’s define the y-axis as parallel to the river. We can treat the x and y motion independently. We are only interested in the motion in the x-direction (across the river) since this will determine how long the trip takes. She is swimming at a constant velocity of 2.0 m/s, so the time to travel a distance of 300 m is vx = t = t = ∆x t ∆x vx 300 m 2.0 m/s t = 150 s. It will take the same amount of time for her to travel back, so the round trip takes 300 s. Note that the current pushing the woman down the river is completely irrelevant here because it affects her motion in the y-direction (along the river) and this is independent of her motion in the x-direction. 3.2 Motion in two dimensions We have previously studied motion of objects moving in a straight line (one dimension). We will now extend our study to two dimensions. We know that if we break motion up into x and y components, that the motion in the two directions is independent, so that motion in the horizontal direction does not affect motion in the vertical direction and vice versa. This is particularly important when studying something called projectile motion which is the motion of any object thrown in some way. If we throw a ball with some horizontal initial velocity, its motion can be studied by breaking it up into the horizontal and vertical motions. In the vertical direction, the object undergoes acceleration due to gravity just as in free fall. In the horizontal direction, there is no acceleration and the velocity remains constant. The resulting two-dimensional motion is the combination of the two components. 27 Chapter 4 Laws of motion So far, we’ve studied motion by describing what happens without being concerned about what causes the motion. Now, we will start to examine the causes of motion and we will learn the rules that govern changes in motion. 4.1 Newton’s first law Isaac Newton developed the laws of motion in the 1600s when he started thinking about why objects close to the Earth tended to fall to Earth unless something was holding them up in some way. His ideas on motion are summed up in three laws that are based on the idea of forces. You probably have an intuitive sense of a force from everyday life. When you push or pull on an object, you are applying an external force on the object. These are examples of contact forces, forces which are caused by one object being in contact with another object. There are also forces that arise without contact of two objects. While this may seem strange (Newton was also uncomfortable with the idea of action-at-adistance), you are very familiar with one such force. Gravity causes all objects near Earth to fall towards the Earth even though the Earth is not touching the object. This is an example of a field force, so called because scientists use the idea of a force field emanating from an object to explain how it might affect the motion of objects that it hasn’t touched. Essentially a force is something that can change the state of motion of an object. Note that force (contact or field) is a vector — it has both a magnitude and direction. If a force is something that can change the state of motion of an object, will objects move without a force? Obviously, if an object is at rest (not moving), it will just sit there forever unless something pushes or pulls it. Suppose now that the object is given a quick push. It will start moving because of the force that has been applied, but what happens after the initial push? In most cases, the object will start to slow down because there is friction between it and the object on which it moves. But suppose we could eliminate the friction, which is a force and changes the motion of the object? If we completely eliminate friction, then the object would continue moving without speeding up or slowing down. So yes, objects will move without the presence of a force, but with a very specific type of motion. This is the essence of Newton’s first law, “An object moves with a velocity that is constant in magnitude and direction unless a non-zero net force acts on it.” The net force is the vector sum of all the for |
ces acting on the object. So this law can be used in two ways. If we know that there is no net force on the object, then we know that it will continue moving with a constant (possibly zero) velocity. Alternatively, if an object is moving with a constant (possibly zero) velocity, then the net force acting on it must be zero. This law is based on the notion of inertia which is the tendency of an object to continue its state of motion in the absence of a force. The law is sometimes stated as “a body in motion will stay in motion and a body at rest will stay at rest unless acted upon by an outside force.” This is closely related to the idea of 28 mass, which measures an object’s resistance to changes in its velocity due to a force. If the same force acts on two objects with different masses, the object with the smaller mass will experience a bigger change in its velocity than the more massive object. 4.2 Newton’s second law In the absence of a force, an object will keep doing whatever it was doing. What happens if a force acts on the object? Clearly it’s velocity will change in some way. If you push on an object, it will accelerate (change its velocity). If you push harder, it will accelerate faster. So the magnitude of the force is related to the acceleration. In fact, the force is proportional to the acceleration; so if you push twice as hard the acceleration will be twice as large. What other things might affect the acceleration? One quantity that we’ve already discussed is the mass. The same force applied to objects with different masses will result in different accelerations. In this case the relationship is inversely proportional — if the mass is twice as large, the acceleration is halved. Newton put both of these observations together into his second law “The acceleration a of an object is directly proportional to the net force acting on it and inversely proportional to its mass.” We can write this statement more compactly using mathematics a = F m , (4.1) where a is the acceleration (vector) of the object, F is the vector sum of of all the forces acting on the object and m is the mass of the object. To actually use this equation, we break it up into its x and y (and maybe z) components Fx = max Fy = may. (4.2) Note that the unit of force in the SI system is the newton where 1 N = 1 kg · m/s2. 29 Example: Horses pulling a barge Two horses are pulling a barge with mass 2.0 × 103 kg along a canal. The cable connected to the first horse makes an angle of θ1 = 30.0◦ with respect to the direction of the canal, while the cable connected to the second horse makes an angle of θ2 = −45.0◦. Find the initial acceleration of the barge, starting at rest, if each horse exerts a force of magnitude 6.00 × 102 N on the barge. Ignore forces of resistance on the barge. Solution: We’ve been given the mass of an object and the forces acting on it and we’re asked to find accelerations. So we want to use Newton’s second law to try to find the acceleration. Let’s define our coordinate system with the x-axis lying along the canal (so the angles are measured relative to the x-axis). Now we can break down the forces into x and y components, F1x = F cos θ1 = (6.00 × 102 N) cos(30.0◦) = 5.2 × 102 N F1y = F sin θ1 = (6.00 × 102 N) sin(30.0◦) = 3.00 × 102 N F2x = F cos θ2 = (6.00 × 102 N) cos(−45.0◦) = 4.24 × 102 N F2y = F sin θ2 = (6.00 × 102 N) sin(−45.0◦) = −4.24 × 102 N. Newton’s second law tells us to find the net force in both the x and y directions, Fx = F1x + F2x = 5.2 × 102 N + 4.24 × 102 N = 9.44 × 102 N Fy = F1y + F2y = 3.00 × 102 N − 4.24 × 102 N = −1.24 × 102 N. Now Newton’s second law says that the net force is related to the acceleration, ax = ay = Fx m Fy m = = 9.44 × 102 N 2.0 × 103 kg −1.24 × 102 N 2.0 × 103 kg = 0.472 m/s2 = −0.062 m/s2. We have the x and y components of the acceleration, so we can find the magnitude and acceleration, a = x + a2 a2 y = 0.476 m/s2 θ = tan−1 ay ax = −7.46◦. 4.2.1 Weight The weight of an object is not the same as its mass. The weight of an object is the magnitude of the gravitational force acting on an object, so on Earth the weight is Fw = mg. Because g is the same everywhere on Earth, we can use an object’s weight to determine its mass and so the two terms are used interchangeably in everyday language. While mass is a fundamental property of an object and will not change if you move the object to another location, its weight can change depending on the object’s location. 4.3 Newton’s third law Newton’s third law is perhaps the least intuitive of the three laws of motion. According to Newton, all forces come in pairs, 30 “If object 1and object 2 interact, the force F12 exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force F21 exerted by object 2 on object 1” The force exerted by object 1 on object 2 is sometimes called the action force and the force exerted on object 2 by object 1 is called the reaction force. The law is sometimes stated as “every action has an equal and opposite reaction”. Basically anytime two objects interact in some way, there will be two forces, one acting on each object. When you walk, your foot pushes on the floor and the floor pushes back on you. When you lean against a wall, the wall pushes back on you. Every time an object falls towards Earth because Earth’s gravity is pulling it, the object also pulls Earth towards it. This may seem strange, but the Earth is so much more massive than other objects that its acceleration due to this force is negligible. One consequence of this law is a force called the normal force. Every object on Earth is being pulled towards the center of the Earth by gravity. Most objects are not moving downward because they are sitting on some surface and this surface is pushing up on the object. The upwards force is the normal force, so named because it is always perpendicular to the surface; this mean it does not always point straight up even though it is a reaction force to the pull of gravity. Example: Standing on a crate (a) A 38 kg crate rests on a horizontal floor, and a 63 kg person is standing on the crate. Determine the magnitude of the normal force that the crate exerts on the person. (b) Determine the magnitude of the normal force that the floor exerts on the crate. Solution: (a) Let’s consider the forces acting on the person only. There is, of course a downward force due to gravity Fg = mpg. There is also a normal force from the crate pushing up on the person. The man is not moving so these two forces must be in equilibrium (the net force must be zero), −Fg + Fcp = 0 Fcp = mpg. (b) Now let’s look at the forces acting on the crate. Gravity acts on the crate, Fg = mcg, and the floor pushes up on the crate through the normal force, Ff c. The person standing on the crate also pushes down on the crate with a “normal” force that is equal in magnitude and opposite to the force of the crate pushing on the person, Fpc = −Fcp. The crate is not moving,so these forces must be in equilibrium, Ff c − Fg − Fpc = 0 Ff c = mcg + mpg. When we are using Newton’s laws to solve problems, we use several assumptions. We assume that each object is a point mass, or that they are particles without any spatial extent (0-dimensional objects). This means we don’t have to worry about rotation of the objects. If strings or ropes are part of the problem, we assume that their mass is negligible and that any tension in the rope is the same at all points on the rope. When solving force problems, it is useful to draw a free-body diagram. The free body diagram is a drawing of all the forces acting on a particular object. It is very important to only draw the forces acting on the object, any force that the object exerts on its surroundings is not included in the free-body diagram. This diagram helps to isolate the forces of interest for our object and can then be used to apply Newton’s laws. 31 Example: Traffic light A traffic light weighing 1.00 × 102 N hangs from a vertical cable tied to two other cables that are fastened to a support. The upper cables make angles of 37◦ and 53◦ with the horizontal (see Fig. 4.14 in your textbook). Find the tension in each of the three cables. Solution: We start by drawing free-body diagrams. First for the traffic light which has gravity acting downward (the weight, W and tension, T3 from the rope pulling it upward. Because both forces are in the y direction only, this leads to a single equation, T3 − W = 0. Although we now know the value of T3, this does not tell us anything about the tension in the other two ropes. We can also consider the forces acting on the knot. The knot has three ropes pulling on it: tension downwards from T3, tension up to the left from T1 and tension up to the right from T2. Because forces are vectors, we need to break everything into x and y components and apply Newton’s second law along each axis. Note that the knot is in equilibrium, so there is no acceleration in either direction, x − direction : y − direction : T2 cos(53◦) − T1 cos(37◦) = 0 T1 sin(37◦) + T2 sin(53◦) − T3 = 0. We know T3 from the first equation, so we are left T1 and T2 as unknowns. Luckily, we have two equations for our two unknowns. so we can solve one equation, T2 = T1 cos(37◦) cos(53◦) and substitute into the other equation, T1 sin(37◦) + T1 cos(37◦) cos(53◦) sin(53◦) − W = 0 T1(sin(37◦) + cos(37◦) cos(53◦) sin(53◦)) = W T1 = sin(37◦) + cos(37◦) W cos(53◦) sin(53◦) T1 = 60.1 N. We can now find T2 as well, T2 = (60.1 N) T2 = 79.9 N. cos(37◦) cos(53◦) 4.4 Friction An object moving on a surface or through some medium encounters resistance as it moves. This resistance is called friction. Friction is an essential force as it allows us to hold objects, drive a car and walk. The strength of the frictional force depends on whether an object is stationary or moving. You’ve undoubtedly had the experience of trying to push a large heavy object; it takes more effort to get the object moving than to keep it moving once it’s go |
ing. When an object is stationary, the frictional force is called the force 32 of static friction and when the object is moving, the frictional force is called the force of kinetic friction. Friction points opposite to the direction of motion in the kinetic case or the direction of impending motion in the static case. The force of static friction is larger than the force of kinetic friction. It has been shown experimentally that both kinetic and static friction are proportional to the normal force. The only way this can be true is if the two forces have different constants of proportionality. For static friction, we have fs ≤ µsN (4.3) where µs is the coefficient of static friction. This is an inequality because the force of static friction can take on smaller values if less force is needed to hold an object in place. You will almost always use the ‘=’ sign in problems. For the force of kinetic friction, we have where µk is the coefficient of kinetic friction. µk will almost always be less than µs. Note that the friction coefficients do not have any units. fk = µkN (4.4) 33 Example: Block on a ramp Suppose a block with a mass of 2.50 kg is resting on a ramp. If the coefficient of static friction between the block and ramp is 0.350, what maximum angle can the ramp make with the horizontal before the block starts to slip down? Solution: This is a problem involving forces, so we will need a free body diagram. There are three forces acting on the block: gravity pulls straight down, the normal force acts perpendicular to the surface, and the force of friction acts upwards along the ramp because the block would like to slide down the ramp. We will choose a tilted coordinate system such that the x-axis runs along the ramp. If we do this, the only force that needs to be broken into components is the force of gravity (friction pulls entirely along the x axis and the normal force pulls entirely along the y axis). We can now use Newton’s second law for both axes to get two equations, Fx = 0 mg sin θ − f = 0 Fy = 0 N − mg cos θ = 0. Remember that the force of friction is related to the normal force, f = µsN. I have used the equal sign here because the static force of friction will be largest (and therefore equal) just before the object starts to slip. Let’s use one equation to solve for N , and substitute into the other equation, N = mg cos θ mg sin θ − µsmg cos θ = 0 tan θ = µs θ = 19.3◦. The angle at which the block will slide depends only on the coefficient of static friction between the ramp and block. 34 NWxyf Example: Two blocks A block of mass m = 5.00 kg rides on top of a second block of mass M = 10.0 kg. A person attaches a string to the bottom block and pulls the system horizontally across a frictionless surface. Friction between the two blocks keeps the 5.0 kg block from slipping off. If the coefficient of static friction is 0.305, (a) what maximum force can be exerted by the string on the 10.0 kg block without causing the 5.0 kg block to slip? (b) What is the acceleration? Solution: This is a problem involving forces, so we will need a free body diagram. There are five forces acting on block M : gravity pulls straight down, the normal force from the floor pushes up, the normal force from block m pushes down, the tension from the rope, and friction from block m opposes the motion (so f points to the left). There are three forces acting on block m: the normal force from block M pushing up, gravity pulling down and the force of friction that opposes the potential motion of block m (so f points to the right). (a) Now let’s use Newton’s second law on both objects. First block M , and for the second block, Fx = M a T − f = M a Fy = 0 N2 − N1 − M g = 0, Fx = ma f = ma Fy = 0 N1 − mg = 0, Remember that the force of friction is related to the normal force, f = µsN1. I have used the equal sign here because the static force of friction will be largest (and therefore equal) just before the object starts to slip. Now we want to find T , N1 = mg µsN1 = ma a = µsg T − f = M a T − µsmg = M µsg T = (m + M )µsg 35 T = 51.5 N (b) We already have an expression for the acceleration a = µsg a = 3.43 m/s2. NNTffM2m1W1W2N1 Chapter 5 Work and Energy 5.1 Work You probably define work as something that expends some of your energy. Typing up a paper, or writing out a homework assignment is considered work as are more physically demanding tasks such as building furniture or moving heavy objects. In physics, work has a very specific definition that involves motion and forces. Work is done by a force only if that force causes a net displacement of the object. There are two key points here: a force is only responsible for motion if it is in the same direction as that motion, so forces that are perpendicular to motion do not result in work, and there must be a net displacement or no work has been done. The mathematical definition of work done on an object is W = F ∆x cos θ (5.1) where F is the force applied to the object, ∆x is the displacement, θ is the angle between the force and the direction of motion, and W is the work. Work is measured in units of joules (joule ). Note that if the force is doubled, work is doubled or if the object is displaced twice as far, then work is also doubled. Work is a scalar quantity — it has a magnitude, but no direction. Work can, however, be positive or negative; it’s negative when the applied force is opposite to the direction of motion. 36 Example: Work on a block A block of mass m = 2.50 kg is pushed a distance d = 2.20 m along a frictionless horizontal table by a constant applied force of magnitude F = 16.0 N directed at an angle θ = 25.0◦ below the horizontal. Determine the work done by (a) the applied force, (b) the normal force exerted by the table, (c) the force of gravity, and (d) the net force on the block. Solution: (a) The applied force has both horizontal and vertical components, but because the motion is entirely horizontal, only the horizontal component contributes to the force. W = F d cos θ W = (16.0 N)(2.20 m) cos(−25.0◦) W = 32 J. (b) The normal force is perpendicular to the motion, so it does not do any work. (c) The force of gravity is also perpendicular to the motion, so it also does no work. (d) We could calculate the net force by vectorially adding the normal force, the applied force and the force of gravity and then find the work done by that force. Or we can save ourselves some effort by remembering that only forces along the direction of motion contribute to the work. The only force that has a horizontal component is the applied force, and we found the work due to that force in part (a). 5.2 Kinetic energy Energy is an indirectly observed quantity that measures an object’s capacity to do work. Energy comes in many different forms and can easily change from one form to another, but the total amount of energy in the universe (or in an isolated system) stays the same. That means that energy is a conserved quantity. The concept of energy provides an alternative formulation for Newton’s laws. An object’s energy determines it’s potential to do work and the work it can do is related to the net force exerted by the object. If the force is constant, the acceleration is also constant and we can use kinematics equations, namely, Wnet = Fnet∆x = ma∆x. v2 − V 2 0 = 2a∆x a∆x = v2 − v2 0 2 . We can substitute this into the work equation, Wnet = m v2 − v2 0 2 Wnet = 1 2 mv2 − 1 2 mv2 0. (5.2) This equation tells us that the net work done on an object leads to a change in a quantity of the form 1 2 mv2. This term is called the kinetic energy of the object and it is the energy of the motion of the object. The equation tells us that any net work done on an object leads to a change in its kinetic energy and for this reason, the equation is known as the work-energy theorem. It’s important to realize that this is just an alternative formulation of Newton’s second law — two different ways of looking at the same process. Both Newton’s second law and the work-energy theorem tell us that interacting with an object in a certain way (by applying a force in Newton’s view, or doing work in the 37 energy view) will lead to changes in the object’s velocity. Why do we need two ways to describe the same process? It is sometimes more convenient to use one formulation over the other when solving problems. The energy formulation uses scalars rather than vectors, which can be easier for calculations, but it is sometimes hard to determine the net work done on an object without considering forces. Example: Stopping a ship A large cruise ship of mass 6.50 × 107 kg has a speed of 12.0 m/s at some instant. (a) What is the ship’s kinetic energy at this time? (b) How much work is required to stop it? (c) What is the magnitude of the constant force required to stop it as it undergoes a displacement of 2.5 km? Solution: (a) We know the ship’s initial speed and we know that the ship’s kinetic energy is determined by its speed, mv2 K0 = 1 2 1 2 K0 = 4.7 × 109 J. K0 = (6.50 × 107 kg)(12.0 m/s)2 (b) We want the ship’s final velocity (and also its kinetic energy) to be 0. The work-energy theorem tells us how much work is required to change an object’s kinetic energy, W = Kf − K0 W = −4.7 × 109 J. (c) Remember that work is done by a force acting on an object that travels some distance. The force slowing down the ship in this case is the drag or friction force of the water. Remember that friction is always opposite to the direction of motion so θ = 180◦. W = F ∆x cos θ F = F = W ∆x cos θ −4.7 × 109 J 2500 m cos(180◦) F = 1.9 × 106 N. 5.2.1 Conservative and nonconservative forces Forces can be broken up into two types: conservative and non-conservative. Conservative forces are forces where you can easily get back the energy you put into system. Gravity is one example of a conservative force. If you lift a book, you will be doing work against gravity to raise that book. As you lower the book, the book is now doing work on you (the normal force still points the |
same way, but the motion is in the opposite direction, so work is negative) and you will recover all the energy you put into the book to lift it. A nonconservative force converts energy of objects into heat or sound — forms of energy that are hard to convert back to motion. Friction is one example of a nonconservative force — you can’t recapture the energy lost to friction simply by moving the object back to where it started (like we did when we lowered the textbook). The proper physics definition of a conservative force is “A force is conservative if the work it does moving an object between two points is the same no matter what path is taken.” 38 This is based on the idea that, for conservative forces, we can get back the energy we put in simply by moving the object back to its starting point. We can re-write the work energy theorem to specifically separate these two types of forces, Wnc + Wc = ∆KE, (5.3) where we’ve separated the work done on the object into two parts: the work done by conservative forces and the work done by nonconservative forces. 5.3 Gravitational potential energy Conservative forces have the nice property that they essentially “store” energy based on their position. When you lift a book, you’ve done some work on that book and put energy into the book. You can get that energy back by lowering the book, or you can convert that energy to something else (like kinetic energy) by letting go of the book. The book is said to have potential energy because it now has the potential to do work on another object. Let’s figure out how much work is done by gravity as a book of mass m falls from yi to yf . Remember that the formula for work is W = F ∆x cos θ. The force of gravity is Fg = −mg, the displacement is ∆x = yf − yi, and the angle between them is θ = 0. So we have, Wg = −mg(yf − yi). (5.4) Assuming we have no other conservative forces, we can explicitly put the effect of gravity into the work-energy theorem, Wnc + Wg = ∆KE Wnc = ∆KE + mg(yf − yi) Wnc = ∆KE + ∆P E. (5.5) Note that when you are using the work-energy theorem, it does not matter what you choose as your reference point for measuring the height of an object. It is only changes in height (and therefore changes in gravitational potential energy) that matter and the the actual value of the potential energy at an one point. In the absence of any nonconservative forces (which will be the case in most of your homework problems), we have 0 = ∆KE + ∆P E KEi + P Ei = KEf + P Ef . (5.6) This is a conservation law — it tells us that the total amount of kinetic energy and potential energy (sometimes called mechanical energy) for a particular system stays the same all the time. The amount of kinetic energy and potential energy might change as the object moves, but if you add the energies together, you will always get the same number. A ball sitting on the top of a hill has lots of gravitational potential energy and no kinetic energy. As it starts to roll down, it’s potential energy decreases, but it’s kinetic energy increases. When it gets to the bottom of the hill, it has no more potential energy, but lots of kinetic energy. So the amount of each type of energy changes, but the total will always be the same. 39 Example: Platform diver A diver of mass m drops from a board 10.0 m above the water’s surface. Neglect air resistance. (a) Find his speed 5.0 m above the water’s surface. (b) Find his speed as he hits the water. Solution: (a) When solving problems dealing with gravitational potential energy, we need to set a reference point (it doesn’t matter where the reference point is, we just need to be consistent for all measurements). Let’s choose the bottom of the diving board as y = 0. We are told that the diver “drops” from the board, so v0 = 0 which means that his kinetic energy is KEi = 0. The initial position of the diver is at the top of the board where his gravitational potential energy is P Ei = mgyi. The conservation of mechanical energy tells us, KEi + P Ei = KEf + P Ef 0 + mgyi = mv2 f + mgyf 1 2 2g(yi − yf ) vf = vf = 2(9.8 m/s2)(10.0 m − 5.0 m) vf = 9.90 m/s. (b) We use the same procedure as for part (a), but with a different end point where yf = 0. KEi + P Ei = KEf + P Ef 0 + mgyi = 1 2 f + 0 mv2 vf = 2gyi vf = 2(9.8 m/s2)(10.0 m) vf = 14.0 m/s. 40 Example: Waterslides Der Stuka is a waterslide at Six Flags in Dallas named for the German dive bombers of World War II. It is 21.9 m high. (a) Determine the speed of a 60.0 kg woman at the bottom of such a slide, assuming no friction is present. (b) If the woman is clocked at 18.0 m/s at the bottom of the slide, find the work done on the woman by friction. Solution: (a) Let’s choose the bottom of the slide as y = 0. We can assume that the woman starts from rest, so v0 = 0 which means that her kinetic energy is KEi = 0 at the top of the slide. The initial position of the woman is at the top of the board where her gravitational potential energy is P Ei = mgyi. We are interested in the final position where her height is yf = 0 and so her gravitational potential energy is P Ef = 0. The conservation of mechanical energy tells us, KEi + P Ei = KEf + P Ef 0 + mgyi = 1 2 f + 0 mv2 vf = 2gyi vf = 2(9.8 m/s2)(21.9 m) vf = 20.7 m/s. (b) In this case we have to use the full work-energy theorem, Wnc = KEf − KEi + P Ef − P Ei f − 0 + 0 − mgyi mv2 Wnc = 1 2 1 2 Wnc = −3.16 × 103 J. Wnc = (60.0 kg)(18.0 m/s)2 − (60.0 kg)(9.8 m/s2)(21.9 m) Note that the work done by friction is negative because the woman is losing her energy to friction. 5.4 Spring potential energy When you compress or stretch a string, you have to apply a force and therefore do some work on the spring. When you move the spring back to its original position, that energy is given back to you. Like gravity, the spring force is a conservative force — any energy you put into the spring when it is stretched or compressed is returned when the spring moves back to its original position. Springs exert a force on an object when they are stretched or compressed and the more you stretch or compress the spring, the larger the force trying to return the spring to its original position. So the force exerted by a spring is proportional to the displacement, Fs = −k∆x, (5.7) where k is a proportionality constant called the spring constant (units of newtons per meter). This constant is different for each spring. This equation is often called Hooke’s law after Robert Hooke who discovered the relationship. In the case of a spring, we measure the displacement from the equilibrium position of the spring. That is x = 0 is the point at which the spring is neither compressed nor stretched. The spring force is sometimes called a restoring force because it tries to return the spring to equilibrium. Calculating the work done by a spring is not as straightforward as calculating the work done by gravity because the size of the spring force changes as the displacement changes (remember the force of gravity is the same no matter the height of the object). The work done by the spring force when an object moves from 41 xi to xf is Ws = −( 1 2 kx2 f − 1 2 kx2 i ). (5.8) We can include this in the work-energy theorem (if there is a spring involved in our system) as part of the work done by conservative forces, Wnc = ∆KE + ∆P Eg + ∆P Es, (5.9) where the potential energy of a spring is 1 2 kx2. Example: Block on a spring A block with mass of 5.00 kg is attached to a horizontal spring with spring constant k = 4.00 × 102 N/m. The surface the block rests upon is frictionless. If the block is pulled out to xi = 0.05 m and released, (a) find the speed of the block when it first reaches the equilibrium point. (b) Find the speed when x = 0.025 m, and (c) repeat part (a) if friction acts on the block with coefficient µk = 0.150. Solution: (a) In the first part of the problem there is no friction, so there aren’t any nonconservative forces and the work-energy theorem can be written as KEi + P Egi + P Esi = KEf + P Egf + P Esf . We can simplify this a little more by realizing that all the action takes place at the same height, so there are no changes in gravitational potential energy and we can remove that from the equation, KEi + P Esi = KEf + P Esf . The problem states that the block is “released” at a certain point — this means that the initial velocity is vi = 0. So the initial kinetic energy is also 0. The final position of the object is at the equilibrium point (xf = 0), so the spring potential energy at this point is also 0. 0 + 1 2 kx2 i = mv2 f + 0 1 2 vf = vf = kx2 i m (4.00 × 102 N/m)(0.05 m)2 5.00 kg vf = 0.45 m/s. (b) This time we’re asked for the velocity at a non-equilibrium point. The method we use is still the same, we just have a non-zero final potential energy, 0 + 1 2 kx2 i = 1 2 mv2 f + k(x2 kx2 f 1 2 i − x2 f ) m vf = vf = (4.00 × 102 N/m)[(0.05 m)2 − (0.025 m)2] 5.00 kg vf = 0.39 m/s. 42 (c) Now we add friction, so we will need to consider the energy lost to this nonconservative force. First we need to find the magnitude of the frictional force, so we need a free-body diagram of the block. The block has four forces acting on it: gravity pulling down, the normal force pushing up, the spring force pulling towards equilibrium, and the force of friction pulling away from equilibrium. There is no acceleration in the y direction, so we must have Fy = 0 N − mg = 0 N = mg. We know that the frictional force is related to the normal force, The work done by friction then is fk = µkN fk = µkmg. Wf = fk∆x cos θ Wf = −µkmgxi. The work-energy theorem now has to include the nonconservative work, Wnc = ∆KE + ∆P Es −µkmgxi = mv2 f − 1 2 kx2 i 1 2 vf = vf = kx2 i m − 2µkgxi (4.00 × 102 N/m)(0.05 m)2 5.00 kg − 2(0.150)(9.8 m/s2)(0.05 m) vf = 0.230 m/s. 43 Example: Circus acrobat A 50.0 kg circus acrobat drops from a height of 2.0 m straight down onto a springboard with a force constant of 8.00 × 103N/m By what maximum distance does she compress the spring? Solution: There aren’t any nonconservativ |
e forces in this problem, but spring potential, gravitational potential and kinetic energy all play a role. In this problem, we need to be very clear about how we are measuring distances because there are two displacements that are relevant: her change in height and the compression of the spring. Let’s set y = 0 to be the point of maximum spring compression and let’s call the distance that the spring compresses d. We will call the acrobat’s height above the uncompressed springboard h. Now let’s use the work-energy theorem, KEi + P Egi + P Esi = KEf + P Egf + P Esf 0 + mg(h + d kd2 − mgd − mgh = 0. 1 2 kd2 We have a quadratic equation and will need to use the quadratic formula, d = d = mg ± m2g2 + 2kmgh k mg k 1 ± 1 + 2kh mg d = 0.56 m (−0.44). 5.4.1 Power Power is the rate at which energy is transferred from one object to another. Remember that work is the amount of energy transferred from one object to another, so the average power will be the amount of work done over some period of time, P = . (5.10) W ∆t The unit of power is the Watt (W) or joule/second (J/s). We can write the power in another form by using the definition of work W = F ∆x cos θ, P = F ∆x cos θ ∆t P = F v cos θ. We can generalize this equation (if we use calculus) to get an equation for the instantaneous power, P = F v cos θ, where P and v are the instantaneous power and velocity rather than the average power and velocity. (5.11) (5.12) 44 Example: Shamu Killer whales are able to accelerate up to 30 mph in a matter of seconds. Neglecting the drag force of water, calculate the average power a killer whale with mass 8.00 × 103 kg would need to generate to reach a speed of 12.0 m/s in 6.00 s. Solution: To find the power needed by the whale, we need to figure out how much work the whale has to do to reach a speed of 12.0 m/s. We do not know the magnitude of the force needed to generate this acceleration, so we can’t use the definition of work. (We also can’t find the acceleration using kinematics because we can’t assume constant acceleration). The other option is to use the work-energy theorem, Wnet = ∆KE mv2 f − 0 Wnet = 1 2 1 2 Wnet = 5.76 × 105 J. Wnet = (8.00 × 103 kg)(12.0 m/s)2 We know the elapsed time, so we can use the equation defining average power, P = P = W ∆t 5.76 × 105 J 6.00 s P = 9.6 × 104 W. You might be wondering why we did not use the equation P = F v cos θ. That equation uses the average velocity and we are given the final velocity. We can’t find the average velocity without assuming constant acceleration, which we can’t do in this case. 45 Chapter 6 Momentum and Collisions 6.1 Momentum and impulse You probably have an intuitive definition of momentum. Objects with a large amount of momentum are hard to stop, i.e. a larger force is required to stop an object with lots of momentum. In physics, of course, we like to have precise definitions for these concepts, so we define the linear momentum as p = mv. (6.1) The linear momentum is proportional to both mass and velocity. That is the more massive an object, the more momentum it has and the faster an object moves the more momentum it has. The unit of momentum is kilogram meter per second (kg · m/s). Notice that momentum is a vector that points in the same direction as an object’s velocity. As usual, when dealing with vectors, you will break up momentum into its x and y components, px = mvx py = mvy. The momentum is related to the kinetic energy of an object, KE = 1 2 mv2 = (mv)2 2m = p2 2m . (6.2) The concept of momentum is closely tied to the idea of inertia and force. A force is required to change the momentum of an object. We can actually restate Newton’s second law in terms of momentum, Fnet = ma = m ∆v ∆t = ∆p ∆t , (6.3) which tells us that the change in momentum over some time is equal to the net force. Two implications arise from this equation. First, if there is no net force, the momentum does not change. Second, to change an object’s momentum you need to continuously apply a force over some time period (however small). We call the change in momentum of an object the impulse, I = ∆p = Fnet∆t. (6.4) This is called the impulse-momentum theorem. Most forces vary over time, making it difficult to use the idea of impulse without calculus. We can, however, replace a time-varying force with an average force which is a constant force that delivers the same impulse in the time ∆t as the real time-varying force. In this case, ∆p = Fav∆t. 46 (6.5) Example: Car crash In a crash test, a car of mass 1.50 × 103 kg collides with a wall and rebounds. The initial and final velocities are vi = −15.0 m/s and vf = 2.60 m/s. If the collision lasts for 0.150 s, find (a) the impulse delivered to the car due to the collision and (b) the size and direction of the average force exerted on the car. Solution: The impulse is the change in momentum, I = pf − pi I = m(vf − vi) I = (1.50 × 103 kg)(2.60 m/s − (−15.0 m/s)) I = 2.64 × 104 kg · m/s. We know that the impulse is related to the average force, Fav = I ∆t 2.64 × 104 kg · m/s 0.150 s Fav = 1.76 × 105 N. Fav = 6.2 Conservation of momentum If the two objects are isolated, then we can Let’s think about what happens when two objects collide. consider them as a single system. While the two objects will exert forces on each other during the collision, there are no net external forces acting on the system as a whole. If there are no net forces, then the total momentum of the system stays the same throughout the collision process. This is known as the conservation of momentum. Suppose a particle with mass m1 is travelling with velocity v1i towards a particle with mass m2 which is travelling towards the first particle with velocity v2i. These two particles will eventually collide. After they collide, the first particle moves away from the second with a velocity v1f and the second particle moves away with velocity v2f . While they are colliding, there will be a contact force between them. The force from particle 2 on particle 1 will change the momentum of particle 1, F21∆t = m1v1f − m1v1i, and similarly, the force from particle 1 on particle 2 will change the momentum of particle 2, F12∆t = m2v2f − m2v2i. (6.6) (6.7) By Newton’s third law, we know that the contact forces are an action/reaction pair and so they must be equal in magnitude and opposite in direction, F21∆t = − F12∆t m1v1f − m1v1i = −(m2v2f − m2v2i) m1v1i + m2v2i = m2v2f + m1v1f . (6.8) This equation tells us that if we add the momenta of the particles before the collision, that will be the same as the sum of all momenta after the collision. 47 Example: Littering fisherman A 75 kg fisherman in a 125 kg boat throws a package of mass 15 kg horizontally with a speed of 4.5 m/s. Neglecting water resistance, and assuming the boat is at rest before the package is thrown, find the velocity of the boat after the package is thrown. Solution: We will treat the fisherman/boat as a single object for the purpose of this problem; the package will be treated as a separate object. Everything is at rest before the package is thrown, so there is no initial momentum. After the package is thrown, the boat/fisherman will have some recoil velocity, 0 = mbVb + mpvp vb = − vb = − mpvp mb (15 kg)(4.5 m/s) 200 kg vb = −0.38 m/s. 6.2.1 Collisions While momentum is conserved during a collision, kinetic energy is not necessarily conserved. It is important to stress that total energy is always conserved, but certain types of energy, like kinetic energy are not always conserved. During a collision, energy is often lost to friction, sound, heat or deformation of the objects. We can classify collisions into several different types: Elastic collision: An elastic collision is a collision in which both momentum and kinetic energy are con- served. Inelastic collision: An inelastic collision is a collision in which momentum is conserved, but kinetic energy is not. Perfectly inelastic collision: A perfectly inelastic collision is a collision in which momentum is conserved, kinetic energy is not conserved and the two objects stick together and move with the same velocity after the collision. 48 Example: Truck versus compact A pickup truck with mass 1.80 × 103 kg is travelling eastbound at 15.0 m/s, while a compact car with mass 9.00 × 102 kg is travelling westbound at -15.0 m/s. The vehicles collide head-on, becoming entangled. (a) Find the speed of the entangled vehicles after the collision. (b) Find the change in the velocity of each vehicle. (c) Find the change in the kinetic energy of the system consisting of both vehicles. Solution: (a) This problem involves a collision, so we should use conservation of momentum. Before the collision, we know the velocities of both the truck and compact. After the collision, they have the same velocity. mtvt + mcvc = (mt + mc)v mtvt + mcvc mt + mc v = v = (1.80 × 103 kg)(15.0 m/s) − (9.00 × 102 kg)(15.0 m/s) 1.80 × 103 kg + 9.00 × 102 kg v = 5.0 m/s. (b) The change in velocity of the truck is ∆v = v − vt = 5.0 m/s − 15.0 m/s = −10 m/s. The change in velocity of the car is ∆v = v − vc = 5.0 m/s + 15.0 m/s = 20 m/s. (c) The change in kinetic energy is ∆KE = KEf − KEi 1 2 ∆KE = −2.7 × 105 J. ∆KE = (mt + mc)v2 − 1 2 mtv2 t − 1 2 mcv2 c 49 Example: Billiard balls Two billiard balls of identical mass move toward each other. Assume that the collision between them is perfectly elastic. If the initial velocities of the balls are 30.0 cm/s and -20.0 cm/s, what are the velocities of the balls after the collision? Assume friction and rotation are unimportant. Solution: This problem involves a collision, so we should use conservation of momentum. Before the collision, we know the velocities of both balls. After the collision, both velocities are unknown. mv1i + mv2i = mv1f + mv2f v1i + v2i = v1f + v2f v1i − v1f = v2f − v2i. We only have one equation with two unknowns, so we will need to find another equation. We are told that the collision is perfectly elastic, so we know that kinetic energy is conserved, 1 2 mv2 mv2 mv2 1 1f + 2i |
= 2 1f + v2 2i = v2 2f 2f − v2 1f = v2 2i (v1i − v1f )(v1i + v1f ) = (v2f − v2i)(v2f + v2i). 1 1i + 2 v2 1i + v2 1i − v2 v2 mv2 2f 1 2 Let’s divide the two equations, (v1i − v1f )(v1i + v1f ) v1i − v1f = (v2f − v2i)(v2f + v2i) v2f − v2i v1i + v1f = v2f + v2i. This equation is easier to deal with than the one with squared velocities, so let’s solve it for v1f and substitute into the conservation of momentum equation v1f = v2f + v2i − v1i v1i − v2f − v2i + v1i = v2f − v2i v2f = v1i v2f = 30.0 cm/s v1f = v1i + v2i − v1i v1f = v2i v1f = −20.0 cm/s. The balls have swapped their velocities as if they had passed through each other. The linear equation derived in the above example v1i − v2i = −(v1f − v2f ) (6.9) is actually true for any elastic collision, even if the masses are not equal. Instead of using the conservation of kinetic energy, which has quadratic terms and is hard to use, you can use the above linear equation for an elastic collision. 50 6.2.2 Collisions in two dimensions Most collisions are not head-on collisions with both objects travelling along a single line before and after the collision. Although most collisions occur in three dimensions, we will limit ourselves to two-dimensional collisions. When doing two-dimensional momentum problems, we break up momentum into x and y components, just as we did for Newton’s second law. The conservation of momentum tells us that momentum will be conserved in the x direction and in the y direction. That is,we now have two equations m1v1ix + m2v2ix = m1v1f x + m2v2f x m1v1iy + m2v2iy = m1v1f y + m2v2f y. (6.10) There are now three subscripts on the velocity: one telling us which object, one telling us which direction, and one telling us whether it was before or after the collision. Example 6.8: Collision at an intersection A car with mass 1.50 × 103 kg travelling east at a speed of 25.0 m/s collides at an intersection with a 2.50 × 103 kg van travelling north at a speed of 20.0 m/s. Find the magnitude and direction of the velocity of the wreckage after the collision, assuming that the vehicles undergo a perfectly inelastic collision and assuming friction between the vehicles and the road can be neglected. Solution: Before the collision, the car has a velocity only in the x direction and the van has a velocity only in the y direction. After the collision, the velocity of the combined object will have components in both directions, mcvc + 0 = (mc + mv)vf x 0 + mvvv = (mc + mv)vf y. We can use these equations to find the x and y components of the final velocity vf x = vf y = mc mc + mv mv mc + mv vc = vv = 1.50 × 103 kg 1.50 × 103 kg + 2.50 × 103 kg 2.50 × 103 kg 1.50 × 103 kg + 2.50 × 103 kg (25.0 m/s) = 9.38 m/s (20.0 m/s) = 12.5 m/s, from which we can find the magnitude and direction f x + v2 v2 f y v = v = (9.38 m/s)2 + (12.5 m/s)2 v = 15.6 m/s. vf y vf x 12.5 m/s 9.38 m/s tan θ = tan θ = θ = 53◦. 51 Example P54: Colliding blocks Consider a frictionless track as shown in Figure P6.54 of your textbook. A block of mass m1 = 5.00 kg is released from height h = 5.00 m. It makes a head-on elastic collision with a block of mass m2 = 10.0 kg that is initially at rest. Calculate the maximum height to which m1 rises after the collision. Solution: Let’s think about what happens in this problem. The block m1 starts from some height with no initial velocity and will travel to the bottom of the ramp, gaining speed as it falls. At the bottom of the ramp, it will hit block 2 and transfer some of its speed to block 2. We want to know if block 1 will travel backward after the collision and, if so, how far back up the ramp it travels. Let’s consider each step separately. First, the block falls from a height h. How fast is it moving at the bottom of the ramp? This is an energy conservation problem since all of the block’s initial potential energy is converted to kinetic energy (no friction), KEi + P Ei = KEf + P Ef 0 + mgh = 1 2 f + 0 m1v2 vf = 2gh vf = 2(9.8 m/s2)(5.00 m) vf = 9.9 m/s. Now we can look at the collision. Block 1 has initial velocity v1i = 9.9 m/s and an unknown final velocity v1f . The second block is initially at rest v2i = 0 and has some unknown final velocity v2f . Use conservation of momentum because this is a collision m1v1i + 0 = m1v1f + m2v2f . We have one equation with two unknowns, but we also know that the collision is elastic will give us a second equation. We’ll use the linear equation derived earlier instead of the quadratic equation to make the calculation easier, v1i − 0 = −(v1f − v2f ). We’re actually not too interested in what block 2 does, so let’s solve for its final velocity first and substitute into the other equation to find v1f v2f = v1i + v1f m1v1i = m1v1f + m2(v1i + v1f ) v1f (m1 + m2) = v1i(m1 − m2) v1f = v1f = v1i m1 − m2 m1 + m2 5.00 kg − 10.0 kg 5.00 kg + 10.0 kg v1f = −3.3 m/s. (9.9 m/s) Finally, we do the reverse of what we did in the first part to see how high the block will go KEi + P Ei = KEf + P Ef 1 2 1f + 0 = 0 + mgh m1v2 h = 52 h = v2 1f 2g (3.3 m/s)2 2(9.8 m/s2) h = 0.56 m. Chapter 7 Rotational Motion All the motion that we have studied to this point is linear motion. All the objects travelled in a straight line (or a series of straight lines). Objects do not always move in a straight line, they often rotate or move in circles. Luckily, many of the concepts you have learned for linear motion have analogues in rotational motion. 7.1 Angular displacement, speed and acceleration When describing linear motion, the important quantities are displacement ∆x, velocity v and acceleration, a. For rotational motion we use the angular displacement ∆θ, angular velocity ω, and angular acceleration α. For linear motion, the displacement measured the change in linear position of the object. For rotational motion, we want to measure the net change in angle as the object moves around the circle. You are used to measuring angles in degrees, but a more natural unit for measuring angles is the radian. Remember that the circumference of a circle of radius r is s = 2πr. Rearranging this equation a little gives s/r = 2π. This quantity is dimensionless, but it tells us that the displacement around any circle is 2π. A displacement around half the circle is π; a quarter circle is π/2. This forms the basis of the unit of radians. Note that we can convert from degrees to radians using the relation 180◦ = π. Angular quantities in physics must be expressed in radians, so be sure to set your calculators to radian mode when doing rotation problems. We can find the angle travelled by an object rotating at a distance r through an arc length s through θ = s r You might wonder why we bother defining a new type of displacement if we can just measure the arc length and use our linear displacement equations. Consider a solid object, called a rigid body, like a rotating compact disc where the entire object rotates as a unit. As the object rotates, all points on the CD will have the same angular displacement, but the arc length travelled by points on the disc will vary depending on the radius. Using the notion of angular displacement, we can easily describe the motion of the entire CD, but using linear displacement makes it difficult to treat the CD as a single object. (7.1) . Let’s properly define the angular properties. Suppose an object starts at an angle θi and ends at an angle θf after some time ∆t. The angular displacement is determined by the initial and final angles, Note that for a rigid body the angular displacement is the same for all points on the object. The unit for angular displacement is the radian (rad). The average angular velocity of an object is the angular displacement divided by the time, ∆θ = θf − θi. (7.2) ω = ∆θ ∆t . 53 (7.3) For a rigid body, again, all points will have the same angular velocity. The units of angular velocity are radian per second (rad/s). We will use the term angular speed when we are not concerned with the direction, but just using the magnitude of the velocity. A positive angular speed denotes counterclockwise rotation and a negative angular speed denotes clockwise rotation. Angular velocity is a vector and the direction is specified by the right-hand rule. Take your right hand, curl your fingers in the direction of the motion and your thumb will give the direction of the vector. This rule specifies the rotation axis of the spinning object. Example: Spinning wheel A wheel has a radius of 2.0 m. (a) How far does a point on the circumference travel if the wheel is rotated through an angle of 30 rad. (b) If this occurs in 2 s, what is the average angular speed of the wheel? Solution: (a) We are first asked to find the arc length travelled by a point on the edge of the wheel when the wheel rotates through 30 rad. We use the equation relating the arc length to the angle, s = rθ s = (2.0 m)(30 rad) s = 60 m. (b) Angular speed is the angular displacement divided by time, ω = ω = ∆θ ∆t 30 rad 2 s ω = 15 rad/s. We define the instantaneous angular speed by taking the limit (as we did for velocity), We use this instantaneous angular speed to define the average angular acceleration, ω = lim ∆t→0 ∆θ ∆t . α = ∆ω ∆t . (7.4) (7.5) The units of angular acceleration are radian per second squared (rad/s2). A rigid body will have the same angular acceleration at all points on the body. The direction of angular acceleration is in the same direction as angular velocity if the object is accelerating, otherwise it is in the opposite direction. 7.1.1 Constant angular acceleration For linear motion, we developed a number of useful equations when we could assume constant linear acceleration. We can derive similar equations using the angular quantities under the assumption of constant angular acceleration. Linear motion Rotational motion 2 v = vf +vi vf = vi + at ∆x = vit + 1 f = v2 v2 2 at2 i + 2a∆x ω = ωf +ωi ωf = ωi + αt 2 ∆θ = ωit + 1 f = ω2 ω2 2 αt2 i + 2α∆θ 54 Example 7.2: Spinning wheel II A wheel rotates with a constant angular acceleration of 3.5 rad/s2. If the an |
gular speed of the wheel is 2.00 rad/s at t = 0, (a) through what angle does the wheel rotate between t = 0 and t = 2.00 s? Give your answer in radians and revolutions. (b) What is the angular speed of the wheel at 2.00 s? (c) What angular displacement (in revolutions) results while the angular speed of part (b) doubles? Solution: (a) We are told that the angular acceleration is constant, so we can use the equations above. We are given angular acceleration, initial angular speed and a time and we want to find angular displacement, ∆θ = ωit + 1 2 αt2 ∆θ = (2.00 rad/s)(2.00 s) + ∆θ = 11.0 rad. 1 2 (3.5 rad/s2)(2.00 s)2 To convert this to revolutions, we remember that one revolution is 2π rad, 11.0 rad/2π = 1.75. (b) We know want the final angular speed, ωf = ωi + αt ωf = (2.00 rad/s) + (3.5 rad/s2)(2.00 s) ωf = 9.00 rad/s. (c) In this case, we have an initial and final angular speed and we want displacement, f = ω2 ω2 i + 2α∆θ f − ω2 ω2 i 2α ∆θ = ∆θ = (18.00 rad/s)2 − (9.00 rad/s)2 2(3.5 rad/s2) ∆θ = 34.7 rad = 5.52. 7.1.2 Relations between angular and linear quantities Remember that we could relate the distance travelled along an arced path to the angular displacement, ∆s = r∆θ. We can use this relationship to find relationships between other angular and linear quantities. For example, we can divide both sides of this equation by time to get the relationship between angular and linear velocity, = r ∆θ ∆s ∆t ∆t vt = rω. (7.6) I’ve used the subscript t here to denote the tangential velocity. The instantaneous velocity of the rotating object is always tangent to the circle. That is, if the object were no longer forced to rotate, it would continue in a straight line tangent to the circle, as dictated by Newton’s first law. So at every point on the path of a rotating object, the velocity is tangent to the path. 55 We can similarly derive a relationship between the angular and linear accelerations, where again, we use the subscript t to denote the tangential acceleration of the object. In this case, the distinction is very important because, as we shall see shortly, there is a second type of acceleration that is important during rotational motion. at = rα, (7.7) Example 7.3: Germy compact disc A compact disc rotates from rest up to an angular speed of 31.4 rad/s in a time of 0.892 s. (a) What is the angular acceleration of the disc, assuming the angular acceleration is uniform? (b) Through what angle does the disc turn while coming up to speed? (c) If the radius of the disc is 4.45 cm, find the tangential speed of a microbe riding on the rim of the disc when t = 0.892 s. (d) What is the magnitude of the tangential acceleration of the microbe at the given time? Solution: (a) We are told to assume uniform (constant) angular acceleration, so we can use the equations from the previous section. ωf = ωi + αt ωf − ωi t α = α = 31.4 rad/s 0.892 s α = 35.2 rad/s2. (b) Now we want to find the angular displacement, f = ω2 ω2 i + 2α∆θ f − ω2 ω2 i 2α ∆θ = ∆θ = (31.4 rad/s)2 2(35.2 rad/s2) ∆θ = 14.0 rad. (c) Now we use the relationship relating angular velocity and tangential velocity, vt = rω vt = (0.0445 m)(31.4 rad/s) vt = 1.4 m/s. (d) This time we need to relate the angular acceleration and the tangential acceleration, at = rα at = (0.0445 m)(35.2 rad/s2) at = 1.57 m/s2. 7.2 Centripetal acceleration We just discussed the relationship between the tangential acceleration and the angular acceleration. The tangential acceleration of an object is determined by changes in how fast an object is spinning. There is 56 another type of acceleration that is present in all rotational motion, even when the rate of rotation is not changing. Recall that Newton’s first law tells us that objects will continue to move in a straight line unless acted upon by an outside force. In order for an object to rotate, it must be continuously pulled from the straight line path that it wants to take. The direction of motion of the object is constantly changing, which means that there is some kind of acceleration. This acceleration is called the centripetal acceleration. This acceleration is towards the center of rotation and is responsible for changing the direction of motion. The tangential acceleration is responsible for changing the speed of rotation. The centripetal acceleration is given by ac = v2 r , where v is the tangential velocity. We can re-write this formula in terms of the angular speed, ac = (rω)2 r = rω2. (7.8) (7.9) The total acceleration consists of both the tangential and the centripetal acceleration. Since the two are always perpendicular, the magnitude of the total acceleration is given by a = c + a2 a2 t . (7.10) 57 Example 7.5: At the racetrack A race car accelerates uniformly from a speed of 40.0 m/s to a speed of 60.0 m/s in 5.00 s while travelling counterclockwise around a circular track of radius 4.00 × 102 m. When the car reaches a speed of 50.0 m/s, find (a) the magnitude of the car’s centripetal acceleration, (b) the angular speed, (c) the magnitude of the tangential acceleration, and (d) the magnitude of the total acceleration. Solution: (a) The centripetal acceleration can be determined from the tangential speed, ac = ac = v2 r (50.0 m/s)2 4.00 × 102 m ac = 6.25 m/s2. (b) The angular speed is related to the tangential speed, ω = v r ω = 50.0 m/s 4.00 × 102 m ω = 0.125 rad/s. (c) The acceleration is uniform, so we can use the constant acceleration equations, at = at = vf − vi ∆t 60.0 m/s − 40.0 m/s 5.00 s at = 4.00 m/s2. (d) The total acceleration is found from the centripetal and tangential accelerations a2 t + a2 c a = a = (4.00 m/s2)2 + (6.25 m/s2)2 a = 7.42 m/s2 . Centripetal force Since there is an acceleration that is directed towards the center, there must be some force pulling objects towards the center of rotation. This is often called the centripetal force, but this is not actually a new force. The centripetal force is one of the forces you are already familiar with (friction, normal force, tension, gravity) that happens to be pulling an object towards the center of rotation. In the case of planets orbiting the sun, the centripetal force is gravity. In the case of a yo-yo being swung in a circle, the centripetal force is tension. When solving rotational motion problems, it is often useful to set up a coordinate system based on the radial and tangential directions of motion. In this case, the net force in the radial direction is the centripetal 58 force and gives rise to the centripetal acceleration, Fc = mac = m v2 r , (7.11) where we have used the formula for centripetal acceleration. If the centripetal force were to disappear, the spinning object would continue travelling in a straight line tangent to the circle. Centrifugal force Many people refer to the centrifugal force when discussing rotational motion. The centrifugal force is a fictitious force; it does not exist. When we experience rotational motion, we often flee like we are being pushed out from the center of rotation, but this feeling comes about because your body would like to continue in a straight line, tangent to the circle, and you have to exert a centripetal force to continue the rotation. Example 7.6: Car in a turn A car travels at a constant speed of 13.4 m/s on a level circular turn of radius 50.0 m. What minimum coefficient of static friction between the tires and the roadway will allow the car to make the circular turn without sliding? Solution: There are three forces acting on the car: gravity, the normal force and friction. There is a frictional force pointing opposite to the direction of motion of the car, but we are not concerned with that kinetic frictional force. There is another frictional force, static friction in this case, due to the circular path of the car. Due to inertia, the car would like to move tangent to the circle, but friction prevents it from doing so. In this example, friction is the centripetal force. fs = m µsN = m µsmg = m v2 r v2 r v2 r µs = µs = v2 gr (13.4 m/s)2 (50.0 m)(9.8 m/s2) µs = 0.366. 59 Example 7.8: Riding the tracks A roller coaster car moves around a frictionless circular loop of radius R. (a) What speed must the car have so that it will just make it over the top without any assistance from the track? (b) What speed will the car subsequently have at the bottom of the loop? (c) What will be the normal force on a passenger at the bottom of the loop if the loop has a radius of 10.0 m? Solution: (a) There are two forces acting on the car: gravity and the normal force, both acting downwards when the car is at the top of the loop. Since we want the car to make it through the loop without the assistance of the track, we set the normal force to 0. mg + N = m Fy = mac v2 t R v2 t mg = m R vt = gR. (b) We can use conservation of energy to find the car’s speed at the bottom. At the top, the car has both potential and kinetic energy; at the bottom it only has kinetic energy, KEi + P Ei = KEf + P Ef 1 2 mv2 b + 0 mv2 t + mg(2R) = 1 2 1 2 5 v2 b = 2 vb = 5gR. gR (c) At the bottom of the loop the normal force and the force of gravity are in opposite directions, N − mg = m Fy = mac v2 b R 5gR R N = 6mg. N = m + mg 7.3 Gravitation One of the reasons we are interested in circular motion is because we know that astronomical bodies move in (roughly) circular orbits. The centripetal force that causes this rotation is the gravitational force first written by Isaac Newton. His law of universal gravitation states that any two objects will exert an attractive force because of their mass. If two particles with masses m1 and m2 are separated by a distance r, a gravitational force F acts along a line joining them, with magnitude given by where G = 6.673 × 10−11 kg−1 · m3 · s−2 is a constant of proportionality called the constant of universal gravitation. F = G m1m2 r2 (7.12) 60 It is important to notice that both objects feel an attractive force; gravitational forces form an action-reaction pair. Example 7.10: Ceres An astronaut standing on the sur |
face of Ceres, the largest asteroid, drops a rock from a height of 10.0 m. It takes 8.06 s to hit the ground. (a) Calculate the acceleration of gravity on Ceres. (b) Find the mass of Ceres, given that the radius of Ceres is RC = 5.1 × 102 km. (c) Calculate the gravitational acceleration 50.0 km from the surface of Ceres. Solution: (a) We can use kinematics to find the acceleration, ∆y = v0t + 1 2 at2 a = a = 2∆y t2 2(10.0 m) (8.06 s)2 a = 0.308 m/s2. (b) This acceleration is caused by the force of gravity pulling on the rock, mM R2 C aR2 C G ma = G M = M = (0.308 m/s2)(5.1 × 102 km)2 6.673 × 10−11 kg−1 · m3 · s−2 M = 1.2 × 1021 kg. (c) From the previous part, we have ma = G a = G mM R2 M R2 a = (6.673 × 10−11 kg−1 · m3 · s−2) 1.2 × 1021 kg (5.1 × 102 km + 50 km)2 a = 0.255 m/s2. 7.3.1 Kepler’s Laws Before Isaac Newton discovered the law of gravity, another astronomer, Johannes Kepler discovered laws that helped him describe the motion of the planets in our solar system. Kepler’s three laws state 1. All planets move in elliptical orbits with the Sun at one of the focal points. 2. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals, 3. The square of the orbital period of any planet is proportional to the cube of the average distance from the planet to the Sun. It turns out that these laws are a consequence of Newton’s gravitational force. 61 First Law It turns out that any object moving under the influence of an inverse-square force (force varies as 1/r2) will move in an elliptical orbit. Ellipses are curves drawn such that the sum of the distances from any point on the curve to the two foci is constant. For planets in our solar system, the Sun is always at one focus, so a planets distance from the Sun will vary as the planet orbits. Second Law Kepler’s second law has an interesting consequence. Since the planet is closer to the Sun at some points of its orbit than at other points, in order to sweep out equal area in equal time, it must change its speed as it moves around the Sun. The planet will move more slowly when it is far from the Sun and more quickly when it is close to the Sun. Third Law We can derive Kepler’s third law from the law of gravity. Suppose that an object (Mp) is moving in a circular orbit around an object (Ms) with a constant velocity (Is this possible given the second law?). We know that the object undergoes centripetal acceleration and that the force causing this acceleration is the gravitational force, Mpac = Mpv2 r = GMpMs r2 . The speed of the object is simply the circumference divided by the time required for one revolution (period) Substituting, we find v = 2πr T . Mp(4π2r2 rT 2 = GMpMs r2 T 2 = 4π2 GMs r3. (7.13) (7.14) 62 Example 7.13: Geosynchronous orbit From a telecommunications point of view, it’s advantageous for satellites to remain at the same location relative to a location on Earth. This can occur only if the satellite’s orbital period is the same as the Earth’s period of rotation, 24.0 h. (a) At what distance from the center of the Earth can this geosynchronous orbit be found? (b) What’s the orbital speed of the satellite? Solution: (a) We can use Kepler’s third law to find the radius of the orbit, T 2 = 4π2 r3 GMs GME 4π2 r = 3 T 2 (6.673 × 10−11 kg−1 · m3 · s−2)(5.98 × 1024 kg (86400 s)2 r = 3 4(3.14)2 r = 4.23 × 107 m. (b) The speed is simply the circumference divided by the period, v = 2πr T 2(3.14)(4.23 × 107 m) 86400 s v = 3.08 × 103 m/s. v = 7.4 Torque When we studied forces, we treated all objects as point masses and assumed that it didn’t matter where on the object a force was applied. The force would simply move linearly in response to the applied force. This is an extreme simplification of reality, it actually does matter at which point on the object the force is applied. Forces applied near the edges of an extended object will tend to rotate the object rather than move it forward. If you lay your textbook on the table and push it with a force applied near the center of the book, it will slide forward. If you push the book with a force applied near the edge, it will rotate rather than move forward. Remember that forces cause an acceleration. Similarly, we define something called a torque which causes an angular acceleration. Forces and torques are not completely independent — forces cause torques, but torques also depend on where the force is applied and on the angle at which the force is applied. Let F be a force acting on an object, and let r be a position vector from the point of rotation to the point of application of the force. The magnitude of the torque τ exerted by the force F is τ = rF sin θ (7.15) where r is the length of the position vector, F is the magnitude of the force and θ is the angle between r and F . The unit of torque is newton-meter. Torque is a vector with the direction given by the right hand rule. Point your fingers in the direction of r and curl them toward the direction of F . Your thumb will point in the direction of the torque. This will be perpendicular to the plane that contains both r and F . When your fingers point in the direction of the torque, your fingers will curl in the direction of rotation that the torque will cause. 63 Example 8.2: Swinging door (a) A man applies a force of F = 3.00 × 102 N at an angle of 60.0◦ to the door 2.00 m from the hinges. Find the torque on the door, choosing the position of the hinges as the axis of rotation. (b) Suppose a wedge is placed at 1.50 m from the hinges on the other side of the door. What minimum force must the wedge exert so that the force applied in part (a) won’t open the door? Solution: (a) We use the above equation to calculate the torque on the door τ = F r sin θ τ = (F = 3.00 × 102 N)(2.00 m) sin(60.0◦) τ = 520 N · m. The direction of torque is out of the board/page (towards top of door). (b) We don’t want the door to rotate, so there must be no net torque on the door. We need to identify all the torques/forces acting on the door. There are three forces acting on the door: the applied force, the force of the wedge and the force of the hinges. Although the hinges apply a force to the door, they do not exert a torque because they act at the point of rotation. τhinges + τwedge + τ = 0 0 + Fwedgerwedge sin(−90◦) + τ = 0 Fwedge = Fwedge = τ rwedge sin(−90◦) 520 N · m (1.50 m) sin(−90◦) Fwedge = 347 N. 7.4.1 Equilibrium An object that is in equilibrium must satisfy two conditions, 1. The net force must be zero, i.e. no linear acceleration ( F = 0). 2. The net torque must be zero, i.e. no angular acceleration ( τ = 0). This does not mean that the object is not moving or rotating, it can be moving at a constant velocity or rotating at a constant angular speed. If an object is in equilibrium, we can choose the axis of rotation so we should choose one which makes the calculation convenient. An axis where at least one torque is zero makes calculations easier. 64 Example 8.3: Balancing act A woman of mass m = 55.0 kg sits of the left side of a seesaw — a plank of length L = 4.00 m, pivoted in the middle. (a) First compute the torques on the seesaw about an axis that passes through the pivot point. Where should a man of mass M = 75.0 kg sit if the system is to be balanced? (b) Find the normal force exerted by the pivot if the plank has a mass of mp = 12.0 kg. (c) Repeat part (a), but this time compute the torques about an axis through the left end of the plank. Solution: (a) We need to identify all the forces acting on the plank and their point of application. The woman and the man will push down on the plank with a force equal to their respective weights. We know the woman’s distance from the center of the plank, but the man’s distance is unknown. Gravity also acts on the plank itself, and will pull down on the center of the plank as if all the plank’s mass was concentrated at that one point. Finally, there is a normal force that pushes upwards from the pivot point. We want the seesaw to be in equilibrium, so the sum of all torques about some axis of rotation (the pivot point in this case) must be zero, τ = 0 τN + τg + τw + τm = 0 0 + 0 + mg L 2 − M gx = 0 x = x = mL 2M (55.0 kg)(4.00 m) 2(75.0 kg) x = 1.47 m. (b) We now want to find the normal force. We can use the first condition of equilibrium to do this, F = 0 −M g − mg − mpg + N = 0 N = (M + m + mp)g N = (75.0 kg + 55.0 kg + 12.0 kg)(9.8 m/s2) N = 1.39 × 103 N. (c) We will now use the left end of the plank (where the woman sits) as the axis of rotation, τ = 0 τN + τg + τw + τm = 0 −N L 2 + mpg L 2 + 0 + M g(N − mpg − M g) L 2 M g (M g + mg + mpg − mpg − M g) L 2 M g mL 2M x = 1.47 m. 65 7.4.2 Center of gravity As we saw in the previous problem, we treat gravity as if it acts on a single point of an extended body. This point is called the center of gravity. Suppose we have an object with some arbitrary shape. We can treat this object as if it is divided into very small pieces of weights m1g, m2g . . . at locations (x1,y1), (x2,y2). . . . Each piece contributes some torque about the axis of rotation due to its weight. For example, τ1 = m1gx1 and so forth. The center of gravity is the point where we apply a single force of magnitude F = i mig which has the same effect on the rotation of the object as all the individual little pieces. F xcg = ( i mig)xcg = xcg = i τi migxi i i mixi i mi . (7.16) This gives us the x coordinate of the center of gravity. We can find the y and z coordinates in a similar fashion, ycg = zcg = i miyi i mi i mizi i mi . (7.17) If an object is symmetric, the center of gravity will lie on the axis of symmetry, so it is sometimes possible to guess where the center of gravity is for such objects (like we did for the plank in the example problem). Example 8.4: Center of gravity Three objects are located on the x-axis as follows: a 5.00 kg mass sits at x = −0.500 m, a 2.00 kg mass sits at the origin, and a 4.00 kg mass sits at x = 1.00 m. Find the center of gravity. (b) How does the answer change |
if the object on the left is displaced upward by 1.00 m and the object on the right is displaced downward by 0.50 m? Solution: (a) We simply apply the formula for center of gravity that we just derived, xcg = xcg = i mixi i mi (5.00 kg)(−0.500 m) + (2.00 kg)(0) + (4.00 kg)(1.00 m) 5.00 kg + 2.00 kg + 4.00 kg xcg = 0.136 m. (b) The x coordinate of the center of gravity will not change since the masses have not been moved along the x-axis. We will, however, have to consider the y axis now, ycg = ycg = i miyi i mi (5.00 kg)(1.00 m) + (2.00 kg)(0) + (4.00 kg)(−0.500 m) 5.00 kg + 2.00 kg + 4.00 kg ycg = 0.273 m. 66 7.5 Torque and angular acceleration When an object is subjected to a torque, it undergoes an angular acceleration. We can derive a law similar to Newton’s second law for the effect of a torque. Suppose we have an object of mass m connected to a very light rod of length r. The rod is pivoted about the end opposite the mass and its movement is confined to a horizontal frictionless table. Suppose a tangential force Ft acts on the mass. This will cause a tangential acceleration, Multiplying both sides of the equation by r, Ft = mat. Ftr = mrat, and substituting for the at = rα for the tangential acceleration gives The left side is simply the torque, Ftr = mr2α. τ = (mr2)α. (7.18) (7.19) This tells us that the torque is proportional to the angular acceleration. The constant of proportionality is mr2 and is called the moment of inertia of the object. Moment of inertia has units of kg · m2 and is denoted by I. So we can write, τ = Iα. (7.20) This is the rotational analog of Newton’s second law. 7.5.1 Moment of inertia The formula for moment of inertia that we just derived is true for a single point mass only. For extended objects, the moment of inertia will be different. Consider a solid object rotating about its axis. We can break this object up into many little pieces like we did to find the center of gravity. The net torque on the object will be the sum of the torques caused by all the small pieces, i τi = ( mir2 i )α. i The moment of inertia of the whole object then is I = mir2 i . i (7.21) (7.22) We can find the moment of inertia of any object or any collection of objects by adding the moments of inertia of its constituents. Notice that the moment of inertia depends not just on the mass of an object, but on how the mass is distributed within the object. Importantly, it matters how the mass is distributed relative to the axis of rotation. 67 Table 7.1: Moments of inertia for various rigid objects of uniform composition Object Hoop or cylindrical shell Solid sphere Solid cylinder or disk Thin spherical shell Long thin rod Long thin rod Axis of Rotation Moment of inertia center center center center center end I = M R2 I = 2 5 M R2 I = 1 2 M R2 I = 2 3 M R2 I = 1 12 M L2 I = 1 3 M L2 Example 8.9: Baton twirler In an effort to be the star of the halftime show, a majorette twirls an unusual baton made up of four spheres fastened to the ends of very light rods. Each rod is 1.0 m long. Two of the spheres have a mass of 0.20 kg and the other two spheres have a mass of 0.30 kg. Spheres of equal masses are placed across from each other. (a) Find the moment of inertia of the baton through the point where the rods cross. (b) The majorette tries spinning her strange baton about the rod holding the 0.2 kg spheres. Calculate the moment of inertia of the baton about this axis. Solution: (a) When the baton is spinning around the point where the rods cross, all four spheres contribute to the moment of inertia. We can treat the spheres as point masses since their radius is small compared to the length of the rods. I = mir2 i i I = 2m1r2 + 2m2r2 I = 2(0.20 kg)(0.5 m)2 + 2(0.30 kg)(0.5 m)2 I = 0.25 kg · m2. (b) In this case only the 0.3 kg spheres contribute to the moment of inertia because the 0.2 kg spheres lie along the axis of rotation (so r = 0). I = mir2 i i I = 2m2r2 I = (0.30 kg)(0.5 m)2 I = 0.15 kg · m2. The moment of inertia for solid extended objects can be calculated using calculus. The moment of inertia for some common objects is given in Table 7.1. Note that the assumption is that the mass is distributed uniformly throughout these objects. Parallel axis theorem A useful property of the moment of inertia is that it is fairly easy to calculate the moment of inertia about an axis parallel to the axis through the center of gravity of the object. This result is called the parallel axis theorem and is as follows, Iz = Icm + M d2, (7.23) 68 where Icm is the moment of inertia of the object rotating about the center of mass, M is the mass of the object and d is the distance between the two parallel axes. 69 Example 8.11: Falling bucket A solid uniform frictionless cylindrical reel of mass M = 3.00 kg and radius R = 0.400 m is used to draw water from a well. A bucket of mass m = 2.00 kg is attached to a cord that is wrapped around the cylinder. (a) Find the tension T in the cord and acceleration a of the bucket. (b) If the bucket starts from rest at the top of the well and falls for 3.00 s before hitting the water, how far does it fall? Solution: (a) We will need free body diagrams for both the wheel and the bucket. The bucket has two forces acting on it: tension pulling up and gravity pulling down. Note that we don’t care where these forces act on the bucket because this object is not rotating. The cylinder has three forces acting on it: gravity acting at the center and pulling down, a normal force (from the bar holding the cylinder) also acting at the center and pushing up, and tension acting at a distance R and pulling down. We know that the bucket is accelerating and the cylinder has an angular acceleration. We can use Newton’s second law on the bucket, F = ma mg − T = ma. We can use the rotational analog of Newton’s second law on the cylinder. In this case, we don’t get to choose the point of rotation because the object is rotating about a specific axis, τ = Iα R2α M Rα. Gravity and the normal force don’t contribute to the torque because they act at the axis of rotation. We now have two equations and three unknowns (a, α and T ). We will need one more equation to solve this problem. Remember that the tangential acceleration is related to the angular acceleration of a rotating object. In this case, the rope is causing the tangential acceleration of the cylinder and we know that the acceleration of the rope is the same as that of the bucket, Using this relationship, we find a = Rα. T = 1 2 M a, which we can use to substitute into the first equation, M a = ma mg − a(m + 1 2 1 2 M ) = mg a = 2 M ) mg (m + 1 (2.00 kg)(9.8 m/s2) 2.00 kg + 1 2 (3.00 kg) a = 5.60 m/s2. a = Now we can also find the tension 70 (3.00 kg)(5.60 m/s2) T = 8.4 N. (b) This is a kinematics problem, ∆y = v0t + 1 2 at2 ∆y = 1 2 (5.60 m/s2)(3.0 s)2 ∆y = 25.2 m. 7.6 Rotational kinetic energy Recall that an object moving through space has kinetic energy. Similarly, a rotating object will have rotational kinetic energy. Consider the mass connected to a light rod rotating on a horizontal frictionless table. The kinetic energy of the mass is KE = mv2. 1 2 We know that the velocity is related to the angular speed, KE = 1 2 m(rω)2 = 1 2 (mr2)ω2 = 1 2 Iω2. (7.24) Notice that again the equations for translational (linear) kinetic energy and rotational kinetic energy are quite similar with moment of inertia replacing mass and angular speed replacing linear velocity. In the case of the rotating mass on a rod, either expression can be used to describe it’s energy because it only undergoes rotational motion. There are cases, however when both expressions are used such as when balls or wheels are rolling. In this case, there is rotation about the center of mass while the center of mass itself is moving through space. The translational kinetic energy refers to the energy of the center of mass’ motion while the rotational kinetic energy refers to the energy of the rotation. This new type of energy needs to be included in the work-energy theorem, Wnc = ∆KEt + ∆KEr + ∆P E. (7.25) 71 Example 8.12: Ball on an incline A ball of mass M and radius R starts from rest at a height of 2.00 m and rolls down a 30◦ slope. What is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping. Solution: We can use energy to solve this problem. Let’s consider the energy at the top and bottom of the ramp, remembering that the ball rolls (rotates) down the ramp, KEti + KEri + P Eg = KEtf + KErf + P E 0 + 0 + M gh = 1 2 M v2 + 1 2 Iω2 + 0. Remember that there is a relationship between the translational velocity and the angular speed. Note that this relationship will only hold if the object “rolls without slipping.” If the ball slips then the center of mass moves while the object is not rotating and the relationship does not hold. 2 5 1 2 M R2 2 v R M gh = gh = v = v = 1 5 v2 M v2 + 1 2 1 v2 + 2 10 7 10 7 gh (9.8 m/s2)(2.00 m) v = 5.29 m/s. The velocity is smaller than the velocity of a block sliding down the incline because some of the gravitational potential energy goes into rotational kinetic energy. 7.7 Angular momentum When we apply a torque to an object, we change its angular acceleration — and we have an equation relating the two. Just as we re-wrote Newton’s second law in terms of momentum, we can re-write the rotational equivalent in terms of angular momentum. τ = Iα = I ∆ω ∆t = I∆ω ∆t = ∆L ∆t , (7.26) where we have defined L = Iω as the angular momentum of an object. If there is no net torque, then the total angular momentum of a system does not change, Li = Lf . The law of conservation of angular momentum explains why figure skaters spin faster when they bring their arms closer to their bodies. As their arms move in, their moment of inertia decreases, so their angular speed must increase to compensate. Note that angular momentum is a vector with the direction determined by the direction of angular velocity. Changes in direction of angular momentum (cha |
nges in the direction of the axis of rotation) are also subject to conservation of momentum. If the axis of rotation changes, there must be a change in the angular momentum of some other part of the system to compensate. 72 Example 8.14: Merry-go-round A merry-go-round modeled as a disk of mass M = 1.00 × 102 kg and radius R = 2.00 m is rotating in a horizontal plane about a frictionless vertical axle. (a) After a student with mass m = 60.0 kg jumps on the rim of the merry-go-round, the system’s angular speed decreases to 2.00 rad/s. If the student walks slowly from the edge toward the center, find the angular speed of the system when she reaches a point 5.00 m from the center. (b) Find the change in the system’s rotational kinetic energy caused by her movement to r = 0.500 m. (c) Find the work done on the student as she walks to r = 0.500 m. Solution: (a) There are two parts to the moment of inertia of the system, the moment of inertia of the disk and the moment of inertia of the person. It is the moment of inertia of the student that changes as she walks towards the center — the moment of inertia of the disk remains the same, ωf = Li = Lf (Id + Ipi)ωi = (Id + Ipf )ωf (Id + Ipi)ωi Id + Ipf ( 1 2 M R2 + mR2)ωi 1 2 M R2 + mr2 ( 1 2 (1.00 × 102 kg)(2.00 m)2 + (60.0 kg)(2.00 m)2)(2.00 rad/s) 1 2 (1.00 × 102 kg)(0.500 m)2 ωf = ωf = ωf = 4.09 rad/s. (b) The change in rotational kinetic energy is ∆KEr = ( ∆KEr = 1 2 1 2 ∆KEr = 920 J. (( − 1 2 1 2 1 2 (Id + Ipf )ω2 f − 1 2 (Id + Ipi)ω2 i (1.00 × 102 kg)(0.500 m)2 + (60.0 kg)(2.00 m)2)(4.09 rad/s)2 (1.00 × 102 kg)(2.00 m)2 + (60.0 kg)(2.00 m)2)(2.00 rad/s))2 (c) When calculating the work done by the student we need to use the change in kinetic energy of the student only, ∆KEr = ∆KEr = 1 2 1 2 Ipf ω2 f − 1 2 Ipiω2 i (60.0 kg)(2.00 m)2(4.09 rad/s)2 − 1 2 (60.0 kg)(2.00 m)2)(2.00 rad/s))2 ∆KEr = −355 J. 73 Chapter 8 Vibrations and Waves We have now studied linear motion and circular motion. There is one more very important type of motion that arises in many aspects of physics. Periodic motion, such as waves or vibrations underlies sound and light and many other physical phenomena. 8.1 Return of springs One simple type of periodic motion is an object attached to a spring. Remember that the force of a spring is given by Hooke’s law, F = −kx. (8.1) This force is sometimes called a restoring force because it likes to pull the object back to the equilibrium position. The negative sign ensures that the force is pulling opposite to the direction of displacement. Suppose we pull the object so that the spring is stretched and let go. The spring force will cause an acceleration back towards the equilibrium position. The object will pick up speed as it moves back towards equilibrium and will overshoot the equilibrium position. Once it passes the equilibrium position the object starts to compress the spring and the force changes direction. The force now decelerates the object, eventually causing the object to stop. When the object stops, the spring is compressed and the force still points towards the equilibrium. So the object will accelerate towards the center again. In this way and object will move back and forth endlessly. This is an example of simple harmonic motion. Simple harmonic motion occurs when the net force along the direction of motion obeys Hooke’s Law. Not all periodic motion is simple harmonic motion. Two people tossing a ball back and forth is not simple harmonic motion even though it is periodic. The force causing the motion of the ball is not of the form of Hooke’s Law, so it cannot be simple harmonic motion. The acceleration of an object undergoing simple harmonic motion can be found using Newton’s second law, F = ma −kx = ma k m a = − x. (8.2) 8.1.1 Energy of simple harmonic motion Let’s consider the energy of an object attached to a spring. Suppose that we pull the object and stretch the spring then release it. Just before the object is released, the spring is at it’s maximum stretch. This is called the amplitude. The energy at this point is E = kA2, (8.3) 1 2 74 where A is the amplitude. Now we release the spring and as it moves it picks up speed. The object now has both potential energy and kinetic energy, so E = 1 2 kx2 + 1 2 mv2. We can use this to find the velocity at any position, 1 2 kA2 = 1 2 1 2 mv2 kx2 + k m v = ± (A2 − x2). (8.4) (8.5) The ± appears because of the square root. The usual convention is that if the object moves to the right, the velocity is positive; if it moves to the left, it is negative. 8.1.2 Connecting simple harmonic motion and circular motion When the object on the spring moves back and forth, it’s similar to an object moving with constant angular velocity around a circle. The object moving around the circle will come back to its original position at regular time intervals, just like the mass on a spring. Remember that the period of a rotating object is T = 2πr v . (8.6) For the rotating object, r is the size of the spatial displacement and corresponds to the amplitude of the mass on a spring, T = 2πA v . The velocity of the mass after it has travelled a distance A can be found from Eq. (8.5) by setting x = 0, Now we can put this into the equation for the period, v = A k m . T = 2π m k . (8.7) This represents the time it takes for a mass on a spring to return to its starting position. A larger mass gives a longer period, while a larger spring constant (stiffer spring) gives a shorter period. The frequency is the inverse of the period f = 1 T = 1 2π k m . (8.8) The units of frequency are cycles per second or Hz. This is related the angular frequency (which is in radians per second), k m . (8.9) ω = 2πf = 75 Example 13.5: Shock absorbers A 1.3 × 103 kg car is constructed on a frame supported by four springs. Each spring has a spring constant of 2.00 × 104 N/m. If two people riding in the car have a combined mass of 1.6 × 102 kg, find the frequency of vibration of the car when it is driven over a pothole. Find also the period and the angular frequency. Assume the weight is evenly distributed. Solution: First we need to find the total mass, mt = mc + mp = 1.3 × 103 kg + 1.6 × 102 kg = 1.46 × 103 kg Each spring will hold up one quarter of the total mass. The frequency is f = f = 1 2π 1 2π k m 2.00 × 104 N/m 365 kg f = 1.18 Hz. The period is the inverse of frequency and the angular frequency is T = 1 f = 1 1.18 Hz = 0.847 s, ω = 2πf = 2π(1.18 Hz) = 7.41 rad/s. 8.2 Position, velocity and acceleration Suppose a mass is moving on a circle with constant angular velocity. If we look at it’s x position as it moves around the circle, we see that it oscillates somewhat like a mass on a spring. The x position of the mass is given by We know that the mass is moving with constant angular speed so x = A cos θ. x = A cos(ωt) = A cos(2πf t). (8.10) (8.11) This equation describes the position of an object undergoing simple harmonic motion as a function of time. We can substitute this into Eq. (8.5) v = ± (A2 − x2) k m k v = ± m v = Aω1 − cos2(2πf t) v = Aω sin(2πf t). (A2 − (A cos(2πf t))2 (8.12) The velocity also oscillates, but it is 90◦ out of phase with the displacement. When the displacement is a maximum or minimum, velocity is zero and vice versa. The maximum value (amplitude) of velocity is Aω 76 (when sin(πf t) = 1). We can also derive an expression for the acceleration a = − k m x a = −Aω2 cos ω(2πf t). (8.13) The acceleration is also sinusoidal and 180◦ out of phase with the displacement. When the displacement is a maximum, acceleration is a minimum and vice versa. The maximum acceleration (amplitude) is Aω2. 77 Example 13.6: Vibrating system (a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is x = (0.250 m) cos π 8.00 t . (b) Find the maximum magnitude of the velocity and acceleration. (c) What are the position, velocity, and acceleration of the object after 1.00 s has elapsed? Solution: (a) Compare the given function to the standard function for simple harmonic motion and we can just read off the amplitude, and the frequency x = A cos(2πf t) A = 0.250 m, 2πf = f = π 8.00 1 16 = 0.0625 Hz. The period is the inverse of frequency (b) The maximum velocity is T = 1 f = 1 0.0625 Hz = 16 s. vmax = Aω vmax = (0.250 m)(2π)(0.0625 Hz) vmax = 0.098 m/s, and the maximum acceleration is amax = Aω2 amax = (0.250 m)(2π)2(0.0625 Hz)2 amax = 0.039 m/s2. (c) We simply substitute into the given equation and for the velocity and for the acceleration x = (0.250 m) cos π 8.00 = 0.231 m, v = −(0.098 m) cos π 8.00 = −0.038 m/s, a = −(0.039 m) cos π 8.00 = −0.036 m/s2. 78 8.3 Motion of a pendulum Another type of periodic motion that you may have observed is that of a pendulum swinging back and forth. To determine if it is simple harmonic motion, we need to figure out whether there is a Hooke’s law type force causing the pendulum to move. There are two forces acting on the pendulum: the force of gravity pulls down and tension pulls towards the center of rotation. If the mass is pulled away from the equilibrium position, then the force trying to pull it back towards equilibrium is F = −mg sin θ, (8.14) where θ gives the angular displacement of the pendulum. We know that the linear displacement is s = Lθ where L is the length of the pendulum (radius of the circle on which the pendulum moves). So the force pulling along the path towards equilibrium is F = −mg sin . s L (8.15) This does not look like Hooke’s law, so in general the motion of a pendulum is not simple harmonic. At small angles, however, the sine of an angle is approximately the same as the angle itself (as long as it’s measured in radians). So for small angles, we can write F = −mg s L = − mg L s. (8.16) Now the equation looks like Hooke’s law. The force is propotional to the linear displacement with the “spring constant” given by k = mg/L. Remember, however, that this is only valid for small |
angles. Recall that the angular frequency for an object undergoing simple harmonic motion is ω = k m . We have an expression for k for a pendulum, so we can substitute, ω = mg/L m = g L . From that we can find the frequency and the period g L f = T = 1 2π ω = 1 f = 2π 1 2π L g . (8.17) (8.18) (8.19) Note that the period depends only on the length of the pendulum and not on its mass or the amplitude of the motion. 79 Example 13.7: Measuring g Using a small pendulum of length 0.171 m, a geophysicist counts 72.0 complete swings in a time of 6.00 s. What is the value of g in this location? Solution: We first need to determine the period of oscillation, T = T = time # of oscillations 6.00 s 72.0 T = 0.833 s. We can use this to find g, T = 2π L g g = 4π2 L T 2 g = 4π2 0.171 m (0.833 s)2 g = 9.73 m/s2. 8.4 Damped oscillations So far we have assumed that the objects will continue oscillating forever. In the real world energy losses due to friction will cause the oscillating object to slow down. In this case the motion is said to be damped. If we consider the mass on the spring, we know that the mass will oscillate for some time but that the amplitude will decrease over time. This scenario is an underdamped oscillation. Suppose now we put the mass on a spring into a liquid. If the liquid is thick enough it will prevent the oscillations and simply allow the spring to come back to its equilibrium position. If the object returns to equilibrium rapidly without oscillating, then the motion is critically damped. If the object returns to equilibrium slowly, the motion is overdamped. 8.5 Waves A wave is typically thought of as a disturbance moving through a medium. When a wave passes through a medium, the individual components of that medium oscillate about some equilibrium point, but they do not move with the wave. Imagine a leaf floating in a pond. You throw a pebble into the pond near the leaf. This creates a wave in the water. When the wave reaches the leaf, it causes the leaf to bo up and down, but it does not carry the leaf with it. The leaf was temporarily disturbed, but once the wave passes it goes back to its original state. 8.5.1 Types of waves Suppose you fix one end of a string to a wall and you hold the other end. If you quickly move your hand up and down, you will create a wave (in this case a pulse) that travels down the string. This is a traveling wave. In the case of the pulse on the string, the individual bits of string move up and down as the pulse goes through. They do not move in the direction of pulse. When the disturbed medium moves perpendicular to the direction of the wave, the wave is called a transverse wave. A longitudinal wave occurs when the disturbed medium oscillates along the direction of travel of the wave. A good example of this is to alternately stretch 80 and compress a spring. The stretched and compressed regions will move down the spring with each coil oscillating along the direction of the wave. While the coils oscillate along this direction they still do not actually move with the wave. It turns out that each point in the medium undergoes simple harmonic motion as the wave passes through. 8.5.2 Velocity of a wave The frequency of a wave is determined by the frequency of the individual oscillating points. The amplitude of the wave is the amplitude of the oscillations. The wavelength of a wave is the distance between two successive points that behave identically (peak to peak, for example). The wavelength is denoted by λ. From these quantities we can determine the speed of the wave. The wave speed is the speed at which a particular part of the wave (like the peak) travels through the medium. Remember that speed is the displacement over time, v = ∆x ∆t . We know that the wave moves a distance of one wavelength in the time it takes for one point on the wave to move through a single cycle (it’s period), v = λ T = f λ. (8.20) 8.5.3 Interference of waves One interesting aspect of waves is how they interact with other waves. Two travelling waves will pass right through each other when they meet. When you throw two pebbles into the water near each other they will each create waves rippling from the point of entry. When those two waves meet they don’t destroy each other. Each wave comes out of the interaction undisturbed. At the point(s) where the two waves meet, they interact with each other in a process called interference. At these points, the motion of the points in space is determined by the principle of superposition: When two or more travelling waves encounter each other while moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point. If the peaks and troughs of two waves occur at the same place at the same time, the waves are in phase and the resulting interference is constructive interference. If the peak of one wave occurs at the same time and place as the trough of another wave then the waves are inverted and the resulting interference is destructive interference. In this case the waves completely cancel each other in the region where they interact (they will re-appear once they pass through each other). 8.5.4 Reflection of waves Waves cannot travel in a particular medium indefinitely. Eventually the waves will reach a boundary. When the waves reach the boundary, some of the wave will be reflected and some of the wave will be transmitted. The reflected wave can sometimes be inverted with respect to the incoming wave. Consider a wave on a string that approaches a wall where the string is fixed. The string will pull on the wall as the wave hits. The wall will exert an equal and opposite force on the string (Newton’s third law), pulling the string in the opposite direction. This causes the reflected wave to be inverted. If the end of the string is free to move, the reflected wave will have the same orientation as the original wave. 81 8.6 Sound waves Sound waves are longitudinal waves that are caused by vibrating objects. When an object vibrates, it pushes the air near it causing alternating compression and stretching of the spacing between molecules (density) in the air. This vibration is picked up by our ears and is interpreted by our brains as sound. In a sound wave, the air molecules oscillate along the direction of travel of the wave (think of the longitudinal wave on a spring). Sound waves can have a range of frequencies. The audible waves have frequencies between 20 and 20000 Hz. Infrasonic waves have frequencies below the audible range while ultrasonic waves have frequencies above the audible range. 8.6.1 Energy and intensity of sound waves Sound waves are created because a vibrating object pushes air molecules. The vibrating object exerts a force on the air and so is doing work on the air. The sound wave carries that energy away from the vibrating object. For waves, we don’t typically measure the total energy in the wave, but instead measure the flow of energy. The average intensity I of a wave on a given surface is defined as the rate at which energy flows through the surface, divided by the surface area, I = 1 A ∆E∆t, (8.21) where the direction of energy flow is perpendicular to the surface. The rate of energy transfer is power, so we can also write this as . (8.22) The units of intensity are W/m2. I = P A The faintest sounds a human ear can hear have an intensity of about 1 × 10−12 W/m2. This is the threshold of hearing. The loudest sounds the ear can tolerate have an intensity of about 1 W/m2. This is the threshold of pain. You’ll notice that the intensities that a human ear can detect vary over a very wide range. The quietest sounds don’t seem to us to be 1×1012 times quieter than the loudest sounds because our brains us an approximately logarithmic scale to determine loudness. This is measured by the intensity level defined by β = 10 log , (8.23) I I0 where the constant I0 = 1 × 10−12 W/m2 is the reference intensity. β is measured in decibels (dB). 82 Example 14.2: Noisy grinding machine A noisy grinding machine in a factory produces a sound intensity of 1.0 × 10−5 W/m2. Calculate (a) the decibel level of this machine and (b) the new intensity level when a second, identical machine is added to the factory. (c) A certain number of additional machines are put into operation alongside these two. The resulting decibel level is 77.0 dB. Find the sound intensity. Solution: (a) For a single grinder β = 10 log I I0 β = 10 log 1.0 × 10−5 W/m2 1.0 × 10−12 W/m2 β = 70.0 dB. (b) With a second grinder the total intensity is 2.0 × 10−5 W/m2. The decibel level is β = 10 log I I0 β = 10 log 2.0 × 10−5 W/m2 1.0 × 10−12 W/m2 β = 73.0 dB. (c) In this case we are given the decibel level and want the intensity β = 10 log I I0 77 dB = 10 log I 1.0 × 10−12 W/m2 I 1.0 × 10−12 W/m2 I = 5.01 × 10−5 W/m2. 107.7 = There are five machines in all. Many sound waves can be thought of as coming from a point source. A point source is small compared to the waves and emits waves symmetrically. The waves emitted by a point source are spherical waves; they spread in a uniform sphere. Suppose that the average power emitted by the source is Pav. Then the intensity at a distance r is I = Pav A = Pav 4πr2 . The average power always remains the same, no matter the distance so we can write Since the average power is the same, we get I1 = I2 = Pav 4πr2 1 Pav 4πr2 2 . I1 I2 = r2 2 r2 1 . 83 (8.24) (8.25) 8.6.2 The doppler effect When a moving object is making a sound, the frequency of the sound changes as as the object moves towards or away from the observer. Think of a train blowing its whistle — the whistle changes in pitch as the train approaches and as it moves away. This is known as the Doppler effect. Suppose that a source is moving through the air with velocity vs towards a stationary observer. Since the source is moving towards the observer, in the same direction as the sound wave, the waves emitted by the source get “squished”. The wavelength measured by the |
observer is shorter than the actual wavelength emitted by the source. During a single vibration, which lasts a time T (the period), the source moves a distance vsT = vs/fs. The wavelength detected by the observer is shortened by this amount, The frequency heard by the observer is λo = λs − vs fs . Rearranging, we get fo = v λo = v λs − vs fs = v − vs fs . v fs fo = fs v v − vs . (8.26) (8.27) The observed frequency increases when the source move towards the observer and decreases when it moves away (vs becomes negative). We can do a similar analysis for the case when the source is stationary and the observer is moving. In fact, both source and observer can be moving and this is covered by the general equation fo = fs v + vo v − vs . (8.28) The sign convention is that velocities are positive when source and observer move towards each other and negative when source and observer move away from each other. Example 14.4: Train whistle A train moving at a speed of 40.0 m/s sounds its whistle, which has a frequency of 5.00 × 102 Hz. Determine the frequency heard by a stationary observer as the train approaches the observer. Solution: The velocity of sound is 345 m/s. We use the equation for the doppler effect, keeping in mind that the train is approaching the observer, fo = fs v + vo v − vs fo = (500 Hz) fo = 566 Hz. 331 m/s 331 m/s − 40 m/s 8.7 Standing waves Suppose we connect one end of a string to a stationary clamp and the other end to a vibrating object. The vibrating object will move down to the end of the string and will be reflected. The reflected wave will interact with the wave originating from the object and the two waves will combine according to the principle If the string vibrates at exactly the right frequency the wave appears to stand still so of superposition. 84 it is called a standing wave. A node occurs when the two travelling waves have the same amplitude but opposite displacement, so the net displacement is zero. Halfway between two nodes there will be an antinode where the string vibrates with the largest amplitude. Note that the distance between two nodes is half the wavelength dN N = λ/2. Suppose we fix both ends of the string, then both ends must be nodes. We can then pluck the string so that we get a single antinode over the length of the string. This is the fundamental or first harmonic and we have half a wavelength on the string. Alternatively, we could set up our wave so that there is another node in the middle of the string. This is the second harmonic and we now have a full wavelength on the string. In fact there are many node/antinode patterns we can set up on the string always keeping in mind that the ends must be nodes. In general, we can set up waves whose wavelengths satisfy the condition λn = 2L n , (8.29) where n is the harmonic of the wave and can be any positive integer. The frequency of the harmonic is fn = v λn = vn 2L . (8.30) The velocity of a wave on a string depends on the tension in the string F and on the mass density of the string µ, F µ . v = This allows us to write fn = F µ . n 2L This series of frequencies forms a harmonic series. Example 14.8: Harmonics of a stretched wire (8.31) (8.32) (a) Find the frequencies of the fundamental and second harmonics of a steel wire 1.00 m long with a mass per unit length of 2.00 × 10−3 kg/m and under a tension of 80.0 N. (b) Find the wavelengths of the sound waves created by the vibrating wire. Assume the speed of sound is 345 m/s. Solution: (a) We use the formula for frequency with n = 1 and n = 2 for the fundamental and second harmonics, f1 = f1 = F µ F µ 1 2L 2 2L = = 1 2(1.00 m) 80.0 N 2.00 × 10−3 kg/m = 100 Hz 1 (1.00 m) 80.0 N 2.00 × 10−3 kg/m = 200 Hz. (b) The frequency of the vibrating string will be transferred to the air. The wave will then move at the speed of sound, λ1 = λ2 = v f1 v f2 = = 345 m/s 100 Hz 345 m/s 200 Hz = 3.45 m = 1.73 m 85 Standing waves can also be set up with sound waves in a pipe. Even if the end of the pipe is open, some of the sound wave will be reflected back into the pipe by the edges of the pipe. The reflected wave will interfere with the original wave and, if the frequency is right, a standing wave can be established. For pipes, the possible standing wave frequencies will depend on whether one end of the pipe is closed or if both ends are open. If both ends are open, then there must be antinodes at either end of the pipe. The first harmonic will have a single node in middle. If the length of the pipe is L, then the wavelength is λ1 = 2L. The second harmonic will have two nodes and a wavelength of λ2 = L. In general, the wavelength will be λn = 2L/n. The frequency will be fn = v λn nv 2L = , (8.33) where v is the speed of sound. For a pipe that is closed at one end, there must be a node at the closed end and an antinode at the open end. In this case, the first harmonic has only a quarter wavelength inside the pipe, λ1 = 4L. The next possible harmonic will have one node inside the pipe, but not at the center. In this case, the wavelength is λ3 = 4L/3. There are no even harmonics in a pipe with one end closed; only the odd harmonics are possible. In general, the wavelength is λn = 4L/n where n is odd integers only. The frequency is fn = nv 4L . (8.34) 8.8 Beats Standing waves are created by interference between two waves of the same frequency, More often, waves of different frequencies will interfere with each other. When waves of different frequencies interfere, then they will alternately go in and out of phase causing periods of constructive interference followed by periods of destructive interference. If you are listening to two sound waves of different frequencies, you will hear a sound that alternates between loud and soft. This loud/soft pattern is known as beats and is also wave. The frequency of the beats is determined by the frquency difference of the two waves fb = |f2 − f1|. (8.35) 86 Example: Out of tune pipes Two pipes of equal length are each open at one end. Each has a fundamental frequency of 480 Hz when the speed of sound is 347 m/s. In one pipe the air temperature is increased so that the speed of sound is now 350 m/s. If the two pipes are sounded together, what beat frequency results? Solution: We will need to find the new fundamental frequency of the second pipe. In order to do this, we need to know the length of the pipe. We know the frequency of the unheated pipe, so we can find the length, f1 = L = L = v 2L v 2f 347 m/s 2(480 Hz L = 0.36 m. Now we can find the new fundamental frequency of the second pipe, f1 = f1 = v 2L 350 m/s 2(0.36 m f1 = 486 Hz. The frequency of beats is the difference between those two frequencies, fb = |f1 − f2| fb = |486 Hz − 480 Hz| fb = 6 Hz. 87 Chapter 9 Solids and Fluids 9.1 States of matter The matter you interact with every day is typically classified as being in one of three states: solid, liquid or gas. There is also a fourth state of matter that you will not ordinarily encounter, plasma. Matter consists of molecules, which are groups of atoms. The properties of particular states of matter are determined by how molecules of a substance interact. At low temperatures, most substances are solid. Macroscopically, solids have a definite volume and shape. In a solid, the molecules are held in (relatively) fixed positions relative to each other. There are usually electrical bonds between molecules that make it difficult for molecules to move away from each other. As temperature increases, substances change from solid to liquid. A liquid has a definite volume, but no fixed shape. In a liquid, the molecules are weakly bound to each other and so can move within the substance with some freedom. Further increases in temperature completely break the bonds between molecules, changing the liquid into a gas. A gas has no definite shape and no definite volume. The molecules of a gas are far from each other (relative to the size of the molecules) and very rarely interact. This means that the gas can expand to fill an volume. At extremely high temperatures (such as those encountered inside stars) the molecules and atoms of the substance are torn apart. Positive and negatively charged particles are free to move around within the substance creating long-range electrical and magnetic forces. This is a plasma. 9.1.1 Characterizing matter Even though substances may be in the same state and will have some broad general characteristics in common, they are by no means identical. For example, two equal masses of different substances may not take up the same volume. This property is the density of a substance and is defined as the mass divided by its volume, ρ = . (9.1) M V The SI unit for density is kg/m3. The density of a liquid or solid varies slightly with temperature and pressure. The density of a gas is very sensitive to temperature and pressure. Liquids are generally, but not always, less dense than solids. Gasses are about 1000 times less dense than liquids or solids. It is sometimes convenient to standardize density by comparing it to some standard. The specific gravity of a substance is the ratio of its density to the density of water at 4 ◦C, which is 1.0 × 103 kg/m3. 88 9.2 Deformation of solids While a solid tends to have a definite shape, the shape can often be altered with the application of a force. While a strong enough force will permanently alter the shape, often when the force is removed, the substance will return to its original shape. This is called elastic behaviour. Different substances have different elastic properties, so we will need some way to quantify or characterize this. The stress on a material is the force per unit area that is causing some deformation. The strain is a measure of the amount of deformation in the material. For small stresses, stress is proportional to the strain with the constant of proportionality depending on the properties of the material. The proportionality constant is called the elastic modulus. The equation for the elasticity of a substance is |
similar to Hooke’s law, F = −k∆x and the elastic modulus can be thought of as a spring constant. It is a measure of the stiffness of a material. A substance with a large elastic modulus is hard to deform. 9.2.1 Young’s modulus Suppose we have a long bar of cross-sectional A and length L0 that is clamped at one end. When we apply an external force F along the bar, we can change the length of the bar. At this new length, the external force is balanced by internal forces that resist the stretch. The bar is said to be stressed. The tensile stress is the magnitude of the external force divided by the cross-sectional area. The tensile strain is the ratio of the change in length to the original length. Since we know that stress and strain are proportional, we have In this equation the proportionality constant is called Young’s modulus. We can re-write this equation as F A = Y ∆L L0 . (9.2) F = Y A L0 ∆L, so that it looks like Hooke’s law with a spring constant of k = Y A/L0. The Young’s modulus depends on whether the material is being stretched or compressed. Many materials are easier to stretch than compress. The elastic response is also not quite linear and substances have an elastic limit. The elastic limit is the point at which the stress is no longer proportional to the strain. If stretched (compressed) beyond this limit, substances will not return to their original shape once the force is released. The ultimate strength is the largest stress that the substance can endure and any force beyond that reaches the substance’s breaking point. 9.2.2 Shear modulus Suppose we have a rectangular block of some substance. One side of the rectangle is held in a fixed position. A force is applied to the other side, parallel to the side (think of sliding the cover of a book that is sitting on a table). This is called shear stress. The shear stress is the ratio of the magnitude of the applied force to the area of the face being sheared. The shear strain is the ratio of the horizontal distance moved to the height of the object. F A ∆x h = S , (9.3) where S is the shear modulus of the substance. In this case, the “spring constant” is k = SA/h. A substance with a large shear modulus is difficult to bend. 89 9.2.3 Bulk modulus Suppose we have a block of some substance and we squeeze it uniformaly with a perpendicular force from all sides. This type of squeezing is common when a substance is immersed in a fluid. The volume stress is the ratio of the change in the magnitude of the applied force to the surface area. The volume strain is the ratio of the change in volume to the original volume. ∆F A = −B ∆V V , (9.4) where B is the bulk modulus. The negative sign appears so that B is positive. An increase in the external force (more squeezing) results in a decrease of the volume. Materials with a large bulk modulus are difficult to compress. Example: Shear stress on the spine Between each pair of vertebrae of the spine is a disc of cartilage of thickness 0.5 cm. Assume the disc has a radius of 0.04 m. The shear modulus of cartilage is 1 × 107 N/m2. A shear force of 10 N is applied to one end of the disc while the other end is held fixed. (a) What is the resulting shear strain? (b) How far has one end of the disc moved with respect to the other end? Solution: (a) The shear strain is caused by the shear force, strain = strain = F AS 10 N π(0.04 m)2(1 × 107 N/m2) strain = 1.99 × 10−4. (b) A shear strain is defined as the displacement over the height, strain = ∆x h ∆x = h × strain ∆x = (0.5 cm)(1.99 × 10−4) ∆x = 0.99 µm. 9.3 Pressure and fluids While a force can deform or break a solid, forces applied to a fluid have a different result. When a fluid is at rest, all parts of the fluid are in static equilibrium. This means that the forces are balanced for every point in the fluid. If there was some kind of a force imbalance at one point, then that part of the fluid would move. When discussing fluids, we often don’t consider a force directly, but rather use the pressure which is the force per unit area, since fluids (and the forces that act on them) tend to be extended over some region of space. Mathematically, the pressure is given by the formula, P = F A . (9.5) The units of pressure are newton per meter2 or pascal (Pa). Suppose we have some fluid sitting in equilibrium in a large container. Consider the forces acting on a piece of the fluid extending from y1 to y2 (y = 0 is the top of the fluid) and having a cross-sectional area A. There are three forces acting on this piece of fluid: the force of gravity (M g), the force caused by the 90 pressure of the fluid above this piece pushing down (P1A), and the force caused by the pressure from the fluid below pushing up (P2A). This piece of fluid is not moving, so the forces must balance, P2A − P1A − M g = 0. (9.6) We can find the mass of the water from the density M = ρV = ρA(y1 − y2). Substituting into our equation, we get P2 = P1 + ρg(y1 − y2). (9.7) You’ll notice that the force of the fluid pushing upward is larger than the force of the fluid pushing down (the difference being the weight of the fluid we’re considering). For a liquid near the surface of the earth exposed to the earth’s atmosphere, this equation can give us the pressure at any depth h, P = P0 + ρgh, (9.8) where P0 = 1.013 × 105 Pa is the atmospheric pressure at sea level. The atmospheric pressure arises because air is also a fluid and the large column of air over the surface of any point on earth will exert a downward pressure on the earth. This equation also suggests that if you change the pressure at the surface of a fluid, then the change is transmitted to every point in the fluid. This is known as Pascal’s principle. The change in pressure will also be transmitted to the containers enclosing the fluid. Example: Oil and Water In a huge oil tanker, salt water has flooded an oil tank to a depth h2 = 5.00 m. On top of the water is a layer of oil h1 = 8.00 m deep. The oil has a density of 700 kg/m3 and salt water has a density of 1025 kg/m3. Find the pressure at the bottom of the tank. Solution: The surface of the oil is exposed to air, so the pressure at that point will be atmospheric pressure. We can find the pressure at the bottom of the oil layer using our equation, P1 = P0 + ρoilgh1 P1 = 1.01 × 105 Pa + (700 kg/m3)(9.8 m/s2)(8.00 m) P1 = 1.56 × 105 Pa. This is the pressure at the surface of the water layer. At the bottom of that layer, the pressure is, P2 = P1 + ρwatergh2 P2 = 1.56 × 105 Pa + (1025 kg/m3)(9.8 m/s2)(5.00 m) P2 = 2.06 × 105 Pa. 9.4 Buoyant forces The idea of buoyancy was discovered by the Greek mathematician Archimedes and is known as Archimedes’ principle: Any object completely or partially submerged in a fluid is buoyed up by a force with magnitude equal to the weight of the fluid displaced by the object. Basically, buoyancy is the pressure difference between the fluid below and above an object. We know that the pressure from fluid below is larger than pressure from the fluid above, so the object will feel lighter if we try to lift it (since the fluid is helping us to move the object upward). 91 Suppose we replace the little piece of fluid in our container with a piece of lead of the same volume. The pressure above and below the lead will not change — their difference is still equal to the mass of the fluid. The lead is denser than water, so the piece of lead is heavier and the downward force of gravity is now larger. Since the forces are no longer in equilibrium, the lead will sink. The buoyant force is due to pressure differences in the surrounding fluid and will not change if a new substance is introduced. The buoyant force is given by, B = ρfluidVfluidg, (9.9) where Vfluid is the volume of fluid displaced by the object. 9.4.1 Fully submerged object For a fully submerged object, the buoyant force pushes upwards while the force of gravity pulls the object downwards. M g − B = M a ρfluidVfluidg − ρobjectVobjectg = ρobjectVobjecta a = (ρfluid − ρobject) g ρobject . (9.10) The acceleration will be positive (upwards) if the density of the fluid is larger than the density of the object. It will be negative (downwards) if the density of the object is larger than the density of the fluid. 9.4.2 Partially submerged object In this case, the object is in equilibrium since it is floating in the fluid and not moving either up or down. This means that the forces must be in equilibrium. B = M g ρfluidVfluidg = ρobjectVobjectg ρobject ρfluid = Vfluid Vobject . (9.11) . 92 Example 9.8: Weighing a crown A bargain hunter purchases a “gold” crown at a flea market. After she gets home, she hangs it from a scale and finds its weight to be 7.48 N. She then weighs the crown while it is immersed in water and now the scale reads 6.86 N. Is the crown made of pure gold? Solution: To determine whether the crown is actually made of gold, we need to find the density of the crown. For this, we will need both the volume and mass of the crown. When the crown is weighed in the air, we have, When the crown is in water, the buoyant force needs to be included, Tair − mg = 0. Given these two equations, then, we must have that Twater + B − mg = 0. B = Tair − Twater B = 7.48 N − 6.86 N B = 0.980 N. The buoyant force is equal to the weight of the water displaced, B = ρwaterVobjg Vobj = Vobj = B ρwaterg 0.980 N (1.0 × 103 kg/m3)(9.8 m/s2) Vobj = 1.0 × 10−4 m3. We can easily get the mass from the first equation, m = Tair g 7.48 N 9.8 m/s2 m = 0.800 kg. m = The density of the crown is ρcrown = ρcrown = m Vobj 0.800 kg 1.0 × 10−4 m3 ρcrown = 8.0 × 103 kg/m3. The density of gold is 19.3 × 103 kg/m3, so this crown is definitely not solid gold. 93 9.5 Fluids in motion When fluids move, there are two broad categories for the type of motion. Laminar or streamline motion occurs when every particle that passes a particular point moves along the same smooth path followed by previous particles passing that point. The path itself is called a streamline. During laminar motion, different streamlines will not cross. When the mot |
ion of the fluid becomes irregular, or turbulent, the streamlines disappear and neighbouring particles can end up moving in very different directions. In turbulent flow, you tend to see eddy currents (little whirlpools) and other non-linear patterns. We will only study laminar motion in this course — turbulent motion is very complicated — and we will only consider the motion of an ideal fluid. The ideal fluid has the following properties: • The fluid is non-viscous, which means there is no internal friction between adjacent particles. (The viscosity of a fluid is a measure of the amount of internal friction.) • The fluid is incompressible, which means the density is constant. • The fluid motion is steady, meaning that the velocity and pressure at each point does not change in time. • The fluid moves without turbulence. This condition means that the particles have no rotational motion and no angular velocity — they only move in straight lines. 9.5.1 Equation of continuity Suppose a fluid flows in a pipe whose cross-sectional area increases from left to right, going from A1 at one end to A2 at the other end. Suppose the fluid enters the pipe with a velocity v1. The fluid entering the pipe moves a distance ∆x1 = v1∆t in a time ∆t. The mass of water contained in this region is ∆M1 = ρwaterA1∆x1 = ρwaterA1v1∆t. We can write a similar equation for the mass flowing out of the other end of the pipe, ∆M2 = ρwaterA2v2∆t. Since mass is conserved (the fluid is incompressible), we must have the same amount of mass going in as is coming out, ∆M1 = ∆M2, or A1v1 = A2v2. (9.12) This equation is known as the equation of continuity. It tells us that fluid will speed up or slow down as the area through which they flow changes. Fluid flows faster through a pipe of small area than through a pipe of large area. The product Av is also known as the flow rate. 94 Example 9.12: Garden hose A water hose 2.5 cm in diameter is used by a gardener to fill a 30.0 L bucket. The gardener notices that it takes 1.0 min to fill the bucket. A nozzle with an opening of cross-sectional are 0.500 cm2 is then attached to the hose. The nozzle is held so the water is projected horizontally from a point 1.0 m above the ground. Over what horizontal distance will the water be projected? Solution: We first need to determine the flow rate for the water in the absence of the nozzle. We can figure this out from how long it takes to fill the bucket, A1v1 = 1000 cm3 1 L A1v1 = 5.0 × 10−4 m3/s. 30 L 1.00 min 1 m 100 cm 3 1 min 60 s The flow rate remains constant when the new nozzle is attached, v2 = A1v1 = A2v2 A1v1 A2 5.0 × 10−4 m3/s 0.5 × 10−4 m2 v2 = v2 = 10.0 m/s. This is the initial horizontal velocity of the water. Once the water leaves the hose, it is a projectile undergoing acceleration in the vertical direction (but not horizontally). We can find how long it takes for the water to hit the ground, 1 2 gt2 y = v0yt − 2y g t = t = 2(1.0 m) 9.8 m/s2 t = 0.452 s. Now we can find the horizontal distance travelled, x = v0xt x = (10 m/s)(0.452 s) x = 4.52 m. 9.6 Bernoulli’s equation Suppose that the pipe with varying diameter is now angled upwards. Let’s consider the work done on the fluid in the pipe. The fluid at the lower end is pushed by the fluid behind it. The fluid at the upper end is pushed by the fluid in front of it. The net work done on the fluid in the pipe then is W = F1∆x1 − F2∆x2 W = P1A1∆x1 − P2A2∆x2 W = P1V − P2V. 95 (9.13) The work done on the fluid can do one of two things: it can change the kinetic energy of the fluid, and it can change the gravitational potential energy, W = ∆KE + ∆P E. Combining these two equations we have, Now we can put in expressions for the kinetic and potential energies, P1V − P2V = ∆KE + ∆P E. We can re-write this as P1 − P2 = 1 2 ρv2 2 − 1 2 ρv2 1 + ρgy2 − ρgy1. P1 + 1 2 ρv2 1 + ρgy1 = P2 + 1 2 ρv2 2 + ρgy2. (9.14) (9.15) (9.16) (9.17) This is known as Bernoulli’s equation. Note that this is not really a new concept, it is just the conservation of energy applied to a fluid. This equation is only true, however, for laminar flow. One consequence of this equation is that faster moving fluids exert less pressure than slowly moving fluids. We can see this by considering the pipe with a varying diameter when it is horizontally level. In this case, Bernoulli’s equation simplifies to P1 + 1 2 ρv2 1 = P2 + 1 2 ρv2 2. We know that the fluid moves faster in the narrow region, so the pressure in that region must be lower than in the wide region in order for the two sides of the equation to balance. Example 9.13: Shootout A nearsighted sheriff fires at a cattle rustler with his trust six-shooter. Fortunately for the rustler, the bullet misses him and penetrates the town water tank, causing a leak. If the top of the tank is open to the atmosphere, determine the speed at which the water leaves the hole when the water level is 0.500 m above the hole. Solution: We can use Bernoulli’s equation to find the velocity. Let’s choose the first point to be the top of the tank and the second point will be the hole. The pressure at both of these points is just P0, the standard atmospheric pressure. We assume that the water level drops very slowly, so the velocity of the fluid at the top of the tank is zero. Putting these into Bernoulli’s equation, P1 + 1 2 ρv2 1 + ρgy1 = P2 + 1 2 ρv2 2 + ρgy2 1 2 ρgy1 = ρv2 2 + ρgy2 v = 2g(y1 − y2) v = 2(9.8 m/s2)(0.500 m) v = 3.13 m/s. 96 Chapter 10 Thermal physics There is a form of energy that we have to date neglected to consider in our description of objects. This is primarily because this form of energy is not typically involved in macroscopic motion. All objects have thermal energy, a type of energy which we intuitively detect as the object being hot or cold. 10.1 Temperature Determining whether an object is hot or cold is rather inexact and we would like to find a more quantitative way of measuring thermal energy. We say that two objects are in thermal contact if energy (particularly thermal energy) can be exchanged between them. Two objects are in thermal equilibrium if they are in thermal contact but there is no net exchange of energy between them. The zeroth law of thermodynamics (law of equilibrium) states: If objects A and B are separately in thermal equilibrium with a third object C , then A and B are in thermal equilibrium with each other. This law allows us to use a thermometer to compare the thermal energies of two objects. Suppose we want to compare the thermal energies of two objects A and B, we could put them into thermal contact and try to detect the direction of energy flow. Suppose, however, that we cannot put the two objects into thermal contact directly. We can then use a third object, C, to compare the thermal energies of A and B. We first put the thermometer into thermal contact with A until it reaches thermal equilibrium at which point we read the thermometer. We then put the thermometer into thermal contact with B until it reaches thermal equilibrium and read the temperature again. If the two readings are the same then A and B are also in thermal equilibrium. This property allows us to define temperature — if two objects are in thermal equilibrium, then they have the same temperature. The thermometer used to measure thermal energy must have some physical property that changes with temperature. Most thermometers use the fact that substances (solids, gasses, liquids) expand as temperature increases. This physical change can usually be measured visually allowing us to put a number on the temperature. We must first, however, calibrate the thermometer. That is, we must agree on a measurement scale for temperature. The Celsius temperature scale is defined by measuring the freezing point of water which is set to be 0 ◦C. and the boiling point of water which is set to be 100 ◦C. The scale most commonly used in the US is the Fahrenheit scale. On this scale, the freezing point is at 32◦F and the boiling point is at 212◦F. We can convert between the two using the formula The temperature scale most often used by scientists is the Kelvin scale. One of the problems with both the Celsius and Fahrenheit scales is that the freezing point and boiling point of water depend not only on the TF = 9 5 TC + 32. (10.1) 97 temperature but on the pressure. Scientists removed the pressure dependence by observing that the pressure of all substances goes to zero at a temperature of −273.15 ◦C. This temperature is known as absolute zero and is defined to be 0 K. The second point used to define the Kelvin scale is the triple point of water. This is the temperature and pressure at which water, water vapour, and ice exist in equilibrium. This point occurs at 0.01 ◦C and 4.58 mm of mercury. This temperature is defined to be 273.16 K. This means that the unit size of both the celsius and kelvin scales are the same. We convert between the two using, TC = TK − 273.15. (10.2) 10.2 Thermal expansion Most substances increase in volume as their temperature (thermal energy) increases. The thermal energy of an object is actually a measure of the average velocity of the constituent atoms. As temperature increases, atoms move faster. In solids and liquids, these atoms cannot actually leave the substance, so their vibrational motion increases leading to an increased separation between atoms. Macroscopically, we see this as an increase in volume. If the expansion is small compared to the object’s original size, the expansion in one dimension is approximately linear with temperature, where ∆L is the change in length (not volume), L is the original length of the object, and α is the coefficient of linear expansion for a particular substance. ∆L = αL0∆T, (10.3) Example 10.3: Expansion of a railroad track (a) A steel railroad track has a length of 30.0 m when the temperature is 0 ◦C. What is the length on a hot day when the temperature is 40.0 ◦C? (b) What is the stress caused by this expansion? Solution: (a) The change in length due to the temperature change, ∆L = αL0∆T ∆L = (11 × 10−6 /◦C |
)(30.0 m)(40.0 ◦C) ∆L = 0.013 m. So the new length is 30.013 m. (b) The railroad undergoes a linear expansion, so this is a tensile strain, = Y ∆L L = (2.0 × 1011 Pa = 8.7 × 107 Pa. F A F A F A 0.013 m 30.0 m Since their is a linear expansion of objects with temperature, there must also be a change in their area and volume. Suppose we have a square of material with a length of L0. Each dimension of the square will undergo linear expansion and the new area is A = L2 A = (L0 + αL0∆T )(L0 + αL0∆T ) A = L2 0α∆T + (αL0∆T )2. 0 + 2L2 98 The last term in that equation will be very small, so we will ignore it, We can re-write this so that it looks like the linear expansion equation, A = A0 + 2αA0∆T. ∆A = 2αA0∆T. (10.4) (10.5) We define a new coefficient γ = 2α as the coefficient of area expansion. We can perform the same type of derivation and show that the increase in volume of a substance is given by ∆V = βV0∆T, (10.6) where β is the coefficient of volume expansion and is given by β = 3α. 10.3 Ideal gas law The effect of temperature change on a gas is somewhat more complex than in solids and liquids. A gas will expand to fill a particular container no matter what the temperature. What will change instead as the temperature increases is the pressure. There is usually a fairly complex relationship between the pressure, volume and temperature of gasses, but for an ideal gas, we can derive a simple relationship. An ideal gas is a gas that is maintained at low density or pressure. In an ideal gas, particles of the gas are so far apart that they rarely interact and so we can assume there are no forces acting on any of the particles and no collisions take place. Each particle of the gas moves randomly. Since gases contain large numbers of particles, we usually count the number of particles in moles where one mole is 6.02×1023 gas particles. The number 6.02×1023 is known as Avogadro’s number and is denoted by NA. Avogadro’s number was chosen so that the mass in grams of one mole of an element is numerically the same as the atomic mass units of the element. Carbon 12 has an atomic mass of 12 amu, so one mole of carbon 12 weighs 12 g. For an ideal gas, the relationship between pressure, volume, and temperature is P V = nRT, (10.7) where n is the number of moles of the substance, and R is the universal gas constant with a value of R = 8.31 J/mol · K. The ideal gas law tells us that the pressure is linearly proportional to temperature and inversely proportional to the volume. As temperature increases, pressure increases. As volume increases, pressure decreases. 99 Example 10.6: Expanding gas An ideal gas at 20.0 ◦C and a pressure of 1.50 × 105 Pa is in a container having a volume of 1.0 L. (a) Determine the number of moles of gas in the container. (b) The gas pushes against a piston, expanding to twice its original volume, while the pressure falls to atmospheric pressure. Find the final temperature. Solution: (a) We need to be careful that all quantities are in SI units. We will need to convert the temperature to kelvins: T = 20 + 273 = 293 K. And we need to convert the volume to m3: V = 1.0 L = 1.0 × 10−3 m3. Now we can go ahead and plug the values into the gas law, n = P V = nRT P V RT (1.50 × 105 Pa)(1.0 × 10−3 m3) (8.31 J/mol · K)(293 K) n = n = 6.16 × 10−2 mol. (b) We can find the new temperature from the gas law, T = T = P V nR (1.01 × 105 Pa)(2.0 × 10−3 m3) (6.16 × 10−2 mol)(8.31 J/mol · K) T = 395 K. 100 |
ed by classical physics. One reason for this is that they are small enough to travel at great speeds, near the speed of light. Figure 1.5 Using a scanning tunneling microscope (STM), scientists can see the individual atoms that compose this sheet of gold. (Erwinrossen) At particle colliders (Figure 1.6), such as the Large Hadron Collider on the France-Swiss border, particle physicists can make subatomic particles travel at very high speeds within a 27 kilometers (17 miles) long superconducting tunnel. They can then study the properties of the particles at high speeds, as well as collide them with each other to see how they exchange energy. This has led to many intriguing discoveries such as the Higgs-Boson particle, which gives matter the property of mass, and antimatter, which causes a huge energy release when it comes in contact with matter. Figure 1.6 Particle colliders such as the Large Hadron Collider in Switzerland or Fermilab in the United States (pictured here), have long tunnels that allows subatomic particles to be accelerated to near light speed. (Andrius.v ) Physicists are currently trying to unify the two theories of modern physics, relativity and quantum mechanics, into a single, comprehensive theory called relativistic quantum mechanics. Relating the behavior of subatomic particles to gravity, time, and space will allow us to explain how the universe works in a much more comprehensive way. Application of Physics You need not be a scientist to use physics. On the contrary, knowledge of physics is useful in everyday situations as well as in nonscientific professions. For example, physics can help you understand why you shouldn’t put metal in the microwave (Figure 1.7), why a black car radiator helps remove heat in a car engine, and why a white roof helps keep the inside of a house cool. The operation of a car’s ignition system, as well as the transmission of electrical signals through our nervous system, are much easier to understand when you think about them in terms of the basic physics of electricity. Figure 1.7 Why can't you put metal in the microwave? Microwaves are high-energy radiation that increases the movement of electrons in Access for free at openstax.org. 1.1 • Physics: Definitions and Applications 11 metal. These moving electrons can create an electrical current, causing sparking that can lead to a fire. (= MoneyBlogNewz) Physics is the foundation of many important scientific disciplines. For example, chemistry deals with the interactions of atoms and molecules. Not surprisingly, chemistry is rooted in atomic and molecular physics. Most branches of engineering are also applied physics. In architecture, physics is at the heart of determining structural stability, acoustics, heating, lighting, and cooling for buildings. Parts of geology, the study of nonliving parts of Earth, rely heavily on physics; including radioactive dating, earthquake analysis, and heat transfer across Earth’s surface. Indeed, some disciplines, such as biophysics and geophysics, are hybrids of physics and other disciplines. Physics also describes the chemical processes that power the human body. Physics is involved in medical diagnostics, such as xrays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements (Figure 1.8). Medical therapy Physics also has many applications in biology, the study of life. For example, physics describes how cells can protect themselves using their cell walls and cell membranes (Figure 1.9). Medical therapy sometimes directly involves physics, such as in using X-rays to diagnose health conditions. Physics can also explain what we perceive with our senses, such as how the ears detect sound or the eye detects color. Figure 1.8 Magnetic resonance imaging (MRI) uses electromagnetic waves to yield an image of the brain, which doctors can use to find diseased regions. (Rashmi Chawla, Daniel Smith, and Paul E. Marik) Figure 1.9 Physics, chemistry, and biology help describe the properties of cell walls in plant cells, such as the onion cells seen here. (Umberto Salvagnin) BOUNDLESS PHYSICS The Physics of Landing on a Comet On November 12, 2014, the European Space Agency’s Rosetta spacecraft (shown in Figure 1.10) became the first ever to reach and orbit a comet. Shortly after, Rosetta’s rover, Philae, landed on the comet, representing the first time humans have ever landed a space probe on a comet. 12 Chapter 1 • What is Physics? Figure 1.10 The Rosetta spacecraft, with its large and revolutionary solar panels, carried the Philae lander to a comet. The lander then detached and landed on the comet’s surface. (European Space Agency) After traveling 6.4 billion kilometers starting from its launch on Earth, Rosetta landed on the comet 67P/ChuryumovGerasimenko, which is only 4 kilometers wide. Physics was needed to successfully plot the course to reach such a small, distant, and rapidly moving target. Rosetta’s path to the comet was not straight forward. The probe first had to travel to Mars so that Mars’s gravity could accelerate it and divert it in the exact direction of the comet. This was not the first time humans used gravity to power our spaceships. Voyager 2, a space probe launched in 1977, used the gravity of Saturn to slingshotover to Uranus and Neptune (illustrated in Figure 1.11), providing the first pictures ever taken of these planets. Now, almost 40 years after its launch, Voyager 2 is at the very edge of our solar system and is about to enter interstellar space. Its sister ship, Voyager 1 (illustrated in Figure 1.11), which was also launched in 1977, is already there. To listen to the sounds of interstellar space or see images that have been transmitted back from the Voyager I or to learn more about the Voyager mission, visit the Voyager’s Mission website (https://openstax.org/l/28voyager) . Figure 1.11 a) Voyager 2, launched in 1977, used the gravity of Saturn to slingshot over to Uranus and Neptune. NASA b) A rendering of Voyager 1, the first space probe to ever leave our solar system and enter interstellar space. NASA Both Voyagers have electrical power generators based on the decay of radioisotopes. These generators have served them for almost 40 years. Rosetta, on the other hand, is solar-powered. In fact, Rosetta became the first space probe to travel beyond the asteroid belt by relying only on solar cells for power generation. At 800 million kilometers from the sun, Rosetta receives sunlight that is only 4 percent as strong as on Earth. In addition, it is very cold in space. Therefore, a lot of physics went into developing Rosetta’s low-intensity low-temperature solar cells. In this sense, the Rosetta project nicely shows the huge range of topics encompassed by physics: from modeling the movement of gigantic planets over huge distances within our solar systems, to learning how to generate electric power from low-intensity light. Physics is, by far, the broadest field of science. GRASP CHECK What characteristics of the solar system would have to be known or calculated in order to send a probe to a distant planet, such as Jupiter? Access for free at openstax.org. 1.1 • Physics: Definitions and Applications 13 a. b. c. d. the effects due to the light from the distant stars the effects due to the air in the solar system the effects due to the gravity from the other planets the effects due to the cosmic microwave background radiation In summary, physics studies many of the most basic aspects of science. A knowledge of physics is, therefore, necessary to understand all other sciences. This is because physics explains the most basic ways in which our universe works. However, it is not necessary to formally study all applications of physics. A knowledge of the basic laws of physics will be most useful to you, so that you can use them to solve some everyday problems. In this way, the study of physics can improve your problem-solving skills. Check Your Understanding 1. Which of the following is notan essential feature of scientific explanations? a. They must be subject to testing. b. They strictly pertain to the physical world. c. Their validity is judged based on objective observations. d. Once supported by observation, they can be viewed as a fact. 2. Which of the following does notrepresent a question that can be answered by science? a. How much energy is released in a given nuclear chain reaction? b. Can a nuclear chain reaction be controlled? c. Should uncontrolled nuclear reactions be used for military applications? d. What is the half-life of a waste product of a nuclear reaction? 3. What are the three conditions under which classical physics provides an excellent description of our universe? a. b. c. d. 1. Matter is moving at speeds less than about 1 percent of the speed of light 2. Objects dealt with must be large enough to be seen with the naked eye. 3. Strong electromagnetic fields are involved. 1. Matter is moving at speeds less than about 1 percent of the speed of light. 2. Objects dealt with must be large enough to be seen with the naked eye. 3. Only weak gravitational fields are involved. 1. Matter is moving at great speeds, comparable to the speed of light. 2. Objects dealt with are large enough to be seen with the naked eye. 3. Strong gravitational fields are involved. 1. Matter is moving at great speeds, comparable to the speed of light. 2. Objects are just large enough to be visible through the most powerful telescope. 3. Only weak gravitational fields are involved. 4. Why is the Greek word for nature appropriate in describing the field of physics? a. Physics is a natural science that studies life and living organism on habitable planets like Earth. b. Physics is a natural science that studies the laws and principles of our universe. c. Physics is a physical science that studies the composition, structure, and changes of matter in our universe. d. Physics is a social science that studies the social behavior of living beings on habitable planets like |
Earth. 5. Which aspect of the universe is studied by quantum mechanics? a. objects at the galactic level b. objects at the classical level c. objects at the subatomic level d. objects at all levels, from subatomic to galactic 14 Chapter 1 • What is Physics? 1.2 The Scientific Methods Section Learning Objectives By the end of this section, you will be able to do the following: • Explain how the methods of science are used to make scientific discoveries • Define a scientific model and describe examples of physical and mathematical models used in physics • Compare and contrast hypothesis, theory, and law Section Key Terms experiment hypothesis model observation principle scientific law scientific methods theory universal Scientific Methods Scientists often plan and carry out investigations to answer questions about the universe around us. Such laws are intrinsic to the universe, meaning that humans did not create them and cannot change them. We can only discover and understand them. Their discovery is a very human endeavor, with all the elements of mystery, imagination, struggle, triumph, and disappointment inherent in any creative effort. The cornerstone of discovering natural laws is observation. Science must describe the universe as it is, not as we imagine or wish it to be. We all are curious to some extent. We look around, make generalizations, and try to understand what we see. For example, we look up and wonder whether one type of cloud signals an oncoming storm. As we become serious about exploring nature, we become more organized and formal in collecting and analyzing data. We attempt greater precision, perform controlled experiments (if we can), and write down ideas about how data may be organized. We then formulate models, theories, and laws based on the data we have collected, and communicate those results with others. This, in a nutshell, describes the scientific method that scientists employ to decide scientific issues on the basis of evidence from observation and experiment. An investigation often begins with a scientist making an observation. The scientist observes a pattern or trend within the natural world. Observation may generate questions that the scientist wishes to answer. Next, the scientist may perform some research about the topic and devise a hypothesis. A hypothesis is a testable statement that describes how something in the natural world works. In essence, a hypothesis is an educated guess that explains something about an observation. Scientists may test the hypothesis by performing an experiment. During an experiment, the scientist collects data that will help them learn about the phenomenon they are studying. Then the scientists analyze the results of the experiment (that is, the data), often using statistical, mathematical, and/or graphical methods. From the data analysis, they draw conclusions. They may conclude that their experiment either supports or rejects their hypothesis. If the hypothesis is supported, the scientist usually goes on to test another hypothesis related to the first. If their hypothesis is rejected, they will often then test a new and different hypothesis in their effort to learn more about whatever they are studying. Scientific processes can be applied to many situations. Let’s say that you try to turn on your car, but it will not start. You have just made an observation! You ask yourself, "Why won’t my car start?" You can now use scientific processes to answer this question. First, you generate a hypothesis such as, "The car won’t start because it has no gasoline in the gas tank." To test this hypothesis, you put gasoline in the car and try to start it again. If the car starts, then your hypothesis is supported by the experiment. If the car does not start, then your hypothesis is rejected. You will then need to think up a new hypothesis to test such as, "My car won’t start because the fuel pump is broken." Hopefully, your investigations lead you to discover why the car won’t start and enable you to fix it. Modeling A model is a representation of something that is often too difficult (or impossible) to study directly. Models can take the form of physical models, equations, computer programs, or simulations—computer graphics/animations. Models are tools that are especially useful in modern physics because they let us visualize phenomena that we normally cannot observe with our senses, such as very small objects or objects that move at high speeds. For example, we can understand the structure of an atom using models, despite the fact that no one has ever seen an atom with their own eyes. Models are always approximate, so they are simpler to consider than the real situation; the more complete a model is, the more complicated it must be. Models put the Access for free at openstax.org. 1.2 • The Scientific Methods 15 intangible or the extremely complex into human terms that we can visualize, discuss, and hypothesize about. Scientific models are constructed based on the results of previous experiments. Even still, models often only describe a phenomenon partially or in a few limited situations. Some phenomena are so complex that they may be impossible to model them in their entirety, even using computers. An example is the electron cloud model of the atom in which electrons are moving around the atom’s center in distinct clouds (Figure 1.12), that represent the likelihood of finding an electron in different places. This model helps us to visualize the structure of an atom. However, it does not show us exactly where an electron will be within its cloud at any one particular time. Figure 1.12 The electron cloud model of the atom predicts the geometry and shape of areas where different electrons may be found in an atom. However, it cannot indicate exactly where an electron will be at any one time. As mentioned previously, physicists use a variety of models including equations, physical models, computer simulations, etc. For example, three-dimensional models are often commonly used in chemistry and physics to model molecules. Properties other than appearance or location are usually modelled using mathematics, where functions are used to show how these properties relate to one another. Processes such as the formation of a star or the planets, can also be modelled using computer simulations. Once a simulation is correctly programmed based on actual experimental data, the simulation can allow us to view processes that happened in the past or happen too quickly or slowly for us to observe directly. In addition, scientists can also run virtual experiments using computer-based models. In a model of planet formation, for example, the scientist could alter the amount or type of rocks present in space and see how it affects planet formation. Scientists use models and experimental results to construct explanations of observations or design solutions to problems. For example, one way to make a car more fuel efficient is to reduce the friction or drag caused by air flowing around the moving car. This can be done by designing the body shape of the car to be more aerodynamic, such as by using rounded corners instead of sharp ones. Engineers can then construct physical models of the car body, place them in a wind tunnel, and examine the flow of air around the model. This can also be done mathematically in a computer simulation. The air flow pattern can be analyzed for regions smooth air flow and for eddies that indicate drag. The model of the car body may have to be altered slightly to produce the smoothest pattern of air flow (i.e., the least drag). The pattern with the least drag may be the solution to increasing fuel efficiency of the car. This solution might then be incorporated into the car design. Snap Lab Using Models and the Scientific Processes Be sure to secure loose items before opening the window or door. In this activity, you will learn about scientific models by making a model of how air flows through your classroom or a room in your house. • One room with at least one window or door that can be opened • Piece of single-ply tissue paper 1. Work with a group of four, as directed by your teacher. Close all of the windows and doors in the room you are working in. Your teacher may assign you a specific window or door to study. 16 Chapter 1 • What is Physics? 2. Before opening any windows or doors, draw a to-scale diagram of your room. First, measure the length and width of your room using the tape measure. Then, transform the measurement using a scale that could fit on your paper, such as 5 centimeters = 1 meter. 3. Your teacher will assign you a specific window or door to study air flow. On your diagram, add arrows showing your hypothesis (before opening any windows or doors) of how air will flow through the room when your assigned window or door is opened. Use pencil so that you can easily make changes to your diagram. 4. On your diagram, mark four locations where you would like to test air flow in your room. To test for airflow, hold a strip of single ply tissue paper between the thumb and index finger. Note the direction that the paper moves when exposed to the airflow. Then, for each location, predict which way the paper will move if your air flow diagram is correct. 5. Now, each member of your group will stand in one of the four selected areas. Each member will test the airflow Agree upon an approximate height at which everyone will hold their papers. 6. When you teacher tells you to, open your assigned window and/or door. Each person should note the direction that their paper points immediately after the window or door was opened. Record your results on your diagram. 7. Did the airflow test data support or refute the hypothetical model of air flow shown in your diagram? Why or why not? Correct your model based on your experimental evidence. 8. With your group, discuss how accurate your model is. What limitations did it have? Write down the limitations that your grou |
p agreed upon. GRASP CHECK Your diagram is a model, based on experimental evidence, of how air flows through the room. Could you use your model to predict how air would flow through a new window or door placed in a different location in the classroom? Make a new diagram that predicts the room’s airflow with the addition of a new window or door. Add a short explanation that describes how. a. Yes, you could use your model to predict air flow through a new window. The earlier experiment of air flow would help you model the system more accurately. b. Yes, you could use your model to predict air flow through a new window. The earlier experiment of air flow is not useful for modeling the new system. c. No, you cannot model a system to predict the air flow through a new window. The earlier experiment of air flow would help you model the system more accurately. d. No, you cannot model a system to predict the air flow through a new window. The earlier experiment of air flow is not useful for modeling the new system. Scientific Laws and Theories A scientific law is a description of a pattern in nature that is true in all circumstances that have been studied. That is, physical laws are meant to be universal, meaning that they apply throughout the known universe. Laws are often also concise, whereas theories are more complicated. A law can be expressed in the form of a single sentence or mathematical equation. For example, Newton’s second law of motion, which relates the motion of an object to the force applied (F), the mass of the object (m), and the object’s acceleration (a), is simply stated using the equation Scientific ideas and explanations that are true in many, but not all situations in the universe are usually called principles. An example is Pascal’s principle, which explains properties of liquids, but not solids or gases. However, the distinction between laws and principles is sometimes not carefully made in science. A theory is an explanation for patterns in nature that is supported by much scientific evidence and verified multiple times by multiple researchers. While many people confuse theories with educated guesses or hypotheses, theories have withstood more rigorous testing and verification than hypotheses. As a closing idea about scientific processes, we want to point out that scientific laws and theories, even those that have been supported by experiments for centuries, can still be changed by new discoveries. This is especially true when new technologies emerge that allow us to observe things that were formerly unobservable. Imagine how viewing previously invisible objects with a Access for free at openstax.org. 1.2 • The Scientific Methods 17 microscope or viewing Earth for the first time from space may have instantly changed our scientific theories and laws! What discoveries still await us in the future? The constant retesting and perfecting of our scientific laws and theories allows our knowledge of nature to progress. For this reason, many scientists are reluctant to say that their studies proveanything. By saying supportinstead of prove, it keeps the door open for future discoveries, even if they won’t occur for centuries or even millennia. Check Your Understanding 6. Explain why scientists sometimes use a model rather than trying to analyze the behavior of the real system. a. Models are simpler to analyze. b. Models give more accurate results. c. Models provide more reliable predictions. d. Models do not require any computer calculations. 7. Describe the difference between a question, generated through observation, and a hypothesis. a. They are the same. b. A hypothesis has been thoroughly tested and found to be true. c. A hypothesis is a tentative assumption based on what is already known. d. A hypothesis is a broad explanation firmly supported by evidence. 8. What is a scientific model and how is it useful? a. A scientific model is a representation of something that can be easily studied directly. It is useful for studying things that can be easily analyzed by humans. b. A scientific model is a representation of something that is often too difficult to study directly. It is useful for studying a complex system or systems that humans cannot observe directly. c. A scientific model is a representation of scientific equipment. It is useful for studying working principles of scientific equipment. d. A scientific model is a representation of a laboratory where experiments are performed. It is useful for studying requirements needed inside the laboratory. 9. Which of the following statements is correct about the hypothesis? a. The hypothesis must be validated by scientific experiments. b. The hypothesis must not include any physical quantity. c. The hypothesis must be a short and concise statement. d. The hypothesis must apply to all the situations in the universe. 10. What is a scientific theory? a. A scientific theory is an explanation of natural phenomena that is supported by evidence. b. A scientific theory is an explanation of natural phenomena without the support of evidence. c. A scientific theory is an educated guess about the natural phenomena occurring in nature. d. A scientific theory is an uneducated guess about natural phenomena occurring in nature. 11. Compare and contrast a hypothesis and a scientific theory. a. A hypothesis is an explanation of the natural world with experimental support, while a scientific theory is an educated guess about a natural phenomenon. b. A hypothesis is an educated guess about natural phenomenon, while a scientific theory is an explanation of natural world with experimental support. c. A hypothesis is experimental evidence of a natural phenomenon, while a scientific theory is an explanation of the natural world with experimental support. d. A hypothesis is an explanation of the natural world with experimental support, while a scientific theory is experimental evidence of a natural phenomenon. 18 Chapter 1 • What is Physics? 1.3 The Language of Physics: Physical Quantities and Units Section Learning Objectives By the end of this section, you will be able to do the following: • Associate physical quantities with their International System of Units (SI)and perform conversions among SI units using scientific notation • Relate measurement uncertainty to significant figures and apply the rules for using significant figures in calculations • Correctly create, label, and identify relationships in graphs using mathematical relationships (e.g., slope, y-intercept, inverse, quadratic and logarithmic) Section Key Terms accuracy ampere constant conversion factor dependent variable derived units English units exponential relationship fundamental physical units independent variable inverse relationship inversely proportional kilogram linear relationship logarithmic (log) scale log-log plot meter method of adding percents order of magnitude precision quadratic relationship scientific notation second semi-log plot SI units significant figures slope uncertainty variable y-intercept The Role of Units Physicists, like other scientists, make observations and ask basic questions. For example, how big is an object? How much mass does it have? How far did it travel? To answer these questions, they make measurements with various instruments (e.g., meter stick, balance, stopwatch, etc.). The measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in meters (for sprinters) or kilometers (for long distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way (Figure 1.13). Figure 1.13 Distances given in unknown units are maddeningly useless. All physical quantities in the International System of Units (SI) are expressed in terms of combinations of seven fundamental Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 19 physical units, which are units for: length, mass, time, electric current, temperature, amount of a substance, and luminous intensity. SI Units: Fundamental and Derived Units There are two major systems of units used in the world: SI units (acronym for the French Le Système International d’Unités, also known as the metric system), and English units (also known as the imperial system). English units were historically used in nations once ruled by the British Empire. Today, the United States is the only country that still uses English units extensively. Virtually every other country in the world now uses the metric system, which is the standard system agreed upon by scientists and mathematicians. Some physical quantities are more fundamental than others. In physics, there are seven fundamental physical quantities that are measured in base or physical fundamental units: length, mass, time, electric current temperature, amount of substance, and luminous intensity. Units for other physical quantities (such as force, speed, and electric charge) described by mathematically combining these seven base units. In this course, we will mainly use five of these: length, mass, time, electric current and temperature. The units in which they are measured are the meter, kilogram, second, ampere, kelvin, mole, and candela (Table 1.1). All other units are made by mathematically combining the fundamental units. These are called derived units. Quantity Name Symbol Length Meter m Mass Time Kilogram kg Second Electric current Ampere Temperature Kelvin s a k Amount of substance Mole mol Luminous intensity Candela cd Table 1.1 SI Base Units The Meter The SI unit for length is the meter (m). The definition of the meter has changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter t |
o be the distance between two engraved lines on a platinum-iridium bar. (The bar is now housed at the International Bureau of Weights and Meaures, near Paris). By 1960, some distances could be measured more precisely by comparing them to wavelengths of light. The meter was redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition as the distance light travels in a vacuum in 1/ 299,792,458 of a second (Figure 1.14). Figure 1.14 The meter is defined to be the distance light travels in 1/299,792,458 of a second through a vacuum. Distance traveled is speed multiplied by time. 20 Chapter 1 • What is Physics? The Kilogram The SI unit for mass is the kilogram (kg). It is defined to be the mass of a platinum-iridium cylinder, housed at the International Bureau of Weights and Measures near Paris. Exact replicas of the standard kilogram cylinder are kept in numerous locations throughout the world, such as the National Institute of Standards and Technology in Gaithersburg, Maryland. The determination of all other masses can be done by comparing them with one of these standard kilograms. The Second The SI unit for time, the second (s) also has a long history. For many years it was defined as 1/86,400 of an average solar day. However, the average solar day is actually very gradually getting longer due to gradual slowing of Earth’s rotation. Accuracy in the fundamental units is essential, since all other measurements are derived from them. Therefore, a new standard was adopted to define the second in terms of a non-varying, or constant, physical phenomenon. One constant phenomenon is the very steady vibration of Cesium atoms, which can be observed and counted. This vibration forms the basis of the cesium atomic clock. In 1967, the second was redefined as the time required for 9,192,631,770 Cesium atom vibrations (Figure 1.15). Figure 1.15 An atomic clock such as this one uses the vibrations of cesium atoms to keep time to a precision of one microsecond per year. The fundamental unit of time, the second, is based on such clocks. This image is looking down from the top of an atomic clock. (Steve Jurvetson/Flickr) The Ampere Electric current is measured in the ampere (A), named after Andre Ampere. You have probably heard of amperes, or amps, when people discuss electrical currents or electrical devices. Understanding an ampere requires a basic understanding of electricity and magnetism, something that will be explored in depth in later chapters of this book. Basically, two parallel wires with an electric current running through them will produce an attractive force on each other. One ampere is defined as the amount of 10–7 newton per meter of separation between the two wires (the electric current that will produce an attractive force of 2.7 newton is the derived unit of force). Kelvins The SI unit of temperature is the kelvin (or kelvins, but not degrees kelvin). This scale is named after physicist William Thomson, Lord Kelvin, who was the first to call for an absolute temperature scale. The Kelvin scale is based on absolute zero. This is the point at which all thermal energy has been removed from all atoms or molecules in a system. This temperature, 0 K, is equal to −273.15 °C and −459.67 °F. Conveniently, the Kelvin scale actually changes in the same way as the Celsius scale. For example, the freezing point (0 °C) and boiling points of water (100 °C) are 100 degrees apart on the Celsius scale. These two temperatures are also 100 kelvins apart (freezing point = 273.15 K; boiling point = 373.15 K). Metric Prefixes Physical objects or phenomena may vary widely. For example, the size of objects varies from something very small (like an atom) Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 21 to something very large (like a star). Yet the standard metric unit of length is the meter. So, the metric system includes many prefixes that can be attached to a unit. Each prefix is based on factors of 10 (10, 100, 1,000, etc., as well as 0.1, 0.01, 0.001, etc.). Table 1.2 gives the metric prefixes and symbols used to denote the different various factors of 10 in the metric system. Prefix Symbol Value[1] Example Name Example Symbol Example Value Example Description exa peta tera giga mega kilo hector E P T G M k h deka da 1018 1015 1012 109 106 103 102 101 ____ ____ 100 (=1) deci centi d c milli m micro nano pico femto atto µ n p f a 10–1 10–2 10–3 10–6 10–9 10–12 10–15 10–18 Exameter Em Petasecond Ps Terawatt TW Gigahertz GHz Megacurie MCi Kilometer Hectoliter Dekagram Deciliter Centimeter km hL dag dL Cm Millimeter Mm Micrometer µm Nanogram Picofarad Ng pF 1018 m 1015 s 1012 W 109 Hz 106 Ci 103 m 102 L 101 g 10–1 L 10–2 m 10–3 m 10–6 m 10–9 g 10–12 F Distance light travels in a century 30 million years Powerful laser output A microwave frequency High radioactivity About 6/10 mile 26 gallons Teaspoon of butter Less than half a soda Fingertip thickness Flea at its shoulder Detail in microscope Small speck of dust Small capacitor in radio Femtometer Fm 10–15 m Size of a proton Attosecond as 10–18 s Time light takes to cross an atom Table 1.2 Metric Prefixes for Powers of 10 and Their Symbols Note—Some examples are approximate. [1]See Appendix A for a discussion of powers of 10. The metric system is convenient because conversions between metric units can be done simply by moving the decimal place of a number. This is because the metric prefixes are sequential powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as U.S. customary units, the relationships are less simple—there are 12 inches in a foot, 5,280 feet in a mile, 4 quarts in a gallon, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by switching to the most-appropriate metric prefix. For example, distances in meters are suitable for building construction, but kilometers are used to describe road construction. Therefore, with the metric system, there is no need to invent new units when measuring very small or very large objects—you just have to move the decimal 22 Chapter 1 • What is Physics? point (and use the appropriate prefix). Known Ranges of Length, Mass, and Time Table 1.3 lists known lengths, masses, and time measurements. You can see that scientists use a range of measurement units. This wide range demonstrates the vastness and complexity of the universe, as well as the breadth of phenomena physicists study. As you examine this table, note how the metric system allows us to discuss and compare an enormous range of phenomena, using one system of measurement (Figure 1.16 and Figure 1.17). Length (m) Phenomenon Measured 10–18 Present experimental limit to smallest observable detail Mass (Kg) 10–30 Phenomenon Measured[1] Mass of an electron (9.11 10–31 kg) 10–15 Diameter of a proton 10–27 Mass of a hydrogen atom 10–27 kg) (1.67 Time (s) 10–23 10–22 1014 Diameter of a uranium nucleus 10–15 Mass of a bacterium 10–15 10–10 Diameter of a hydrogen atom 10–5 Mass of a mosquito 10–13 Phenomenon Measured[1] Time for light to cross a proton Mean life of an extremely unstable nucleus Time for one oscillation of a visible light Time for one vibration of an atom in a solid Time for one oscillation of an FM radio wave 10–8 10–6 10–3 1 102 104 107 1011 1016 1021 Thickness of membranes in cell of living organism 10–2 Mass of a hummingbird 10–8 Wavelength of visible light Size of a grain of sand Height of a 4-year-old child Length of a football field Greatest ocean depth Diameter of Earth Distance from Earth to the sun Distance traveled by light in 1 year (a light year) 1 102 103 108 1012 1015 1023 1025 Diameter of the Milky Way Galaxy 1030 Mass of a liter of water (about a quart) 10–3 Duration of a nerve impulse 1 105 107 109 Time for one heartbeat One day (8.64 104 s) One year (3.16 107 s) About half the life expectancy of a human 1011 Recorded history 1017 Age of Earth 1018 Age of the universe Mass of a person Mass of a car Mass of a large ship Mass of a large iceberg Mass of the nucleus of a comet Mass of the moon (7.35 1022 kg) Mass of Earth (5.97 kg) 1024 Mass of the Sun (1.99 kg) 1024 Table 1.3 Approximate Values of Length, Mass, and Time [1] More precise values are in parentheses. Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 23 Length (m) 1022 1026 Phenomenon Measured Mass (Kg) Phenomenon Measured[1] Time (s) Phenomenon Measured[1] Distance from Earth to the nearest large galaxy (Andromeda) Distance from Earth to the edges of the known universe 1042 1053 Mass of the Milky Way galaxy (current upper limit) Mass of the known universe (current upper limit) Table 1.3 Approximate Values of Length, Mass, and Time [1] More precise values are in parentheses. Figure 1.16 Tiny phytoplankton float among crystals of ice in the Antarctic Sea. They range from a few micrometers to as much as 2 millimeters in length. (Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collections) Figure 1.17 Galaxies collide 2.4 billion light years away from Earth. The tremendous range of observable phenomena in nature challenges the imagination. (NASA/CXC/UVic./A. Mahdavi et al. Optical/lensing: CFHT/UVic./H. Hoekstra et al.) Using Scientific Notation with Physical Measurements Scientific notation is a way of writing numbers that are too large or small to be conveniently written as a decimal. For example, consider the number 840,000,000,000,000. It’s a rather large number to write out. The scientific notation for this number is 8.40 1014. Scientific notation follows this general format In this format xis the value of the measurement with all placeholder zeros removed. In the example above, xis 8.4. The xis multiplied by a factor, 10y, which indicates the number of placeholder zeros in the measurement. |
Placeholder zeros are those at the end of a number that is 10 or greater, and at the beginning of a decimal number that is less than 1. In the example above, the factor is 1014. This tells you that you should move the decimal point 14 positions to the right, filling in placeholder zeros as you go. In this case, moving the decimal point 14 places creates only 13 placeholder zeros, indicating that the actual measurement value is 840,000,000,000,000. 24 Chapter 1 • What is Physics? Numbers that are fractions can be indicated by scientific notation as well. Consider the number 0.0000045. Its scientific notation is 4.5 10–6. Its scientific notation has the same format Here, xis 4.5. However, the value of yin the 10yfactor is negative, which indicates that the measurement is a fraction of 1. 10–6, the decimal point would be Therefore, we move the decimal place to the left, for a negative y. In our example of 4.5 moved to the left six times to yield the original number, which would be 0.0000045. The term order of magnitude refers to the power of 10 when numbers are expressed in scientific notation. Quantities that have the same power of 10 when expressed in scientific notation, or come close to it, are said to be of the same order of magnitude. For example, the number 800 can be written as 8 same value for y. Therefore, 800 and 450 are of the same order of magnitude. Similarly, 101 and 99 would be regarded as the same order of magnitude, 102. Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10−9 m, while the diameter of the sun is on the order of 109 m. These two values are 18 orders of magnitude apart. 102, and the number 450 can be written as 4.5 102. Both numbers have the Scientists make frequent use of scientific notation because of the vast range of physical measurements possible in the universe, such as the distance from Earth to the moon (Figure 1.18), or to the nearest star. Figure 1.18 The distance from Earth to the moon may seem immense, but it is just a tiny fraction of the distance from Earth to our closest neighboring star. (NASA) Unit Conversion and Dimensional Analysis It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook in the United States, some quantities may be expressed in liters and you need to convert them to cups. A Canadian tourist driving through the United States might want to convert miles to kilometers, to have a sense of how far away his next destination is. A doctor in the United States might convert a patient’s weight in pounds to kilograms. Let’s consider a simple example of how to convert units within the metric system. How can we want to convert 1 hour to seconds? Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. A conversion factor is simply a fraction which equals 1. You can multiply any number by 1 and get the same value. When you multiply a number by a conversion factor, you are simply multiplying it by one. For example, the following are conversion factors: (1 foot)/(12 inches) = 1 to convert inches to feet, (1 meter)/(100 centimeters) = 1 to convert centimeters to meters, (1 minute)/(60 seconds) = 1 to convert seconds to minutes. In this case, we know that there are 1,000 meters in 1 kilometer. Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor (1 km/1,000m) = 1, so we are simply multiplying 80m by 1: When there is a unit in the original number, and a unit in the denominator (bottom) of the conversion factor, the units cancel. In this case, hours and minutes cancel and the value in seconds remains. You can use this method to convert between any types of unit, including between the U.S. customary system and metric system. Notice also that, although you can multiply and divide units algebraically, you cannot add or subtract different units. An expression like 10 km + 5 kgmakes no sense. Even adding two lengths in different units, such as 10 km + 20 mdoes not make sense. You express both lengths in the same unit. See Appendix C for a more complete list of conversion factors. 1.1 Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 25 WORKED EXAMPLE Unit Conversions: A Short Drive Home Suppose that you drive the 10.0 km from your university to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note—Average speed is distance traveled divided by time of travel.) Strategy First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place. Solution for (a) 1. Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form, 2. Substitute the given values for distance and time. 3. Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is . Thus, Discussion for (a) To check your answer, consider the following: 1. Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows which are obviously not the desired units of km/h. 2. Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units. 3. Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/h does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is definedto be 60 min, so the precision of the conversion factor is perfect. 4. Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable. Solution (b) There are several ways to convert the average speed into meters per second. 1. Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters. 2. Multiplying by these yields 26 Chapter 1 • What is Physics? Discussion for (b) If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s. You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? WORKED EXAMPLE Using Physics to Evaluate Promotional Materials A commemorative coin that is 2″ in diameter is advertised to be plated with 15 mg of gold. If the density of gold is 19.3 g/cc, and the amount of gold around the edge of the coin can be ignored, what is the thickness of the gold on the top and bottom faces of the coin? Strategy To solve this problem, the volume of the gold needs to be determined using the gold’s mass and density. Half of that volume is distributed on each face of the coin, and, for each face, the gold can be represented as a cylinder that is 2″ in diameter with a height equal to the thickness. Use the volume formula for a cylinder to determine the thickness. Solution The mass of the gold is given by the formula where and Vis the volume. Solving for the volume gives If tis the thickness, the volume corresponding to half the gold is where the 1″ radius has been converted to cm. Solving for the thickness gives Discussion The amount of gold used is stated to be 15 mg, which is equivalent to a thickness of about 0.00019 mm. The mass figure may make the amount of gold sound larger, both because the number is much bigger (15 versus 0.00019), and because people may have a more intuitive feel for how much a millimeter is than for how much a milligram is. A simple analysis of this sort can clarify the significance of claims made by advertisers. Accuracy, Precision and Significant Figures Science is based on experimentation that requires good measurements. The validity of a measurement can be described in terms of its accuracy and its precision (see Figure 1.19 and Figure 1.20). Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard piece of printer paper. The packaging in which you purchased the paper states that it is 11 inches long, and suppose this stated value is correct. You measure the length of the paper three times and obtain the following measurements: 11.1 inches, 11.2 inches, and 10.9 inches. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate. This is why measuring instruments are calibrated based on a known measurement. If the instrument consistently returns the correct value of the known measurement, it is safe for use in finding unknown values. Access for free at openstax.org. 1.3 • The Language of Physics: Physical Qua |
ntities and Units 27 Figure 1.19 A double-pan mechanical balance is used to compare different masses. Usually an object with unknown mass is placed in one pan and objects of known mass are placed in the other pan. When the bar that connects the two pans is horizontal, then the masses in both pans are equal. The known masses are typically metal cylinders of standard mass such as 1 gram, 10 grams, and 100 grams. (Serge Melki) Figure 1.20 Whereas a mechanical balance may only read the mass of an object to the nearest tenth of a gram, some digital scales can measure the mass of an object up to the nearest thousandth of a gram. As in other measuring devices, the precision of a scale is limited to the last measured figures. This is the hundredths place in the scale pictured here. (Splarka, Wikimedia Commons) Precision states how well repeated measurements of something generate the same or similar results. Therefore, the precision of measurements refers to how close together the measurements are when you measure the same thing several times. One way to analyze the precision of measurements would be to determine the range, or difference between the lowest and the highest measured values. In the case of the printer paper measurements, the lowest value was 10.9 inches and the highest value was 11.2 inches. Thus, the measured values deviated from each other by, at most, 0.3 inches. These measurements were reasonably precise because they varied by only a fraction of an inch. However, if the measured values had been 10.9 inches, 11.1 inches, and 11.9 inches, then the measurements would not be very precise because there is a lot of variation from one measurement to another. The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull’s-eye target. Then think of each GPS attempt to locate the restaurant as a black dot on the bull’s eye. In Figure 1.21, you can see that the GPS measurements are spread far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in Figure 1.22, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system. Finally, in Figure 1.23, the GPS is both precise and accurate, allowing the restaurant to be located. 28 Chapter 1 • What is Physics? Figure 1.21 A GPS system attempts to locate a restaurant at the center of the bull’s-eye. The black dots represent each attempt to pinpoint the location of the restaurant. The dots are spread out quite far apart from one another, indicating low precision, but they are each rather close to the actual location of the restaurant, indicating high accuracy. (Dark Evil) Figure 1.22 In this figure, the dots are concentrated close to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (Dark Evil) Figure 1.23 In this figure, the dots are concentrated close to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (Dark Evil) Uncertainty The accuracy and precision of a measuring system determine the uncertainty of its measurements. Uncertainty is a way to describe how much your measured value deviates from the actual value that the object has. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 inches plus or minus 0.2 inches or 11.0 ± 0.2 inches. The uncertainty in a Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 29 measurement, A, is often denoted as δA("delta A"), The factors contributing to uncertainty in a measurement include the following: 1. Limitations of the measuring device 2. The skill of the person making the measurement Irregularities in the object being measured 3. 4. Any other factors that affect the outcome (highly dependent on the situation) In the printer paper example uncertainty could be caused by: the fact that the smallest division on the ruler is 0.1 inches, the person using the ruler has bad eyesight, or uncertainty caused by the paper cutting machine (e.g., one side of the paper is slightly longer than the other.) It is good practice to carefully consider all possible sources of uncertainty in a measurement and reduce or eliminate them, Percent Uncertainty One method of expressing uncertainty is as a percent of the measured value. If a measurement, A, is expressed with uncertainty, δA, the percent uncertainty is 1.2 WORKED EXAMPLE Calculating Percent Uncertainty: A Bag of Apples A grocery store sells 5-lb bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements: • Week 1 weight: • Week 2 weight: • Week 3 weight: • Week 4 weight: You determine that the weight of the 5 lb bag has an uncertainty of ±0.4 lb. What is the percent uncertainty of the bag’s weight? Strategy First, observe that the expected value of the bag’s weight, following equation to determine the percent uncertainty of the weight , is 5 lb. The uncertainty in this value, , is 0.4 lb. We can use the Solution Plug the known values into the equation Discussion We can conclude that the weight of the apple bag is 5 lb ± 8 percent. Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100 percent. If you do not do this, you will have a decimal quantity, not a percent value. Uncertainty in Calculations There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the both the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements in the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of 4.00 m and a width of 3.00 m, with uncertainties of 2 percent and 1 30 Chapter 1 • What is Physics? percent, respectively, then the area of the floor is 12.0 m2 and has an uncertainty of 3 percent (expressed as an area this is 0.36 m2, which we round to 0.4 m2 since the area of the floor is given to a tenth of a square meter). For a quick demonstration of the accuracy, precision, and uncertainty of measurements based upon the units of measurement, try this simulation (http://openstax.org/l/28precision) . You will have the opportunity to measure the length and weight of a desk, using milli- versus centi- units. Which do you think will provide greater accuracy, precision and uncertainty when measuring the desk and the notepad in the simulation? Consider how the nature of the hypothesis or research question might influence how precise of a measuring tool you need to collect data. Precision of Measuring Tools and Significant Figures An important factor in the accuracy and precision of measurements is the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, consider measuring the thickness of a coin. A standard ruler can measure thickness to the nearest millimeter, while a micrometer can measure the thickness to the nearest 0.005 millimeter. The micrometer is a more precise measuring tool because it can measure extremely small differences in thickness. The more precise the measuring tool, the more precise and accurate the measurements can be. When we express measured values, we can only list as many digits as we initially measured with our measuring tool (such as the rulers shown in Figure 1.24). For example, if you use a standard ruler to measure the length of a stick, you may measure it with a decimeter ruler as 3.6 cm. You could not express this value as 3.65 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36 mm and 37 mm. He or she must estimate the value of the last digit. The rule is that the last digit written down in a measurement is the first digit with some uncertainty. For example, the last measured value 36.5 mm has three digits, or three significant figures. The number of significant figures in a measurement indicates the precision of the measuring tool. The more pr |
ecise a measuring tool is, the greater the number of significant figures it can report. Figure 1.24 Three metric rulers are shown. The first ruler is in decimeters and can measure point three decimeters. The second ruler is in centimeters long and can measure three point six centimeters. The last ruler is in millimeters and can measure thirty-six point five millimeters. Zeros Special consideration is given to zeros when counting significant figures. For example, the zeros in 0.053 are not significant because they are only placeholders that locate the decimal point. There are two significant figures in 0.053—the 5 and the 3. However, if the zero occurs between other significant figures, the zeros are significant. For example, both zeros in 10.053 are significant, as these zeros were actually measured. Therefore, the 10.053 placeholder has five significant figures. The zeros in 1300 may or may not be significant, depending on the style of writing numbers. They could mean the number is known to the last zero, or the zeros could be placeholders. So 1300 could have two, three, or four significant figures. To avoid this ambiguity, Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 31 write 1300 in scientific notation as 1.3 × 103. Only significant figures are given in the xfactor for a number in scientific notation (in the form ). Therefore, we know that 1 and 3 are the only significant digits in this number. In summary, zeros are significant except when they serve only as placeholders. Table 1.4 provides examples of the number of significant figures in various numbers. Number Significant Figures Rationale 4 4 3 6 3 7 4 4 1.657 0.4578 0.000458 2000.56 45,600 15895000 5.457 1013 6.520 10–23 Table 1.4 There are no zeros and all non-zero numbers are always significant. The first zero is only a placeholder for the decimal point. The first four zeros are placeholders needed to report the data to the ten-thoudsandths place. The three zeros are significant here because they occur between other significant figures. With no underlines or scientific notation, we assume that the last two zeros are placeholders and are not significant. The two underlined zeros are significant, while the last zero is not, as it is not underlined. In scientific notation, all numbers reported in front of the multiplication sign are significant In scientific notation, all numbers reported in front of the multiplication sign are significant, including zeros. Significant Figures in Calculations When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and another rule for addition and subtraction, as discussed below. 1. For multiplication and division: The answer should have the same number of significant figures as the starting value with . Let us see the fewest significant figures. For example, the area of a circle can be calculated from its radius using how many significant figures the area will have if the radius has only two significant figures, for example, r= 2.0 m. Then, using a calculator that keeps eight significant figures, you would get But because the radius has only two significant figures, the area calculated is meaningful only to two significant figures or even though the value of is meaningful to at least eight digits. 2. For addition and subtraction: The answer should have the same number places (e.g. tens place, ones place, tenths place, etc.) as the least-precise starting value. Suppose that you buy 7.56 kg of potatoes in a grocery store as measured with a scale having a precision of 0.01 kg. Then you drop off 6.052 kg of potatoes at your laboratory as measured by a scale with a precision of 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with a precision of 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction: 32 Chapter 1 • What is Physics? The least precise measurement is 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer should be rounded to the tenths place, giving 15.2 kg. The same is true for non-decimal numbers. For example, We cannot report the decimal places in the answer because 2has no decimal places that would be significant. Therefore, we can only report to the ones place. It is a good idea to keep extra significant figures while calculating, and to round off to the correct number of significant figures only in the final answers. The reason is that small errors from rounding while calculating can sometimes produce significant errors in the final answer. As an example, try calculating to obtain a final answer to only two significant figures. Keeping all significant during the calculation gives 48. Rounding to two significant figures in the middle of the calculation changes it to which is way off. You would similarly avoid rounding in the middle of the calculation in counting and in doing accounting, where many small numbers need to be added and subtracted accurately to give possibly much larger final numbers. Significant Figures in this Text In this textbook, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, such as optics, more than three significant figures will be used. Finally, if a number is exact, such as the 2in the formula, significant figures in a calculation. , it does not affect the number of WORKED EXAMPLE Approximating Vast Numbers: a Trillion Dollars The U.S. federal deficit in the 2008 fiscal year was a little greater than $10 trillion. Most of us do not have any concept of how much even one trillion actually is. Suppose that you were given a trillion dollars in $100 bills. If you made 100-bill stacks, like that shown in Figure 1.25, and used them to evenly cover a football field (between the end zones), make an approximation of how high the money pile would become. (We will use feet/inches rather than meters here because football fields are measured in yards.) One of your friends says 3 in., while another says 10 ft. What do you think? Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 33 Figure 1.25 A bank stack contains one hundred $100 bills, and is worth $10,000. How many bank stacks make up a trillion dollars? (Andrew Magill) Strategy When you imagine the situation, you probably envision thousands of small stacks of 100 wrapped $100 bills, such as you might see in movies or at a bank. Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of 100 bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height. Solution 1. Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is 2. Calculate the number of stacks. Note that a trillion dollars is equal to , and a stack of one-hundred bills is equal to or . The number of stacks you will have is 1.3 3. Calculate the area of a football field in square inches. The area of a football field is , which gives . Because we are working in inches, we need to convert square yards to square inches This conversion gives us calculations.) for the area of the field. (Note that we are using only one significant figure in these 4. Calculate the total volume of the bills. The volume of all the $100-bill stacks is 5. Calculate the height. To determine the height of the bills, use the following equation 34 Chapter 1 • What is Physics? The height of the money will be about 100 in. high. Converting this value to feet gives Discussion The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough guesstimatesversus carefully calculated approximations? In the example above, the final approximate value is much higher than the first friend’s early estimate of 3 in. However, the other friend’s early estimate of 10 ft. (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise suggest about the value of rough guesstimatesversus carefully calculated approximations? Graphing in Physics Most results in science are presented in scientific journal articles using graphs. Graphs present data in a way that is easy to visualize for humans in general, especially someone unfamiliar with what is being studied. They are also useful for presenting large amounts of data or data with complicated trends in an easily-readable way. One commonly-used graph in physics and other sciences is the line graph, probably because it is the best graph for showing how one quantity changes in response to the other. Let’s build a line graph based on the data in Table 1.5, which shows the measured distance that a train travels from its station versus time. Our two variables, or things that change along the graph, are time in minutes, and distance from the station, in kilometers. Remember that measured data may not ha |
ve perfect accuracy. Time (min) Distance from Station (km) 0 24 36 60 84 97 116 140 0 10 20 30 40 50 60 70 Table 1.5 1. Draw the two axes. The horizontal axis, or x-axis, shows the independent variable, which is the variable that is controlled or manipulated. The vertical axis, or y-axis, shows the dependent variable, the non-manipulated variable that changes with (or is dependent on) the value of the independent variable. In the data above, time is the independent variable and should be plotted on the x-axis. Distance from the station is the dependent variable and should be plotted on the y-axis. Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 35 2. Label each axes on the graph with the name of each variable, followed by the symbol for its units in parentheses. Be sure to leave room so that you can number each axis. In this example, use Time (min)as the label for the x-axis. 3. Next, you must determine the best scale to use for numbering each axis. Because the time values on the x-axis are taken every 10 minutes, we could easily number the x-axis from 0 to 70 minutes with a tick mark every 10 minutes. Likewise, the y-axis scale should start low enough and continue high enough to include all of the distance from stationvalues. A scale from 0 km to 160 km should suffice, perhaps with a tick mark every 10 km. In general, you want to pick a scale for both axes that 1) shows all of your data, and 2) makes it easy to identify trends in your data. If you make your scale too large, it will be harder to see how your data change. Likewise, the smaller and more fine you make your scale, the more space you will need to make the graph. The number of significant figures in the axis values should be coarser than the number of significant figures in the measurements. 4. Now that your axes are ready, you can begin plotting your data. For the first data point, count along the x-axis until you find the 10 min tick mark. Then, count up from that point to the 10 km tick mark on the y-axis, and approximate where 22 km is along the y-axis. Place a dot at this location. Repeat for the other six data points (Figure 1.26). Figure 1.26 The graph of the train’s distance from the station versus time from the exercise above. 5. Add a title to the top of the graph to state what the graph is describing, such as the y-axis parameter vs. the x-axis parameter. In the graph shown here, the title is train motion. It could also be titled distance of the train from the station vs. time. 6. Finally, with data points now on the graph, you should draw a trend line (Figure 1.27). The trend line represents the dependence you think the graph represents, so that the person who looks at your graph can see how close it is to the real data. In the present case, since the data points look like they ought to fall on a straight line, you would draw a straight line as the trend line. Draw it to come closest to all the points. Real data may have some inaccuracies, and the plotted points may not all fall on the trend line. In some cases, none of the data points fall exactly on the trend line. 36 Chapter 1 • What is Physics? Figure 1.27 The completed graph with the trend line included. Analyzing a Graph Using Its Equation One way to get a quick snapshot of a dataset is to look at the equation of its trend line. If the graph produces a straight line, the equation of the trend line takes the form The bin the equation is the y-intercept while the min the equation is the slope. The y-intercept tells you at what yvalue the line intersects the y-axis. In the case of the graph above, the y-intercept occurs at 0, at the very beginning of the graph. The y-intercept, therefore, lets you know immediately where on the y-axis the plot line begins. The min the equation is the slope. This value describes how much the line on the graph moves up or down on the y-axis along the line’s length. The slope is found using the following equation In order to solve this equation, you need to pick two points on the line (preferably far apart on the line so the slope you calculate describes the line accurately). The quantities Y2 and Y1 represent the y-values from the two points on the line (not data points) that you picked, while X2 and X1 represent the two x-values of the those points. What can the slope value tell you about the graph? The slope of a perfectly horizontal line will equal zero, while the slope of a perfectly vertical line will be undefined because you cannot divide by zero. A positive slope indicates that the line moves up the y-axis as the x-value increases while a negative slope means that the line moves down the y-axis. The more negative or positive the slope is, the steeper the line moves up or down, respectively. The slope of our graph in Figure 1.26 is calculated below based on the two endpoints of the line Equation of line: Because the xaxis is time in minutes, we would actually be more likely to use the time tas the independent (x-axis) variable and write the equation as Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 37 only applies to linear relationships, or ones that produce a straight line. Another common type of line The formula in physics is the quadratic relationship, which occurs when one of the variables is squared. One quadratic relationship in physics is the relation between the speed of an object its centripetal acceleration, which is used to determine the force needed to keep an object moving in a circle. Another common relationship in physics is the inverse relationship, in which one variable decreases whenever the other variable increases. An example in physics is Coulomb’s law. As the distance between two charged objects increases, the electrical force between the two charged objects decreases. Inverse proportionality, such the relation between xand yin the equation 1.4 for some number k, is one particular kind of inverse relationship. A third commonly-seen relationship is the exponential relationship, in which a change in the independent variable produces a proportional change in the dependent variable. As the value of the dependent variable gets larger, its rate of growth also increases. For example, bacteria often reproduce at an exponential rate when grown under ideal conditions. As each generation passes, there are more and more bacteria to reproduce. As a result, the growth rate of the bacterial population increases every generation (Figure 1.28). Figure 1.28 Examples of (a) linear, (b) quadratic, (c) inverse, and (d) exponential relationship graphs. Using Logarithmic Scales in Graphing Sometimes a variable can have a very large range of values. This presents a problem when you’re trying to figure out the best scale to use for your graph’s axes. One option is to use a logarithmic (log) scale. In a logarithmic scale, the value each mark labels 38 Chapter 1 • What is Physics? is the previous mark’s value multiplied by some constant. For a log base 10 scale, each mark labels a value that is 10 times the value of the mark before it. Therefore, a base 10 logarithmic scale would be numbered: 0, 10, 100, 1,000, etc. You can see how the logarithmic scale covers a much larger range of values than the corresponding linear scale, in which the marks would label the values 0, 10, 20, 30, and so on. If you use a logarithmic scale on one axis of the graph and a linear scale on the other axis, you are using a semi-log plot. The Richter scale, which measures the strength of earthquakes, uses a semi-log plot. The degree of ground movement is plotted on a logarithmic scale against the assigned intensity level of the earthquake, which ranges linearly from 1-10 (Figure 1.29 (a)). If a graph has both axes in a logarithmic scale, then it is referred to as a log-log plot. The relationship between the wavelength and frequency of electromagnetic radiation such as light is usually shown as a log-log plot (Figure 1.29 (b)). Log-log plots are also commonly used to describe exponential functions, such as radioactive decay. Figure 1.29 (a) The Richter scale uses a log base 10 scale on its y-axis(microns of amplified maximum ground motion). (b) The relationship between the frequency and wavelength of electromagnetic radiation can be plotted as a straight line if a log-log plot is used. Virtual Physics Graphing Lines In this simulation you will examine how changing the slope and y-intercept of an equation changes the appearance of a plotted line. Select slope-intercept form and drag the blue circles along the line to change the line’s characteristics. Then, play the line game and see if you can determine the slope or y-intercept of a given line. Click to view content (https://phet.colorado.edu/sims/html/graphing-lines/latest/graphing-lines_en.html) GRASP CHECK How would the following changes affect a line that is neither horizontal nor vertical and has a positive slope? 1. 2. increase the slope but keeping the y-intercept constant increase the y-intercept but keeping the slope constant a. Increasing the slope will cause the line to rotate clockwise around the y-intercept. Increasing the y-intercept will cause the line to move vertically up on the graph without changing the line’s slope. Increasing the slope will cause the line to rotate counter-clockwise around the y-intercept. Increasing the y-intercept will cause the line to move vertically up on the graph without changing the line’s slope. Increasing the slope will cause the line to rotate clockwise around the y-intercept. Increasing the y-intercept will cause the line to move horizontally right on the graph without changing the line’s slope. b. c. Access for free at openstax.org. 1.3 • The Language of Physics: Physical Quantities and Units 39 d. Increasing the slope will cause the line to rotate counter-clockwise around the y-intercept. Increasing the y-intercept will cause the line to move horizontally right on the graph without c |
hanging the line’s slope. Check Your Understanding 12. Identify some advantages of metric units. a. Conversion between units is easier in metric units. b. Comparison of physical quantities is easy in metric units. c. Metric units are more modern than English units. d. Metric units are based on powers of 2. 13. The length of an American football field is , excluding the end zones. How long is the field in meters? Round to the . nearest a. b. c. d. 14. The speed limit on some interstate highways is roughly . How many miles per hour is this if is about ? a. 0.1 mi/h b. 27.8 mi/h c. 62 mi/h 160 mi/h d. 15. Briefly describe the target patterns for accuracy and precision and explain the differences between the two. a. Precision states how much repeated measurements generate the same or closely similar results, while accuracy states how close a measurement is to the true value of the measurement. b. Precision states how close a measurement is to the true value of the measurement, while accuracy states how much repeated measurements generate the same or closely similar result. c. Precision and accuracy are the same thing. They state how much repeated measurements generate the same or closely similar results. d. Precision and accuracy are the same thing. They state how close a measurement is to the true value of the measurement. 40 Chapter 1 • Key Terms KEY TERMS accuracy how close a measurement is to the correct value for that measurement ampere the SI unit for electrical current atom smallest and most basic units of matter classical physics physics, as it developed from the Renaissance to the end of the nineteenth century constant a quantity that does not change conversion factor a ratio expressing how many of one unit are equal to another unit century to the present, involving the theories of relativity and quantum mechanics observation step where a scientist observes a pattern or trend within the natural world order of magnitude the size of a quantity in terms of its power of 10 when expressed in scientific notation physics science aimed at describing the fundamental aspects of our universe—energy, matter, space, motion, and time dependent variable the vertical, or y-axis, variable, which precision how well repeated measurements generate the changes with (or is dependent on) the value of the independent variable same or closely similar results principle description of nature that is true in many, but not derived units units that are derived by combining the all situations fundamental physical units quadratic relationship relation between variables that can English units (also known as the customary or imperial system) system of measurement used in the United States; includes units of measurement such as feet, gallons, degrees Fahrenheit, and pounds experiment process involved with testing a hypothesis exponential relationship relation between variables in which a constant change in the independent variable is accompanied by change in the dependent variable that is proportional to the value it already had be expressed in the form produces a curved line when graphed , which quantum mechanics major theory of modern physics which describes the properties and nature of atoms and their subatomic particles science the study or knowledge of how the physical world operates, based on objective evidence determined through observation and experimentation scientific law pattern in nature that is true in all fundamental physical units the seven fundamental circumstances studied thus far physical units in the SI system of units are length, mass, time, electric current, temperature, amount of a substance, and luminous intensity hypothesis testable statement that describes how something in the natural world works independent variable the horizontal, or x-axis, variable, which is not influence by the second variable on the graph, the dependent variable inverse proportionality a relation between two variables where k expressible by an equation of the form stays constant when xand ychange; the special form of inverse relationship that satisfies this equation inverse relationship any relation between variables where one variable decreases as the other variable increases kilogram the SI unit for mass, abbreviated (kg) linear relationships relation between variables that produce a straight line when graphed log-log plot a plot that uses a logarithmic scale in both axes logarithmic scale a graphing scale in which each tick on an axis is the previous tick multiplied by some value the SI unit for length, abbreviated (m) meter method of adding percents calculating the percent uncertainty of a quantity in multiplication or division by adding the percent uncertainties in the quantities being added or divided model system that is analogous to the real system of interest in essential ways but more easily analyzed modern physics physics as developed from the twentieth Access for free at openstax.org. scientific methods techniques and processes used in the constructing and testing of scientific hypotheses, laws, and theories, and in deciding issues on the basis of experiment and observation scientific notation way of writing numbers that are too large or small to be conveniently written in simple decimal form; the measurement is multiplied by a power of 10, which indicates the number of placeholder zeros in the measurement SI units second the SI unit for time, abbreviated (s) semi-log plot A plot that uses a logarithmic scale on one axis of the graph and a linear scale on the other axis. International System of Units (SI); the international system of units that scientists in most countries have agreed to use; includes units such as meters, liters, and grams; also known as the metric system significant figures when writing a number, the digits, or number of digits, that express the precision of a measuring tool used to measure the number slope the ratio of the change of a graph on the yaxis to the change along the x-axis, the value of min the equation of a line, theory explanation of patterns in nature that is supported by much scientific evidence and verified multiple times by various groups of researchers theory of relativity theory constructed by Albert Einstein which describes how space, time and energy are different for different observers in relative motion uncertainty a quantitative measure of how much measured values deviate from a standard or expected value universal applies throughout the known universe y-intercept the point where a plot line intersects the y-axis Chapter 1 • Section Summary 41 SECTION SUMMARY 1.1 Physics: Definitions and Applications • Physics is the most fundamental of the sciences, concerning itself with energy, matter, space and time, and their interactions. • Modern physics involves the theory of relativity, which describes how time, space and gravity are not constant in our universe can be different for different observers, and quantum mechanics, which describes the behavior of subatomic particles. • Physics is the basis for all other sciences, such as chemistry, biology and geology, because physics describes the fundamental way in which the universe functions. 1.2 The Scientific Methods • Science seeks to discover and describe the underlying order and simplicity in nature. • The processes of science include observation, hypothesis, experiment, and conclusion. • Theories are scientific explanations that are supported by a large body experimental results. • Scientific laws are concise descriptions of the universe that are universally true. 1.3 The Language of Physics: Physical Quantities and Units • Physical quantities are a characteristic or property of an KEY EQUATIONS 1.3 The Language of Physics: Physical Quantities and Units slope intercept form quadratic formula CHAPTER REVIEW Concept Items 1.1 Physics: Definitions and Applications 1. Which statement best compares and contrasts the aims and topics of natural philosophy had versus physics? object that can be measured or calculated from other measurements. • The four fundamental units we will use in this textbook are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature. • Unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units. • Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value. • Precision of measured values refers to how close the agreement is between repeated measurements. • Significant figures express the precision of a measuring tool. • When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value. • When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value. positive exponential formula negative exponential formula a. Natural philosophy included all aspects of nature including physics. b. Natural philosophy included all aspects of nature excluding physics. c. Natural philosophy and physics are different. d. Natural philosophy and physics are essentially the 42 Chapter 1 • Chapter Review same thing. 2. Which of the following is not an underlying assumption essential to scientific understanding? a. Characteristics of the physical universe can be perceived and objectively measured by human beings. b. Explanations of natural phenomena can be established with absolute certainty. c. Fundamental physical processes dictate how characteristics of the physical universe evolve. d. The fundamental processes of nature operate the same way everywhere |
and at all times. 3. Which of the following questions regarding a strain of genetically modified rice is not one that can be answered by science? a. How does the yield of the genetically modified rice b. compare with that of existing rice? Is the genetically modified rice more resistant to infestation than existing rice? c. How does the nutritional value of the genetically modified rice compare to that of existing rice? d. Should the genetically modified rice be grown commercially and sold in the marketplace? 4. What conditions imply that we can use classical physics without considering special relativity or quantum mechanics? a. 1. matter is moving at speeds of less than roughly 1 percent the speed of light, 2. objects are large enough to be seen with the 3. naked eye, and there is the involvement of a strong gravitational field. b. 1. matter is moving at speeds greater than roughly 1 percent the speed of light, 2. objects are large enough to be seen with the 3. naked eye, and there is the involvement of a strong gravitational field. c. 1. matter is moving at speeds of less than roughly 1 percent the speed of light, 2. objects are too small to be seen with the naked 3. eye, and there is the involvement of only a weak gravitational field. 5. How could physics be useful in weather prediction? a. Physics helps in predicting how burning fossil fuel releases pollutants. b. Physics helps in predicting dynamics and movement of weather phenomena. c. Physics helps in predicting the motion of tectonic plates. d. Physics helps in predicting how the flowing water affects Earth’s surface. 6. How do physical therapists use physics while on the job? Explain. a. Physical therapists do not require knowledge of physics because their job is mainly therapy and not physics. b. Physical therapists do not require knowledge of physics because their job is more social in nature and unscientific. c. Physical therapists require knowledge of physics know about muscle contraction and release of energy. d. Physical therapists require knowledge of physics to know about chemical reactions inside the body and make decisions accordingly. 7. What is meant when a physical law is said to be universal? a. The law can explain everything in the universe. b. The law is applicable to all physical phenomena. c. The law applies everywhere in the universe. d. The law is the most basic one and all laws are derived from it. 8. What subfield of physics could describe small objects traveling at high speeds or experiencing a strong gravitational field? a. general theory of relativity b. classical physics c. quantum relativity d. special theory of relativity 9. Why is Einstein’s theory of relativity considered part of modern physics, as opposed to classical physics? a. Because it was considered less outstanding than the classics of physics, such as classical mechanics. b. Because it was popular physics enjoyed by average people today, instead of physics studied by the elite. c. Because the theory deals with very slow-moving objects and weak gravitational fields. d. Because it was among the new 19th-century d. 1. matter is moving at speeds of less than roughly 1 discoveries that changed physics. percent the speed of light, 2. objects are large enough to be seen with the 1.2 The Scientific Methods 3. naked eye, and there is the involvement of a weak gravitational field. 10. Describe the difference between an observation and a hypothesis. Access for free at openstax.org. Chapter 1 • Chapter Review 43 a. An observation is seeing what happens; a hypothesis is a testable, educated guess. b. An observation is a hypothesis that has been confirmed. c. Hypotheses and observations are independent of each other. d. Hypotheses are conclusions based on some observations. 11. Describe how modeling is useful in studying the structure of the atom. a. Modeling replaces the real system by something similar but easier to examine. b. Modeling replaces the real system by something more interesting to examine. c. Modeling replaces the real system by something with more realistic properties. a result. d. The dependent and independent variables are fixed by a convention and hence they are the same. 15. What could you conclude about these two lines? 1. Line A has a slope of 2. Line B has a slope of a. Line A is a decreasing line while line B is an increasing line, with line A being much steeper than line B. b. Line A is a decreasing line while line B is an increasing line, with line B being much steeper than line A. c. Line B is a decreasing line while line A is an increasing line, with line A being much steeper than line B. d. Modeling includes more details than are present in d. Line B is a decreasing line while line A is an the real system. 12. How strongly is a hypothesis supported by evidence compared to a theory? a. A theory is supported by little evidence, if any, at first, while a hypothesis is supported by a large amount of available evidence. b. A hypothesis is supported by little evidence, if any, at first. A theory is supported by a large amount of available evidence. c. A hypothesis is supported by little evidence, if any, at first. A theory does not need any experiments in support. d. A theory is supported by little evidence, if any, at first. A hypothesis does not need any experiments in support. 1.3 The Language of Physics: Physical Quantities and Units 13. Which of the following does not contribute to the uncertainty? a. b. c. d. other factors that affect the outcome (depending on the limitations of the measuring device the skill of the person making the measurement the regularities in the object being measured the situation) 14. How does the independent variable in a graph differ from the dependent variable? a. The dependent variable varies linearly with the independent variable. b. The dependent variable depends on the scale of the axis chosen while independent variable does not. c. The independent variable is directly manipulated or controlled by the person doing the experiment, while dependent variable is the one that changes as increasing line, with line B being much steeper than line A. 16. Velocity, or speed, is measured using the following where vis velocity, dis the distance formula: travelled, and tis the time the object took to travel the distance. If the velocity-time data are plotted on a graph, which variable will be on which axis? Why? a. Time would be on the x-axis and velocity on the yaxis, because time is an independent variable and velocity is a dependent variable. b. Velocity would be on the x-axis and time on the yaxis, because time is the independent variable and velocity is the dependent variable. c. Time would be on the x-axis and velocity on the yaxis, because time is a dependent variable and velocity is a independent variable. d. Velocity would be on x-axis and time on the y-axis, because time is a dependent variable and velocity is a independent variable. 17. The uncertainty of a triple-beam balance is . What is the percent uncertainty in a measurement of ? a. b. c. d. 18. What is the definition of uncertainty? a. Uncertainty is the number of assumptions made prior to the measurement of a physical quantity. b. Uncertainty is a measure of error in a measurement due to the use of a non-calibrated instrument. c. Uncertainty is a measure of deviation of the measured value from the standard value. d. Uncertainty is a measure of error in measurement 44 Chapter 1 • Chapter Review due to external factors like air friction and temperature. Critical Thinking Items 1.1 Physics: Definitions and Applications 19. In what sense does Einstein’s theory of relativity illustrate that physics describes fundamental aspects of our universe? a. It describes how speed affects different observers’ measurements of time and space. It describes how different parts of the universe are far apart and do not affect each other. It describes how people think of other people’s views from their own frame of reference. It describes how a frame of reference is necessary to describe position or motion. b. c. d. 20. Can classical physics be used to accurately describe a satellite moving at a speed of 7500 m/s? Explain why or why not. a. No, because the satellite is moving at a speed much smaller than the speed of the light and is not in a strong gravitational field. b. No, because the satellite is moving at a speed much smaller than the speed of the light and is in a strong gravitational field. c. Yes, because the satellite is moving at a speed much smaller than the speed of the light and it is not in a strong gravitational field. d. Yes, because the satellite is moving at a speed much smaller than the speed of the light and is in a strong gravitational field. 21. What would be some ways in which physics was involved in building the features of the room you are in right now? a. Physics is involved in structural strength, not age very much yourself? a. by traveling at a speed equal to the speed of light b. by traveling at a speed faster than the speed of light c. by traveling at a speed much slower than the speed of light d. by traveling at a speed slightly slower than the speed of light 1.2 The Scientific Methods 24. You notice that the water level flowing in a stream near your house increases when it rains and the water turns brown. Which of these are the best hypothesis to explain why the water turns brown. Assume you have all of the means to test the contents of the stream water. a. The water in the stream turns brown because molecular forces between water molecules are stronger than mud molecules b. The water in the stream turns brown because of the breakage of a weak chemical bond with the hydrogen atom in the water molecule. c. The water in the stream turns brown because it picks up dirt from the bank as the water level increases when it rains. d. The water in the stream turns brown because the density of the water increases with increase in water level. 25. Light travels |
as waves at an approximate speed of 300,000,000 m/s (186,000 mi/s). Designers of devices that use mirrors and lenses model the traveling light by straight lines, or light rays. Describe why it would be useful to model the light as rays of light instead of describing them accurately as electromagnetic waves. a. A model can be constructed in such a way that the dimensions, etc., of the room. speed of light decreases. b. Physics is involved in the air composition inside the b. Studying a model makes it easier to analyze the room. path that the light follows. c. Physics is involved in the desk arrangement inside c. Studying a model will help us to visualize why light the room. travels at such great speed. d. Physics is involved in the behavior of living beings d. Modeling cannot be used to study traveling light as inside the room. our eyes cannot track the motion of light. 22. What theory of modern physics describes the interrelationships between space, time, speed, and gravity? a. atomic theory b. nuclear physics c. quantum mechanics d. general relativity 23. According to Einstein’s theory of relativity, how could you effectively travel many years into Earth’s future, but 26. A friend says that he doesn’t trust scientific explanations because they are just theories, which are basically educated guesses. What could you say to convince him that scientific theories are different from the everyday use of the word theory? a. A theory is a scientific explanation that has been repeatedly tested and supported by many experiments. b. A theory is a hypothesis that has been tested and Access for free at openstax.org. supported by some experiments. c. A theory is a set of educated guesses, but at least one of the guesses remain true in each experiment. d. A theory is a set of scientific explanations that has at least one experiment in support of it. 27. Give an example of a hypothesis that cannot be tested experimentally. a. The structure of any part of the broccoli is similar to the whole structure of the broccoli. b. Ghosts are the souls of people who have died. c. The average speed of air molecules increases with temperature. d. A vegetarian is less likely to be affected by night blindness. 28. Would it be possible to scientifically prove that a supreme being exists or not? Briefly explain your answer. a. It can be proved scientifically because it is a testable hypothesis. It cannot be proved scientifically because it is not a testable hypothesis. It can be proved scientifically because it is not a testable hypothesis. It cannot be proved scientifically because it is a testable hypothesis. b. c. d. 1.3 The Language of Physics: Physical Quantities and Units 29. A marathon runner completes a course in , , and . There is an uncertainty of in the distance traveled and an uncertainty of elapsed time. in the 1. Calculate the percent uncertainty in the distance. 2. Calculate the uncertainty in the elapsed time. 3. What is the average speed in meters per second? 4. What is the uncertainty in the average speed? a. b. , , , , , , Problems 1.3 The Language of Physics: Physical Quantities and Units 34. A commemorative coin that sells for $40 is advertised to be plated with 15 mg of gold. Suppose gold is worth about $1,300 per ounce. Which of the following best represents the value of the gold in the coin? a. $0.33 b. $0.69 Chapter 1 • Chapter Review 45 c. d. , , , , , , 30. A car engine moves a piston with a circular cross section of diameter a distance of to compress the gas in the cylinder. By what amount did the gas decrease in volume in cubic centimeters? Find the uncertainty in this volume. a. b. c. d. 31. What would be the slope for a line passing through the two points below? Point 1: (1, 0.1) Point 2: (7, 26.8) a. b. c. d. 32. The sides of a small rectangular box are measured and long and high. Calculate its volume and uncertainty in cubic centimeters. Assume the measuring device is accurate to a. b. c. d. . 33. Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint—The mass of a hydrogen atom is on the order of 10−27 kg and the mass of a bacterium is on the order of 10−15 kg .) a. b. c. d. 1010 atoms 1011 atoms 1012 atoms 1013 atoms c. $3.30 d. $6.90 35. If a marathon runner runs in another direction and in one direction, in a third direction, how much distance did the runner run? Be sure to report your answer using the proper number of significant figures. a. b. c. 46 Chapter 1 • Test Prep d. 37. The length and width of a rectangular room are 36. The speed limit on some interstate highways is roughly . What is this in meters per second? How many , miles per hour is this? a. b. c. d. , , , Performance Task 1.3 The Language of Physics: Physical Quantities and Units 38. a. Create a new system of units to describe something that interests you. Your unit should be described using at least two subunits. For example, you can decide to measure the quality of songs using a new unit called song awesomeness. Song awesomeness TEST PREP Multiple Choice 1.1 Physics: Definitions and Applications 39. Modern physics could best be described as the combination of which theories? a. quantum mechanics and Einstein’s theory of relativity b. quantum mechanics and classical physics c. Newton’s laws of motion and classical physics d. Newton’s laws of motion and Einstein’s theory of relativity 40. Which of the following could be studied accurately using classical physics? a. b. c. d. the strength of gravity within a black hole the motion of a plane through the sky the collisions of subatomic particles the effect of gravity on the passage of time 41. Which of the following best describes why knowledge of physics is necessary to understand all other sciences? a. Physics explains how energy passes from one object to another. b. Physics explains how gravity works. c. Physics explains the motion of objects that can be seen with the naked eye. d. Physics explains the fundamental aspects of the universe. 42. What does radiation therapy, used to treat cancer patients, have to do with physics? a. Understanding how cells reproduce is mainly about Access for free at openstax.org. by measured to be . Calculate the area of the room and its uncertainty in square meters. a. b. c. d. is measured by: the number of songs downloaded and the number of times the song was used in movies. b. Create an equation that shows how to calculate your unit. Then, using your equation, create a sample dataset that you could graph. Are your two subunits related linearly, quadratically, or inversely? physics. b. Predictions of the side effects from the radiation therapy are based on physics. c. The devices used for generating some kinds of radiation are based on principles of physics. d. Predictions of the life expectancy of patients receiving radiation therapy are based on physics. 1.2 The Scientific Methods 43. The free-electron model of metals explains some of the important behaviors of metals by assuming the metal’s electrons move freely through the metal without repelling one another. In what sense is the free-electron theory based on a model? a. Its use requires constructing replicas of the metal wire in the lab. It involves analyzing an imaginary system simpler than the real wire it resembles. It examines a model, or ideal, behavior that other metals should imitate. It attempts to examine the metal in a very realistic, or model, way. b. c. d. 44. A scientist wishes to study the motion of about 1,000 molecules of gas in a container by modeling them as tiny billiard balls bouncing randomly off one another. Which of the following is needed to calculate and store data on their detailed motion? a. a group of hypotheses that cannot be practically tested in real life b. a computer that can store and perform calculations on large data sets c. a large amount of experimental results on the molecules and their motion d. a collection of hypotheses that have not yet been tested regarding the molecules 45. When a large body of experimental evidence supports a hypothesis, what may the hypothesis eventually be considered? a. observation insight b. conclusion c. law d. 46. While watching some ants outside of your house, you notice that the worker ants gather in a specific area on your lawn. Which of the following is a testable hypothesis that attempts to explain why the ants gather in that specific area on the lawn. a. The worker thought it was a nice location. b. because ants may have to find a spot for the queen to lay eggs c. because there may be some food particles lying there d. because the worker ants are supposed to group together at a place. 1.3 The Language of Physics: Physical Quantities and Units 47. Which of the following would describe a length that is of a meter? a. b. kilometers megameters Short Answer 1.1 Physics: Definitions and Applications 51. Describe the aims of physics. a. Physics aims to explain the fundamental aspects of our universe and how these aspects interact with one another. b. Physics aims to explain the biological aspects of our universe and how these aspects interact with one another. c. Physics aims to explain the composition, structure and changes in matter occurring in the universe. d. Physics aims to explain the social behavior of living beings in the universe. 52. Define the fields of magnetism and electricity and state how are they are related. a. Magnetism describes the attractive force between a Chapter 1 • Test Prep 47 c. d. millimeters micrometers 48. Suppose that a bathroom scale reads a person’s mass as 65 kg with a 3 percent uncertainty. What is the uncertainty in their mass in kilograms? a. a. 2 kg b. b. 98 kg c. d. d. 0 c. 5 kg 49. Which of the following best describes a variable? a. a trend that shows an exponential relationship b. something whose value can change over multiple measurements c. a measure of how much a plot line changes alo |
ng d. the y-axis something that remains constant over multiple measurements 50. A high school track coach has just purchased a new stopwatch that has an uncertainty of ±0.05 s . Runners on the team regularly clock 100-m sprints in 12.49 s to 15.01 s . At the school’s last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in at 12.07 s . Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not? a. No, the uncertainty in the stopwatch is too large to effectively differentiate between the sprint times. b. No, the uncertainty in the stopwatch is too small to effectively differentiate between the sprint times. c. Yes, the uncertainty in the stopwatch is too large to effectively differentiate between the sprint times. d. Yes, the uncertainty in the stopwatch is too small to effectively differentiate between the sprint times. magnetized object and a metal like iron. Electricity involves the study of electric charges and their movements. Magnetism is not related to the electricity. b. Magnetism describes the attractive force between a magnetized object and a metal like iron. Electricity involves the study of electric charges and their movements. Magnetism is produced by a flow electrical charges. c. Magnetism involves the study of electric charges and their movements. Electricity describes the attractive force between a magnetized object and a metal. Magnetism is not related to the electricity. d. Magnetism involves the study of electric charges and their movements. Electricity describes the attractive force between a magnetized object and a metal. Magnetism is produced by the flow electrical charges. 48 Chapter 1 • Test Prep 53. Describe what two topics physicists are trying to unify with relativistic quantum mechanics. How will this unification create a greater understanding of our universe? a. Relativistic quantum mechanics unifies quantum mechanics with Einstein’s theory of relativity. The unified theory creates a greater understanding of our universe because it can explain objects of all sizes and masses. b. Relativistic quantum mechanics unifies classical mechanics with Einstein’s theory of relativity. The unified theory creates a greater understanding of our universe because it can explain objects of all sizes and masses. c. Relativistic quantum mechanics unifies quantum mechanics with Einstein’s theory of relativity. The unified theory creates a greater understanding of our universe because it is unable to explain objects of all sizes and masses. d. Relativistic quantum mechanics unifies classical mechanics with the Einstein’s theory of relativity. The unified theory creates a greater understanding of our universe because it is unable to explain objects of all sizes and masses. a. An understanding of force, pressure, heat, electricity, etc., which all involve physics, will help me design a sound and energy-efficient house. b. An understanding of the air composition, chemical composition of matter, etc., which all involves physics, will help me design a sound and energyefficient house. c. An understanding of material cost and economic factors involving physics will help me design a sound and energy-efficient house. d. An understanding of geographical location and social environment which involves physics will help me design a sound and energy-efficient house. 57. What aspects of physics would a chemist likely study in trying to discover a new chemical reaction? a. Physics is involved in understanding whether the reactants and products dissolve in water. b. Physics is involved in understanding the amount of energy released or required in a chemical reaction. c. Physics is involved in what the products of the reaction will be. d. Physics is involved in understanding the types of ions produced in a chemical reaction. 54. The findings of studies in quantum mechanics have 1.2 The Scientific Methods been described as strange or weird compared to those of classical physics. Explain why this would be so. a. It is because the phenomena it explains are outside the normal range of human experience which deals with much larger objects. It is because the phenomena it explains can be perceived easily, namely, ordinary-sized objects. It is because the phenomena it explains are outside the normal range of human experience, namely, the very large and the very fast objects. It is because the phenomena it explains can be perceived easily, namely, the very large and the very fast objects. b. c. d. 55. How could knowledge of physics help you find a faster way to drive from your house to your school? a. Physics can explain the traffic on a particular street and help us know about the traffic in advance. b. Physics can explain about the ongoing construction of roads on a particular street and help us know about delays in the traffic in advance. c. Physics can explain distances, speed limits on a particular street and help us categorize faster routes. d. Physics can explain the closing of a particular street and help us categorize faster routes. 56. How could knowledge of physics help you build a sound and energy-efficient house? Access for free at openstax.org. 58. You notice that it takes more force to get a large box to start sliding across the floor than it takes to get the box sliding faster once it is already moving. Create a testable hypothesis that attempts to explain this observation. a. The floor has greater distortions of space-time for moving the sliding box faster than for the box at rest. b. The floor has greater distortions of space-time for the box at rest than for the sliding box. c. The resistance between the floor and the box is less when the box is sliding then when the box is at rest. d. The floor dislikes having objects move across it and therefore holds the box rigidly in place until it cannot resist the force. 59. Design an experiment that will test the following hypothesis: driving on a gravel road causes greater damage to a car than driving on a dirt road. a. To test the hypothesis, compare the damage to the car by driving it on a smooth road and a gravel road. b. To test the hypothesis, compare the damage to the car by driving it on a smooth road and a dirt road. c. To test the hypothesis, compare the damage to the car by driving it on a gravel road and the dirt road. d. This is not a testable hypothesis. 60. How is a physical model, such as a spherical mass held in place by springs, used to represent an atom vibrating in a solid, similar to a computer-based model, such as that predicting how gravity affects the orbits of the planets? a. Both a physical model and a computer-based model should be built around a hypothesis and could be able to test the hypothesis. b. Both a physical model and a computer-based model should be built around a hypothesis but they cannot be used to test the hypothesis. c. Both a physical model and a computer-based model should be built around the results of scientific studies and could be used to make predictions about the system under study. d. Both a physical model and a computer-based model should be built around the results of scientific studies but cannot be used to make predictions about the system under study. 61. Explain the advantages and disadvantages of using a model to predict a life-or-death situation, such as whether or not an asteroid will strike Earth. a. The advantage of using a model is that it provides predictions quickly, but the disadvantage of using a model is that it could make erroneous predictions. b. The advantage of using a model is that it provides accurate predictions, but the disadvantage of using a model is that it takes a long time to make predictions. c. The advantage of using a model is that it provides predictions quickly without any error. There are no disadvantages of using a scientific model. d. The disadvantage of using models is that it takes longer time to make predictions and the predictions are inaccurate. There are no advantages to using a scientific model. 62. A friend tells you that a scientific law cannot be changed. State whether or not your friend is correct and then briefly explain your answer. a. Correct, because laws are theories that have been proved true. b. Correct, because theories are laws that have been c. d. proved true. Incorrect, because a law is changed if new evidence contradicts it. Incorrect, because a law is changed when a theory contradicts it. 63. How does a scientific law compare to a local law, such as that governing parking at your school, in terms of whether or not laws can be changed, and how universal a law is? a. A local law applies only in a specific area, but a Chapter 1 • Test Prep 49 scientific law is applicable throughout the universe. Both the local law and the scientific law can change. b. A local law applies only in a specific area, but a scientific law is applicable throughout the universe. A local law can change, but a scientific law cannot be changed. c. A local law applies throughout the universe but a scientific law is applicable only in a specific area. Both the local and the scientific law can change. d. A local law applies throughout the universe, but a scientific law is applicable only in a specific area. A local law can change, but a scientific law cannot be changed. 64. Can the validity of a model be limited, or must it be universally valid? How does this compare to the required validity of a theory or a law? a. Models, theories and laws must be universally valid. b. Models, theories, and laws have only limited validity. c. Models have limited validity while theories and laws are universally valid. d. Models and theories have limited validity while laws are universally valid. 1.3 The Language of Physics: Physical Quantities and Units 65. The speed of sound is measured at on a certain ? Report your answer in day. What is this in scientific notation. a. b. c. d. 66. Describe the main difference between the metric system and |
the U.S. Customary System. a. In the metric system, unit changes are based on powers of 10, while in the U.S. customary system, each unit conversion has unrelated conversion factors. In the metric system, each unit conversion has unrelated conversion factors, while in the U.S. customary system, unit changes are based on powers of 10. In the metric system, unit changes are based on powers of 2, while in the U.S. customary system, each unit conversion has unrelated conversion factors. In the metric system, each unit conversion has unrelated conversion factors, while in the U.S. customary system, unit changes are based on b. c. d. 50 Chapter 1 • Test Prep powers of 2. 67. An infant’s pulse rate is measured to be . What is the percent uncertainty in this measurement? a. b. c. d. 68. Explain how the uncertainty of a measurement relates to the accuracy and precision of the measuring device. Include the definitions of accuracy and precision in your answer. a. A decrease in the precision of a measurement increases the uncertainty of the measurement, while a decrease in accuracy does not. b. A decrease in either the precision or accuracy of a measurement increases the uncertainty of the measurement. c. An increase in either the precision or accuracy of a measurement will increase the uncertainty of that measurement. d. An increase in the accuracy of a measurement will increase the uncertainty of that measurement, while an increase in precision will not. 69. Describe all of the characteristics that can be and determined about a straight line with a slope of a y-intercept of on a graph. a. Based on the information, the line has a negative slope. Because its y-intercept is 50 and its slope is negative, this line gradually rises on the graph as the x-value increases. b. Based on the information, the line has a negative slope. Because its y-intercept is 50 and its slope is negative, this line gradually moves downward on Extended Response 1.2 The Scientific Methods 71. You wish to perform an experiment on the stopping distance of your new car. Create a specific experiment to measure the distance. Be sure to specifically state how you will set up and take data during your experiment. a. Drive the car at exactly 50 mph and then press harder on the accelerator pedal until the velocity reaches the speed 60 mph and record the distance this takes. b. Drive the car at exactly 50 mph and then apply the brakes until it stops and record the distance this takes. c. Drive the car at exactly 50 mph and then apply the brakes until it stops and record the time it takes. Access for free at openstax.org. the graph as the x-value increases. c. Based on the information, the line has a positive slope. Because its y-intercept is 50 and its slope is positive, this line gradually rises on the graph as the x-value increases. d. Based on the information, the line has a positive slope. Because its y-intercept is 50 and its slope is positive, this line gradually moves downward on the graph as the x-value increases. 70. The graph shows the temperature change over time of a heated cup of water. What is the slope of the graph between the time period 2 min and 5 min? a. –15 ºC/min b. –0.07 ºC/min c. 0.07 ºC/min 15 ºC/min d. d. Drive the car at exactly 50 mph and then apply the accelerator until it reaches the speed of 60 mph and record the time it takes. 72. You wish to make a model showing how traffic flows around your city or local area. Describe the steps you would take to construct your model as well as some hypotheses that your model could test and the model’s limitations in terms of what could not be tested. a. 1. Testable hypotheses like the gravitational pull on each vehicle while in motion and the average speed of vehicles is 40 mph 2. Non-testable hypotheses like the average number of vehicles passing is 935 per day and carbon emission from each of the moving vehicle b. 1. Testable hypotheses like the average number of vehicles passing is 935 per day and the average speed of vehicles is 40 mph 2. Non-testable hypotheses like the gravitational pull on each vehicle while in motion and the carbon emission from each of the moving vehicle c. 1. Testable hypotheses like the average number of vehicles passing is 935 per day and the carbon emission from each of the moving vehicle 2. Non-testable hypotheses like the gravitational pull on each vehicle while in motion and the average speed of the vehicles is 40 mph d. 1. Testable hypotheses like the average number of vehicles passing is 935 per day and the gravitational pull on each vehicle while in motion 2. Non-testable hypotheses like the average speed of vehicles is 40 mph and the carbon emission from each of the moving vehicle 73. What would play the most important role in leading to an experiment in the scientific world becoming a scientific law? a. Further testing would need to show it is a universally followed rule. b. The observation would have to be described in a Chapter 1 • Test Prep 51 published scientific article. c. The experiment would have to be repeated once or twice. d. The observer would need to be a well-known scientist whose authority was accepted. 1.3 The Language of Physics: Physical Quantities and Units at this speed? What is its speed in 74. Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of . What distance does it move in kilometers per million years? Report all of your answers using scientific notation. a. b. c. d. 75. At x = 3, a function f(x) has a positive value, with a positive slope that is decreasing in magnitude with increasing x. Which option could correspond to f(x)? a. b. c. d. 52 Chapter 1 • Test Prep Access for free at openstax.org. CHAPTER 2 Motion in One Dimension Figure 2.1 Shanghai Maglev. At this rate, a train traveling from Boston to Washington, DC, a distance of 439 miles, could make the trip in under an hour and a half. Presently, the fastest train on this route takes over six hours to cover this distance. (Alex Needham, Public Domain) Chapter Outline 2.1 Relative Motion, Distance, and Displacement 2.2 Speed and Velocity 2.3 Position vs. Time Graphs 2.4 Velocity vs. Time Graphs Unless you have flown in an airplane, you have probably never traveled faster than 150 mph. Can you imagine INTRODUCTION traveling in a train like the one shown in Figure 2.1 that goes over 300 mph? Despite the high speed, the people riding in this train may not notice that they are moving at all unless they look out the window! This is because motion, even motion at 300 mph, is relative to the observer. In this chapter, you will learn why it is important to identify a reference frame in order to clearly describe motion. For now, the motion you describe will be one-dimensional. Within this context, you will learn the difference between distance and displacement as well as the difference between speed and velocity. Then you will look at some graphing and problem-solving techniques. 54 Chapter 2 • Motion in One Dimension 2.1 Relative Motion, Distance, and Displacement Section Learning Objectives By the end of this section, you will be able to do the following: • Describe motion in different reference frames • Define distance and displacement, and distinguish between the two • Solve problems involving distance and displacement Section Key Terms displacement distance kinematics magnitude position reference frame scalar vector Defining Motion Our study of physics opens with kinematics—the study of motion without considering its causes. Objects are in motion everywhere you look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. Even in inanimate objects, atoms are always moving. How do you know something is moving? The location of an object at any particular time is its position. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to Earth as a whole, while a professor’s position could be described in terms of where she is in relation to the nearby white board. In other cases, we use reference frames that are not stationary but are in motion relative to Earth. To describe the position of a person in an airplane, for example, we use the airplane, not Earth, as the reference frame. (See Figure 2.2.) Thus, you can only know how fast and in what direction an object's position is changing against a background of something else that is either not moving or moving with a known speed and direction. The reference frame is the coordinate system from which the positions of objects are described. Figure 2.2 Are clouds a useful reference frame for airplane passengers? Why or why not? (Paul Brennan, Public Domain) Your classroom can be used as a reference frame. In the classroom, the walls are not moving. Your motion as you walk to the door, can be measured against the stationary background of the classroom walls. You can also tell if other things in the classroom are moving, such as your classmates entering the classroom or a book falling off a desk. You can also tell in what direction something is moving in the classroom. You might say, “The teacher is moving toward the door.” Your reference frame allows you to determine not only that something is moving but also the direction of motion. You could also serve as a reference frame for others’ movement. If you remained seated as your classmates left the room, you would measure their movement away from your stationary location. If you and your classmates left the room together, then your perspective of their motion would be change. You, as the reference frame, would be moving in the same dire |
ction as your other moving classmates. As you will learn in the Snap Lab, your description of motion can be quite different when viewed from different reference frames. Access for free at openstax.org. 2.1 • Relative Motion, Distance, and Displacement 55 Snap Lab Looking at Motion from Two Reference Frames In this activity you will look at motion from two reference frames. Which reference frame is correct? • Choose an open location with lots of space to spread out so there is less chance of tripping or falling due to a collision and/or loose basketballs. • 1 basketball Procedure 1. Work with a partner. Stand a couple of meters away from your partner. Have your partner turn to the side so that you are looking at your partner’s profile. Have your partner begin bouncing the basketball while standing in place. Describe the motion of the ball. 2. Next, have your partner again bounce the ball, but this time your partner should walk forward with the bouncing ball. You will remain stationary. Describe the ball's motion. 3. Again have your partner walk forward with the bouncing ball. This time, you should move alongside your partner while continuing to view your partner’s profile. Describe the ball's motion. 4. Switch places with your partner, and repeat Steps 1–3. GRASP CHECK How do the different reference frames affect how you describe the motion of the ball? a. The motion of the ball is independent of the reference frame and is same for different reference frames. b. The motion of the ball is independent of the reference frame and is different for different reference frames. c. The motion of the ball is dependent on the reference frame and is same for different reference frames. d. The motion of the ball is dependent on the reference frames and is different for different reference frames. LINKS TO PHYSICS History: Galileo's Ship Figure 2.3 Galileo Galilei (1564–1642) studied motion and developed the concept of a reference frame. (Domenico Tintoretto) The idea that a description of motion depends on the reference frame of the observer has been known for hundreds of years. The 17th-century astronomer Galileo Galilei (Figure 2.3) was one of the first scientists to explore this idea. Galileo suggested the following thought experiment: Imagine a windowless ship moving at a constant speed and direction along a perfectly calm sea. Is there a way that a person inside the ship can determine whether the ship is moving? You can extend this thought experiment 56 Chapter 2 • Motion in One Dimension by also imagining a person standing on the shore. How can a person on the shore determine whether the ship is moving? Galileo came to an amazing conclusion. Only by looking at each other can a person in the ship or a person on shore describe the motion of one relative to the other. In addition, their descriptions of motion would be identical. A person inside the ship would describe the person on the land as moving past the ship. The person on shore would describe the ship and the person inside it as moving past. Galileo realized that observers moving at a constant speed and direction relative to each other describe motion in the same way. Galileo had discovered that a description of motion is only meaningful if you specify a reference frame. GRASP CHECK Imagine standing on a platform watching a train pass by. According to Galileo’s conclusions, how would your description of motion and the description of motion by a person riding on the train compare? a. b. c. d. I would see the train as moving past me, and a person on the train would see me as stationary. I would see the train as moving past me, and a person on the train would see me as moving past the train. I would see the train as stationary, and a person on the train would see me as moving past the train. I would see the train as stationary, and a person on the train would also see me as stationary. Distance vs. Displacement As we study the motion of objects, we must first be able to describe the object’s position. Before your parent drives you to school, the car is sitting in your driveway. Your driveway is the starting position for the car. When you reach your high school, the car has changed position. Its new position is your school. Figure 2.4 Your total change in position is measured from your house to your school. Physicists use variables to represent terms. We will use d to represent car’s position. We will use a subscript to differentiate between the initial position, d0, and the final position, df. In addition, vectors, which we will discuss later, will be in bold or will have an arrow above the variable. Scalars will be italicized. TIPS FOR SUCCESS In some books, x or s is used instead of d to describe position. In d0, said d naught, the subscript 0 stands for initial. When we begin to talk about two-dimensional motion, sometimes other subscripts will be used to describe horizontal position, dx, or vertical position, dy. So, you might see references to d0x and dfy. Now imagine driving from your house to a friend's house located several kilometers away. How far would you drive? The distance an object moves is the length of the path between its initial position and its final position. The distance you drive to your friend's house depends on your path. As shown in Figure 2.5, distance is different from the length of a straight line between two points. The distance you drive to your friend's house is probably longer than the straight line between the two houses. Access for free at openstax.org. 2.1 • Relative Motion, Distance, and Displacement 57 Figure 2.5 A short line separates the starting and ending points of this motion, but the distance along the path of motion is considerably longer. We often want to be more precise when we talk about position. The description of an object’s motion often includes more than just the distance it moves. For instance, if it is a five kilometer drive to school, the distance traveled is 5 kilometers. After dropping you off at school and driving back home, your parent will have traveled a total distance of 10 kilometers. The car and your parent will end up in the same starting position in space. The net change in position of an object is its displacement, or The Greek letter delta, , means change in. Figure 2.6 The total distance that your car travels is 10 km, but the total displacement is 0. Snap Lab Distance vs. Displacement In this activity you will compare distance and displacement. Which term is more useful when making measurements? 1 recorded song available on a portable device 1 tape measure 3 pieces of masking tape • • • • A room (like a gym) with a wall that is large and clear enough for all pairs of students to walk back and forth without running into each other. Procedure 1. One student from each pair should stand with their back to the longest wall in the classroom. Students should stand at least 0.5 meters away from each other. Mark this starting point with a piece of masking tape. 2. The second student from each pair should stand facing their partner, about two to three meters away. Mark this point 58 Chapter 2 • Motion in One Dimension with a second piece of masking tape. 3. Student pairs line up at the starting point along the wall. 4. The teacher turns on the music. Each pair walks back and forth from the wall to the second marked point until the music stops playing. Keep count of the number of times you walk across the floor. 5. When the music stops, mark your ending position with the third piece of masking tape. 6. Measure from your starting, initial position to your ending, final position. 7. Measure the length of your path from the starting position to the second marked position. Multiply this measurement by the total number of times you walked across the floor. Then add this number to your measurement from step 6. 8. Compare the two measurements from steps 6 and 7. GRASP CHECK 1. Which measurement is your total distance traveled? 2. Which measurement is your displacement? 3. When might you want to use one over the other? a. Measurement of the total length of your path from the starting position to the final position gives the distance traveled, and the measurement from your initial position to your final position is the displacement. Use distance to describe the total path between starting and ending points,and use displacement to describe the shortest path between starting and ending points. b. Measurement of the total length of your path from the starting position to the final position is distance traveled, and the measurement from your initial position to your final position is displacement. Use distance to describe the shortest path between starting and ending points, and use displacement to describe the total path between starting and ending points. c. Measurement from your initial position to your final position is distance traveled, and the measurement of the total length of your path from the starting position to the final position is displacement. Use distance to describe the total path between starting and ending points, and use displacement to describe the shortest path between starting and ending points. d. Measurement from your initial position to your final position is distance traveled, and the measurement of the total length of your path from the starting position to the final position is displacement. Use distance to describe the shortest path between starting and ending points, and use displacement to describe the total path between starting and ending points. If you are describing only your drive to school, then the distance traveled and the displacement are the same—5 kilometers. When you are describing the entire round trip, distance and displacement are different. When you describe distance, you only include the magnitude, the size or amount, of the distance traveled. However, when you describe the displacement, you take into account both the magnitude of the change in position and the direction |
of movement. In our previous example, the car travels a total of 10 kilometers, but it drives five of those kilometers forward toward school and five of those kilometers back in the opposite direction. If we ascribe the forward direction a positive (+) and the opposite direction a negative (–), then the two quantities will cancel each other out when added together. A quantity, such as distance, that has magnitude (i.e., how big or how much) but does not take into account direction is called a scalar. A quantity, such as displacement, that has both magnitude and direction is called a vector. WATCH PHYSICS Vectors & Scalars This video (http://openstax.org/l/28vectorscalar) introduces and differentiates between vectors and scalars. It also introduces quantities that we will be working with during the study of kinematics. Click to view content (https://www.khanacademy.org/embed_video?v=ihNZlp7iUHE) Access for free at openstax.org. GRASP CHECK 2.1 • Relative Motion, Distance, and Displacement 59 How does this video (https://www.khanacademy.org/science/ap-physics-1/ap-one-dimensional-motion/ap-physicsfoundations/v/introduction-to-vectors-and-scalars) help you understand the difference between distance and displacement? Describe the differences between vectors and scalars using physical quantities as examples. a. It explains that distance is a vector and direction is important, whereas displacement is a scalar and it has no direction attached to it. It explains that distance is a scalar and direction is important, whereas displacement is a vector and it has no direction attached to it. It explains that distance is a scalar and it has no direction attached to it, whereas displacement is a vector and direction is important. It explains that both distance and displacement are scalar and no directions are attached to them. b. c. d. Displacement Problems Hopefully you now understand the conceptual difference between distance and displacement. Understanding concepts is half the battle in physics. The other half is math. A stumbling block to new physics students is trying to wade through the math of physics while also trying to understand the associated concepts. This struggle may lead to misconceptions and answers that make no sense. Once the concept is mastered, the math is far less confusing. So let’s review and see if we can make sense of displacement in terms of numbers and equations. You can calculate an object's displacement by subtracting its original position, d0, from its final position df. In math terms that means If the final position is the same as the initial position, then . To assign numbers and/or direction to these quantities, we need to define an axis with a positive and a negative direction. We also need to define an origin, or O. In Figure 2.6, the axis is in a straight line with home at zero and school in the positive direction. If we left home and drove the opposite way from school, motion would have been in the negative direction. We would have assigned it a negative value. In the round-trip drive, df and d0 were both at zero kilometers. In the one way trip to school, df was at 5 kilometers and d0 was at zero km. So, was 5 kilometers. TIPS FOR SUCCESS You may place your origin wherever you would like. You have to make sure that you calculate all distances consistently from your zero and you define one direction as positive and the other as negative. Therefore, it makes sense to choose the easiest axis, direction, and zero. In the example above, we took home to be zero because it allowed us to avoid having to interpret a solution with a negative sign. WORKED EXAMPLE Calculating Distance and Displacement A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her displacement? 60 Chapter 2 • Motion in One Dimension Strategy To solve this problem, we need to find the difference between the final position and the initial position while taking care to note the direction on the axis. The final position is the sum of the two displacements, and . Solution a. Displacement: The rider’s displacement is b. Distance: The distance traveled is 3 km + 2 km = 5 km. c. The magnitude of the displacement is 1 km. . Discussion The displacement is negative because we chose east to be positive and west to be negative. We could also have described the displacement as 1 km west. When calculating displacement, the direction mattered, but when calculating distance, the direction did not matter. The problem would work the same way if the problem were in the north–south or y-direction. TIPS FOR SUCCESS Physicists like to use standard units so it is easier to compare notes. The standard units for calculations are called SIunits (International System of Units). SI units are based on the metric system. The SI unit for displacement is the meter (m), but sometimes you will see a problem with kilometers, miles, feet, or other units of length. If one unit in a problem is an SI unit and another is not, you will need to convert all of your quantities to the same system before you can carry out the calculation. Practice Problems 1. On an axis in which moving from right to left is positive, what is the displacement and distance of a student who walks 32 m to the right and then 17 m to the left? a. Displacement is -15 m and distance is -49 m. b. Displacement is -15 m and distance is 49 m. c. Displacement is 15 m and distance is -49 m. d. Displacement is 15 m and distance is 49 m. 2. Tiana jogs 1.5 km along a straight path and then turns and jogs 2.4 km in the opposite direction. She then turns back and jogs 0.7 km in the original direction. Let Tiana’s original direction be the positive direction. What are the displacement and distance she jogged? a. Displacement is 4.6 km,and distance is -0.2 km. b. Displacement is -0.2 km, and distance is 4.6 km. c. Displacement is 4.6 km, and distance is +0.2 km. d. Displacement is +0.2 km, and distance is 4.6 km. WORK IN PHYSICS Mars Probe Explosion Figure 2.7 The Mars Climate Orbiter disaster illustrates the importance of using the correct calculations in physics. (NASA) Access for free at openstax.org. 2.1 • Relative Motion, Distance, and Displacement 61 Physicists make calculations all the time, but they do not always get the right answers. In 1998, NASA, the National Aeronautics and Space Administration, launched the Mars Climate Orbiter, shown in Figure 2.7, a $125-million-dollar satellite designed to monitor the Martian atmosphere. It was supposed to orbit the planet and take readings from a safe distance. The American scientists made calculations in English units (feet, inches, pounds, etc.) and forgot to convert their answers to the standard metric SI units. This was a very costly mistake. Instead of orbiting the planet as planned, the Mars Climate Orbiter ended up flying into the Martian atmosphere. The probe disintegrated. It was one of the biggest embarrassments in NASA’s history. GRASP CHECK In 1999 the Mars Climate Orbiter crashed because calculation were performed in English units instead of SI units. At one point the orbiter was just 187,000 feet above the surface, which was too close to stay in orbit. What was the height of the orbiter at this time in kilometers? (Assume 1 meter equals 3.281 feet.) 16 km a. 18 km b. 57 km c. d. 614 km Check Your Understanding 3. What does it mean when motion is described as relative? a. b. c. d. It means that motion of any object is described relative to the motion of Earth. It means that motion of any object is described relative to the motion of any other object. It means that motion is independent of the frame of reference. It means that motion depends on the frame of reference selected. 4. If you and a friend are standing side-by-side watching a soccer game, would you both view the motion from the same reference frame? a. Yes, we would both view the motion from the same reference point because both of us are at rest in Earth’s frame of reference. b. Yes, we would both view the motion from the same reference point because both of us are observing the motion from two points on the same straight line. c. No, we would both view the motion from different reference points because motion is viewed from two different points; the reference frames are similar but not the same. d. No, we would both view the motion from different reference points because response times may be different; so, the motion observed by both of us would be different. 5. What is the difference between distance and displacement? a. Distance has both magnitude and direction, while displacement has magnitude but no direction. b. Distance has magnitude but no direction, while displacement has both magnitude and direction. c. Distance has magnitude but no direction, while displacement has only direction. d. There is no difference. Both distance and displacement have magnitude and direction. 6. Which situation correctly identifies a race car’s distance traveled and the magnitude of displacement during a one-lap car race? a. The perimeter of the race track is the distance, and the shortest distance between the start line and the finish line is the magnitude of displacement. b. The perimeter of the race track is the magnitude of displacement, and the shortest distance between the start and finish line is the distance. c. The perimeter of the race track is both the distance and magnitude of displacement. d. The shortest distance between the start line and the finish line is both the distance and magnitude of displacement. 7. Why is it important to specify a reference frame when describing motion? a. Because Earth is continuously in motion; an object at rest on Earth will be in motion when viewed from outer space. b. Because the position of a moving object can be defined only when there is a fixed reference frame. 62 Chapter 2 • Motion in One Dimension c. Because motion is a relative term; it |
appears differently when viewed from different reference frames. d. Because motion is always described in Earth’s frame of reference; if another frame is used, it has to be specified with each situation. 2.2 Speed and Velocity Section Learning Objectives By the end of this section, you will be able to do the following: • Calculate the average speed of an object • Relate displacement and average velocity Section Key Terms average speed average velocity instantaneous speed instantaneous velocity speed velocity Speed There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we will look at time, speed, and velocity to expand our understanding of motion. A description of how fast or slow an object moves is its speed. Speed is the rate at which an object changes its location. Like distance, speed is a scalar because it has a magnitude but not a direction. Because speed is a rate, it depends on the time interval of motion. You can calculate the elapsed time or the change in time, and the beginning time , of motion as the difference between the ending time The SI unit of time is the second (s), and the SI unit of speed is meters per second (m/s), but sometimes kilometers per hour (km/h), miles per hour (mph) or other units of speed are used. When you describe an object's speed, you often describe the average over a time period. Average speed, vavg, is the distance traveled divided by the time during which the motion occurs. You can, of course, rearrange the equation to solve for either distance or time Suppose, for example, a car travels 150 kilometers in 3.2 hours. Its average speed for the trip is A car's speed would likely increase and decrease many times over a 3.2 hour trip. Its speed at a specific instant in time, however, is its instantaneous speed. A car's speedometer describes its instantaneous speed. Access for free at openstax.org. 2.2 • Speed and Velocity 63 Figure 2.8 During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, because there was no net change in position. WORKED EXAMPLE Calculating Average Speed A marble rolls 5.2 m in 1.8 s. What was the marble's average speed? Strategy We know the distance the marble travels, 5.2 m, and the time interval, 1.8 s. We can use these values in the average speed equation. Solution Discussion Average speed is a scalar, so we do not include direction in the answer. We can check the reasonableness of the answer by estimating: 5 meters divided by 2 seconds is 2.5 m/s. Since 2.5 m/s is close to 2.9 m/s, the answer is reasonable. This is about the speed of a brisk walk, so it also makes sense. Practice Problems 8. A pitcher throws a baseball from the pitcher’s mound to home plate in 0.46 s. The distance is 18.4 m. What was the average speed of the baseball? a. 40 m/s b. - 40 m/s c. 0.03 m/s d. 8.5 m/s 9. Cassie walked to her friend’s house with an average speed of 1.40 m/s. The distance between the houses is 205 m. How long did the trip take her? a. 146 s b. 0.01 s c. 2.50 min d. 287 s Velocity The vector version of speed is velocity. Velocity describes the speed and direction of an object. As with speed, it is useful to describe either the average velocity over a time period or the velocity at a specific moment. Average velocity is displacement divided by the time over which the displacement occurs. 64 Chapter 2 • Motion in One Dimension Velocity, like speed, has SI units of meters per second (m/s), but because it is a vector, you must also include a direction. Furthermore, the variable v for velocity is bold because it is a vector, which is in contrast to the variable vfor speed which is italicized because it is a scalar quantity. TIPS FOR SUCCESS It is important to keep in mind that the average speed is not the same thing as the average velocity without its direction. Like we saw with displacement and distance in the last section, changes in direction over a time interval have a bigger effect on speed and velocity. Suppose a passenger moved toward the back of a plane with an average velocity of –4 m/s. We cannot tell from the average velocity whether the passenger stopped momentarily or backed up before he got to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals such as those shown in Figure 2.9. If you consider infinitesimally small intervals, you can define instantaneous velocity, which is the velocity at a specific instant in time. Instantaneous velocity and average velocity are the same if the velocity is constant. Figure 2.9 The diagram shows a more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip. Earlier, you have read that distance traveled can be different than the magnitude of displacement. In the same way, speed can be different than the magnitude of velocity. For example, you drive to a store and return home in half an hour. If your car’s odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero because your displacement for the round trip is zero. WATCH PHYSICS Calculating Average Velocity or Speed This video (http://openstax.org/l/28avgvelocity) reviews vectors and scalars and describes how to calculate average velocity and average speed when you know displacement and change in time. The video also reviews how to convert km/h to m/s. Click to view content (https://www.khanacademy.org/embed_video?v=MAS6mBRZZXA) GRASP CHECK Which of the following fully describes a vector and a scalar quantity and correctly provides an example of each? Access for free at openstax.org. 2.2 • Speed and Velocity 65 a. A scalar quantity is fully described by its magnitude, while a vector needs both magnitude and direction to fully describe it. Displacement is an example of a scalar quantity and time is an example of a vector quantity. b. A scalar quantity is fully described by its magnitude, while a vector needs both magnitude and direction to fully describe it. Time is an example of a scalar quantity and displacement is an example of a vector quantity. c. A scalar quantity is fully described by its magnitude and direction, while a vector needs only magnitude to fully describe it. Displacement is an example of a scalar quantity and time is an example of a vector quantity. d. A scalar quantity is fully described by its magnitude and direction, while a vector needs only magnitude to fully describe it. Time is an example of a scalar quantity and displacement is an example of a vector quantity. WORKED EXAMPLE Calculating Average Velocity A student has a displacement of 304 m north in 180 s. What was the student's average velocity? Strategy We know that the displacement is 304 m north and the time is 180 s. We can use the formula for average velocity to solve the problem. Solution 2.1 Discussion Since average velocity is a vector quantity, you must include direction as well as magnitude in the answer. Notice, however, that the direction can be omitted until the end to avoid cluttering the problem. Pay attention to the significant figures in the problem. The distance 304 m has three significant figures, but the time interval 180 s has only two, so the quotient should have only two significant figures. TIPS FOR SUCCESS Note the way scalars and vectors are represented. In this book d represents distance and displacement. Similarly, v represents speed, and v represents velocity. A variable that is not bold indicates a scalar quantity, and a bold variable indicates a vector quantity. Vectors are sometimes represented by small arrows above the variable. WORKED EXAMPLE Solving for Displacement when Average Velocity and Time are Known Layla jogs with an average velocity of 2.4 m/s east. What is her displacement after 46 seconds? Strategy We know that Layla's average velocity is 2.4 m/s east, and the time interval is 46 seconds. We can rearrange the average velocity formula to solve for the displacement. Solution 2.2 Discussion The answer is about 110 m east, which is a reasonable displacement for slightly less than a minute of jogging. A calculator shows the answer as 110.4 m. We chose to write the answer using scientific notation because we wanted to make it clear that we only 66 Chapter 2 • Motion in One Dimension used two significant figures. TIPS FOR SUCCESS Dimensional analysis is a good way to determine whether you solved a problem correctly. Write the calculation using only units to be sure they match on opposite sides of the equal mark. In the worked example, you have m = (m/s)(s). Since seconds is in the denominator for the average velocity and in the numerator for the time, the unit cancels out leaving only m and, of course, m = m. WORKED EXAMPLE Solving for Time when Displacement and Average Velocity are Known Phillip walks along a straight path from his house to his school. How long will it take him to get to school if he walks 428 m west with an average velocity of 1.7 m/s west? Strategy We know that Phillip's displacement is 428 m west, and his average velocity is 1.7 m/s west. We can calculate the time required for the trip by rearranging the average velocity equation. Solution 2.3 Discussion Here again we had to use scientific notation because the answer could only have two significant figures. Since time is a scalar, the answer includes only a magnitude and not a direction. Practice Problems 10. A trucker drives along a straight highway for 0.25 h with a displacement of 16 km south. What is the trucker’s average velocity? a. 4 km/h north b. 4 km/h south c. 64 km/h north d. 64 km/h south 11. A bird flies with an average velocity of 7.5 m/s east from one branch to another in 2.4 s. It then pauses before flyi |
ng with an average velocity of 6.8 m/s east for 3.5 s to another branch. What is the bird’s total displacement from its starting point? a. 42 m west b. 6 m west c. 6 m east d. 42 m east Virtual Physics The Walking Man In this simulation you will put your cursor on the man and move him first in one direction and then in the opposite direction. Keep the Introductiontab active. You can use the Chartstab after you learn about graphing motion later in this chapter. Carefully watch the sign of the numbers in the position and velocity boxes. Ignore the acceleration box for now. See if you can make the man’s position positive while the velocity is negative. Then see if you can do the opposite. Access for free at openstax.org. 2.3 • Position vs. Time Graphs 67 Click to view content (https://archive.cnx.org/specials/e2ca52af-8c6b-450e-ac2f-9300b38e8739/moving-man/) GRASP CHECK Which situation correctly describes when the moving man’s position was negative but his velocity was positive? a. Man moving toward 0 from left of 0 b. Man moving toward 0 from right of 0 c. Man moving away from 0 from left of 0 d. Man moving away from 0 from right of 0 Check Your Understanding 12. Two runners travel along the same straight path. They start at the same time, and they end at the same time, but at the halfway mark, they have different instantaneous velocities. Is it possible for them to have the same average velocity for the trip? a. Yes, because average velocity depends on the net or total displacement. b. Yes, because average velocity depends on the total distance traveled. c. No, because the velocities of both runners must remain the exactly same throughout the journey. d. No, because the instantaneous velocities of the runners must remain same midway but can be different elsewhere. 13. If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity, and under what circumstances are these two quantities the same? a. Average speed. Both are the same when the car is traveling at a constant speed and changing direction. b. Average speed. Both are the same when the speed is constant and the car does not change its direction. c. Magnitude of average velocity. Both are same when the car is traveling at a constant speed. d. Magnitude of average velocity. Both are same when the car does not change its direction. 14. Is it possible for average velocity to be negative? a. Yes, in cases when the net displacement is negative. b. Yes, if the body keeps changing its direction during motion. c. No, average velocity describes only magnitude and not the direction of motion. d. No, average velocity describes only the magnitude in the positive direction of motion. 2.3 Position vs. Time Graphs Section Learning Objectives By the end of this section, you will be able to do the following: • Explain the meaning of slope in position vs. time graphs • Solve problems using position vs. time graphs Section Key Terms dependent variable independent variable tangent Graphing Position as a Function of Time A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information, they also reveal relationships between physical quantities. In this section, we will investigate kinematics by analyzing graphs of position over time. Graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against each other, the horizontal axis is usually considered the independent variable, and the vertical axis is the dependent variable. In algebra, you would have referred to the horizontal axis as the x-axis and the vertical axis as the y-axis. As in Figure 2.10, a straight-line graph has the general form . 68 Chapter 2 • Motion in One Dimension Here mis the slope, defined as the rise divided by the run (as seen in the figure) of the straight line. The letter bis the y-intercept which is the point at which the line crosses the vertical, y-axis. In terms of a physical situation in the real world, these quantities will take on a specific significance, as we will see below. (Figure 2.10.) Figure 2.10 The diagram shows a straight-line graph. The equation for the straight line is yequals mx+ b. In physics, time is usually the independent variable. Other quantities, such as displacement, are said to depend upon it. A graph of position versus time, therefore, would have position on the vertical axis (dependent variable) and time on the horizontal axis (independent variable). In this case, to what would the slope and y-intercept refer? Let’s look back at our original example when studying distance and displacement. The drive to school was 5 km from home. Let’s assume it took 10 minutes to make the drive and that your parent was driving at a constant velocity the whole time. The position versus time graph for this section of the trip would look like that shown in Figure 2.11. Figure 2.11 A graph of position versus time for the drive to school is shown. What would the graph look like if we added the return trip? As we said before, d0 = 0 because we call home our Oand start calculating from there. In Figure 2.11, the line starts at d = 0, as well. This is the bin our equation for a straight line. Our initial position in a position versus time graph is always the place where the graph crosses the x-axis at t= 0. What is the slope? The riseis the change in position, (i.e., displacement) and the runis the change in time. This relationship can also be written This relationship was how we defined average velocity. Therefore, the slope in a d versus tgraph, is the average velocity. TIPS FOR SUCCESS Sometimes, as is the case where we graph both the trip to school and the return trip, the behavior of the graph looks different during different time intervals. If the graph looks like a series of straight lines, then you can calculate the average velocity for each time interval by looking at the slope. If you then want to calculate the average velocity for the entire trip, you can do a 2.4 Access for free at openstax.org. weighted average. Let’s look at another example. Figure 2.12 shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada. 2.3 • Position vs. Time Graphs 69 Figure 2.12 The diagram shows a graph of position versus time for a jet-powered car on the Bonneville Salt Flats. Using the relationship between dependent and independent variables, we see that the slope in the graph in Figure 2.12 is average velocity, vavg and the intercept is displacement at time zero—that is, d0. Substituting these symbols into y= mx+ bgives or 2.5 2.6 Thus a graph of position versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation. From the figure we can see that the car has a position of 400 m at t= 0 s, 650 m at t= 1.0 s, and so on. And we can learn about the object’s velocity, as well. Snap Lab Graphing Motion In this activity, you will release a ball down a ramp and graph the ball’s displacement vs. time. • Choose an open location with lots of space to spread out so there is less chance for tripping or falling due to rolling balls. 1 ball • • 1 board • 2 or 3 books 1 stopwatch • • 1 tape measure • 6 pieces of masking tape 1 piece of graph paper • 1 pencil • Procedure 1. Build a ramp by placing one end of the board on top of the stack of books. Adjust location, as necessary, until there is no obstacle along the straight line path from the bottom of the ramp until at least the next 3 m. 2. Mark distances of 0.5 m, 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m from the bottom of the ramp. Write the distances on the tape. 70 Chapter 2 • Motion in One Dimension 3. Have one person take the role of the experimenter. This person will release the ball from the top of the ramp. If the ball does not reach the 3.0 m mark, then increase the incline of the ramp by adding another book. Repeat this Step as necessary. 4. Have the experimenter release the ball. Have a second person, the timer, begin timing the trial once the ball reaches the bottom of the ramp and stop the timing once the ball reaches 0.5 m. Have a third person, the recorder, record the time in a data table. 5. Repeat Step 4, stopping the times at the distances of 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m from the bottom of the ramp. 6. Use your measurements of time and the displacement to make a position vs. time graph of the ball’s motion. 7. Repeat Steps 4 through 6, with different people taking on the roles of experimenter, timer, and recorder. Do you get the same measurement values regardless of who releases the ball, measures the time, or records the result? Discuss possible causes of discrepancies, if any. GRASP CHECK True or False: The average speed of the ball will be less than the average velocity of the ball. a. True b. False Solving Problems Using Position vs. Time Graphs So how do we use graphs to solve for things we want to know like velocity? WORKED EXAMPLE Using Position–Time Graph to Calculate Average Velocity: Jet Car Find the average velocity of the car whose position is graphed in Figure 1.13. Strategy The slope of a graph of dvs. tis average velocity, since slope equals rise over run. 2.7 Since the slope is constant here, any two points on the graph can be used to find the slope. Solution 1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points.) 2. Substitute the d and tvalues of the chosen points into the equation. Remember in calculating change (Δ) we always use final value minus initial value. 2.8 Discussion This is an impressively high land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 27 m/s or 96 km/h, but considerably shy |
of the record of 343 m/s or 1,234 km/h, set in 1997. But what if the graph of the position is more complicated than a straight line? What if the object speeds up or turns around and goes backward? Can we figure out anything about its velocity from a graph of that kind of motion? Let’s take another look at the jet-powered car. The graph in Figure 2.13 shows its motion as it is getting up to speed after starting at rest. Time starts at zero for this motion (as if measured with a stopwatch), and the displacement and velocity are initially 200 m and 15 m/s, respectively. Access for free at openstax.org. 2.3 • Position vs. Time Graphs 71 Figure 2.13 The diagram shows a graph of the position of a jet-powered car during the time span when it is speeding up. The slope of a distance versus time graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point. Figure 2.14 A U.S. Air Force jet car speeds down a track. (Matt Trostle, Flickr) The graph of position versus time in Figure 2.13 is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a position-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in Figure 2.13. The average velocity is the net displacement divided by the time traveled. WORKED EXAMPLE Using Position–Time Graph to Calculate Average Velocity: Jet Car, Take Two Calculate the instantaneous velocity of the jet car at a time of 25 s by finding the slope of the tangent line at point Q in Figure 2.13. Strategy The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. Solution 1. Find the tangent line to the curve at 2. Determine the endpoints of the tangent. These correspond to a position of 1,300 m at time 19 s and a position of 3120 m at . time 32 s. 72 Chapter 2 • Motion in One Dimension 3. Plug these endpoints into the equation to solve for the slope, v. 2.9 Discussion The entire graph of v versus tcan be obtained in this fashion. Practice Problems 15. Calculate the average velocity of the object shown in the graph below over the whole time interval. a. 0.25 m/s b. 0.31 m/s c. 3.2 m/s d. 4.00 m/s 16. True or False: By taking the slope of the curve in the graph you can verify that the velocity of the jet car is at . a. True b. False Check Your Understanding 17. Which of the following information about motion can be determined by looking at a position vs. time graph that is a straight line? Access for free at openstax.org. 2.4 • Velocity vs. Time Graphs 73 a. frame of reference b. average acceleration c. velocity d. direction of force applied 18. True or False: The position vs time graph of an object that is speeding up is a straight line. a. True b. False 2.4 Velocity vs. Time Graphs Section Learning Objectives By the end of this section, you will be able to do the following: • Explain the meaning of slope and area in velocity vs. time graphs • Solve problems using velocity vs. time graphs Section Key Terms acceleration Graphing Velocity as a Function of Time Earlier, we examined graphs of position versus time. Now, we are going to build on that information as we look at graphs of velocity vs. time. Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity; we will discuss acceleration more in another chapter. These concepts are all very interrelated. Virtual Physics Maze Game In this simulation you will use a vector diagram to manipulate a ball into a certain location without hitting a wall. You can manipulate the ball directly with position or by changing its velocity. Explore how these factors change the motion. If you would like, you can put it on the asetting, as well. This is acceleration, which measures the rate of change of velocity. We will explore acceleration in more detail later, but it might be interesting to take a look at it here. Click to view content (https://archive.cnx.org/specials/30e37034-2fbd-11e5-83a2-03be60006ece/maze-game/) GRASP CHECK Click to view content (https://archive.cnx.org/specials/30e37034-2fbd-11e5-83a2-03be60006ece/maze-game/#sim-mazegame) a. The ball can be easily manipulated with displacement because the arena is a position space. b. The ball can be easily manipulated with velocity because the arena is a position space. c. The ball can be easily manipulated with displacement because the arena is a velocity space. d. The ball can be easily manipulated with velocity because the arena is a velocity space. What can we learn about motion by looking at velocity vs. time graphs? Let’s return to our drive to school, and look at a graph of position versus time as shown in Figure 2.15. 74 Chapter 2 • Motion in One Dimension Figure 2.15 A graph of position versus time for the drive to and from school is shown. We assumed for our original calculation that your parent drove with a constant velocity to and from school. We now know that the car could not have gone from rest to a constant velocity without speeding up. So the actual graph would be curved on either end, but let’s make the same approximation as we did then, anyway. TIPS FOR SUCCESS It is common in physics, especially at the early learning stages, for certain things to be neglected, as we see here. This is because it makes the concept clearer or the calculation easier. Practicing physicists use these kinds of short-cuts, as well. It works out because usually the thing being neglectedis small enough that it does not significantly affect the answer. In the earlier example, the amount of time it takes the car to speed up and reach its cruising velocity is very small compared to the total time traveled. Looking at this graph, and given what we learned, we can see that there are two distinct periods to the car’s motion—the way to school and the way back. The average velocity for the drive to school is 0.5 km/minute. We can see that the average velocity for the drive back is –0.5 km/minute. If we plot the data showing velocity versus time, we get another graph (Figure 2.16): Figure 2.16 Graph of velocity versus time for the drive to and from school. We can learn a few things. First, we can derive a v versus tgraph from a d versus tgraph. Second, if we have a straight-line position–time graph that is positively or negatively sloped, it will yield a horizontal velocity graph. There are a few other interesting things to note. Just as we could use a position vs. time graph to determine velocity, we can use a velocity vs. time graph to determine position. We know that v = d/t. If we use a little algebra to re-arrange the equation, we see that d = v t. In Figure 2.16, we have velocity on the y-axis and time along the x-axis. Let’s take just the first half of the motion. We get 0.5 km/ minute If we calculate the same for the return trip, we get –5 km. If we add them together, we see that the net displacement for the 10 minutes. The units for minutescancel each other, and we get 5 km, which is the displacement for the trip to school. Access for free at openstax.org. 2.4 • Velocity vs. Time Graphs 75 whole trip is 0 km, which it should be because we started and ended at the same place. TIPS FOR SUCCESS You can treat units just like you treat numbers, so a km/km=1 (or, we say, it cancels out). This is good because it can tell us whether or not we have calculated everything with the correct units. For instance, if we end up with m × s for velocity instead of m/s, we know that something has gone wrong, and we need to check our math. This process is called dimensional analysis, and it is one of the best ways to check if your math makes sense in physics. The area under a velocity curve represents the displacement. The velocity curve also tells us whether the car is speeding up. In our earlier example, we stated that the velocity was constant. So, the car is not speeding up. Graphically, you can see that the slope of these two lines is 0. This slope tells us that the car is not speeding up, or accelerating. We will do more with this information in a later chapter. For now, just remember that the area under the graph and the slope are the two important parts of the graph. Just like we could define a linear equation for the motion in a position vs. time graph, we can also define one for a velocity vs. time graph. As we said, the slope equals the acceleration, a. And in this graph, the y-intercept is v0. Thus, . But what if the velocity is not constant? Let’s look back at our jet-car example. At the beginning of the motion, as the car is speeding up, we saw that its position is a curve, as shown in Figure 2.17. Figure 2.17 A graph is shown of the position of a jet-powered car during the time span when it is speeding up. The slope of a d vs. t graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point. You do not have to do this, but you could, theoretically, take the instantaneous velocity at each point on this graph. If you did, you would get Figure 2.18, which is just a straight line with a positive slope. 76 Chapter 2 • Motion in One Dimension Figure 2.18 The graph shows the velocity of a jet-powered car during the time span when it is speeding up. Again, if we take the slope of the velocity vs. time graph, we get the acceleration, the rate of change of the velocity. And, if we take the area under the slope, we get back to the displacement. Solving Problems using Velocity–Time Graphs Most velocity vs. time graphs will be straight lines. When this is the case, our calculations are fairly simple. WORKED EXAMPLE Using Velocity Graph to Calculate Some Stuff: Jet Car Use this figure to (a) find the displacement of the jet car over the |
time shown (b) calculate the rate of change (acceleration) of the velocity. (c) give the instantaneous velocity at 5 s, and (d) calculate the average velocity over the interval shown. Strategy a. The displacement is given by finding the area under the line in the velocity vs. time graph. b. The acceleration is given by finding the slope of the velocity graph. c. The instantaneous velocity can just be read off of the graph. d. To find the average velocity, recall that Solution a. 1. Analyze the shape of the area to be calculated. In this case, the area is made up of a rectangle between 0 and 20 m/s stretching to 30 s. The area of a rectangle is length width. Therefore, the area of this piece is 600 m. 2. Above that is a triangle whose base is 30 s and height is 140 m/s. The area of a triangle is 0.5 length width. The area of this piece, therefore, is 2,100 m. 3. Add them together to get a net displacement of 2,700 m. b. 1. Take two points on the velocity line. Say, t= 5 s and t= 25 s. At t= 5 s, the value of v = 40 m/s. At t= 25 s, v = 140 m/s. 2. Find the slope. c. The instantaneous velocity at t= 5 s, as we found in part (b) is just 40 m/s. 1. Find the net displacement, which we found in part (a) was 2,700 m. d. 2. Find the total time which for this case is 30 s. 3. Divide 2,700 m/30 s = 90 m/s. Access for free at openstax.org. 2.4 • Velocity vs. Time Graphs 77 Discussion The average velocity we calculated here makes sense if we look at the graph. 100m/s falls about halfway across the graph and since it is a straight line, we would expect about half the velocity to be above and half below. TIPS FOR SUCCESS You can have negative position, velocity, and acceleration on a graph that describes the way the object is moving. You should never see a graph with negative time on an axis. Why? Most of the velocity vs. time graphs we will look at will be simple to interpret. Occasionally, we will look at curved graphs of velocity vs. time. More often, these curved graphs occur when something is speeding up, often from rest. Let’s look back at a more realistic velocity vs. time graph of the jet car’s motion that takes this speeding upstage into account. Figure 2.19 The graph shows a more accurate graph of the velocity of a jet-powered car during the time span when it is speeding up. WORKED EXAMPLE Using Curvy Velocity Graph to Calculate Some Stuff: jet car, Take Two Use Figure 2.19 to (a) find the approximate displacement of the jet car over the time shown, (b) calculate the instantaneous acceleration at t= 30 s, (c) find the instantaneous velocity at 30 s, and (d) calculate the approximate average velocity over the interval shown. Strategy a. Because this graph is an undefined curve, we have to estimate shapes over smaller intervals in order to find the areas. b. Like when we were working with a curved displacement graph, we will need to take a tangent line at the instant we are interested and use that to calculate the instantaneous acceleration. c. The instantaneous velocity can still be read off of the graph. d. We will find the average velocity the same way we did in the previous example. Solution a. 1. This problem is more complicated than the last example. To get a good estimate, we should probably break the curve into four sections. 0 → 10 s, 10 → 20 s, 20 → 40 s, and 40 → 70 s. 2. Calculate the bottom rectangle (common to all pieces). 165 m/s 3. Estimate a triangle at the top, and calculate the area for each section. Section 1 = 225 m; section 2 = 100 m + 450 m = 70 s = 11,550 m. 550 m; section 3 = 150 m + 1,300 m = 1,450 m; section 4 = 2,550 m. 4. Add them together to get a net displacement of 16,325 m. b. Using the tangent line given, we find that the slope is 1 m/s2. 78 Chapter 2 • Motion in One Dimension c. The instantaneous velocity at t= 30 s, is 240 m/s. d. 1. Find the net displacement, which we found in part (a), was 16,325 m. 2. Find the total time, which for this case is 70 s. 3. Divide Discussion This is a much more complicated process than the first problem. If we were to use these estimates to come up with the average velocity over just the first 30 s we would get about 191 m/s. By approximating that curve with a line, we get an average velocity of 202.5 m/s. Depending on our purposes and how precise an answer we need, sometimes calling a curve a straight line is a worthwhile approximation. Practice Problems 19. Figure 2.20 Consider the velocity vs. time graph shown below of a person in an elevator. Suppose the elevator is initially at rest. It then speeds up for 3 seconds, maintains that velocity for 15 seconds, then slows down for 5 seconds until it stops. Find the instantaneous velocity at t= 10 s and t= 23 s. a. b. c. d. Instantaneous velocity at t= 10 s and t= 23 s are 0 m/s and 0 m/s. Instantaneous velocity at t= 10 s and t= 23 s are 0 m/s and 3 m/s. Instantaneous velocity at t= 10 s and t= 23 s are 3 m/s and 0 m/s. Instantaneous velocity at t= 10 s and t= 23 s are 3 m/s and 1.5 m/s. Access for free at openstax.org. 20. 2.4 • Velocity vs. Time Graphs 79 Figure 2.21 Calculate the net displacement and the average velocity of the elevator over the time interval shown. a. Net displacement is 45 m and average velocity is 2.10 m/s. b. Net displacement is 45 m and average velocity is 2.28 m/s. c. Net displacement is 57 m and average velocity is 2.66 m/s. d. Net displacement is 57 m and average velocity is 2.48 m/s. Snap Lab Graphing Motion, Take Two In this activity, you will graph a moving ball’s velocity vs. time. • your graph from the earlier Graphing Motion Snap Lab! • • 1 piece of graph paper 1 pencil Procedure 1. Take your graph from the earlier Graphing Motion Snap Lab! and use it to create a graph of velocity vs. time. 2. Use your graph to calculate the displacement. GRASP CHECK Describe the graph and explain what it means in terms of velocity and acceleration. a. The graph shows a horizontal line indicating that the ball moved with a constant velocity, that is, it was not accelerating. b. The graph shows a horizontal line indicating that the ball moved with a constant velocity, that is, it was accelerating. c. The graph shows a horizontal line indicating that the ball moved with a variable velocity, that is, it was not accelerating. d. The graph shows a horizontal line indicating that the ball moved with a variable velocity, that is, it was accelerating. Check Your Understanding 21. What information could you obtain by looking at a velocity vs. time graph? a. acceleration b. direction of motion c. reference frame of the motion 80 Chapter 2 • Motion in One Dimension d. shortest path 22. How would you use a position vs. time graph to construct a velocity vs. time graph and vice versa? a. Slope of position vs. time curve is used to construct velocity vs. time curve, and slope of velocity vs. time curve is used to construct position vs. time curve. b. Slope of position vs. time curve is used to construct velocity vs. time curve, and area of velocity vs. time curve is used to construct position vs. time curve. c. Area of position vs. time curve is used to construct velocity vs. time curve, and slope of velocity vs. time curve is used to construct position vs. time curve. d. Area of position/time curve is used to construct velocity vs. time curve, and area of velocity vs. time curve is used to construct position vs. time curve. Access for free at openstax.org. KEY TERMS acceleration the rate at which velocity changes average speed distance traveled divided by time during which motion occurs average velocity displacement divided by time over which displacement occurs dependent variable the variable that changes as the independent variable changes displacement fixed axis the change in position of an object against a distance the length of the path actually traveled between an initial and a final position independent variable the variable, usually along the horizontal axis of a graph, that does not change based on human or experimental action; in physics this is usually SECTION SUMMARY 2.1 Relative Motion, Distance, and Displacement • A description of motion depends on the reference frame from which it is described. • The distance an object moves is the length of the path along which it moves. • Displacement is the difference in the initial and final positions of an object. 2.2 Speed and Velocity • Average speed is a scalar quantity that describes distance traveled divided by the time during which the motion occurs. • Velocity is a vector quantity that describes the speed and direction of an object. • Average velocity is displacement over the time period during which the displacement occurs. If the velocity is constant, then average velocity and instantaneous KEY EQUATIONS 2.1 Relative Motion, Distance, and Displacement Displacement 2.2 Speed and Velocity Average speed Average velocity Chapter 2 • Key Terms 81 time instantaneous speed speed at a specific instant in time instantaneous velocity velocity at a specific instant in time kinematics causes the study of motion without considering its magnitude size or amount position the location of an object at any particular time reference frame a coordinate system from which the positions of objects are described scalar a quantity that has magnitude but no direction speed rate at which an object changes its location tangent a line that touches another at exactly one point vector a quantity that has both magnitude and direction velocity the speed and direction of an object velocity are the same. 2.3 Position vs. Time Graphs • Graphs can be used to analyze motion. • The slope of a position vs. time graph is the velocity. • For a straight line graph of position, the slope is the average velocity. • To obtain the instantaneous velocity at a given moment for a curved graph, find the tangent line at that point and take its slope. 2.4 Velocity vs. Time Graphs • The slope of a velocity vs. time graph is the acceleration. • The area under a velocity vs. time curve |
is the displacement. • Average velocity can be found in a velocity vs. time graph by taking the weighted average of all the velocities. 2.3 Position vs. Time Graphs Displacement . 2.4 Velocity vs. Time Graphs Velocity Acceleration 82 Chapter 2 • Chapter Review CHAPTER REVIEW Concept Items 2.1 Relative Motion, Distance, and Displacement 1. Can one-dimensional motion have zero distance but a nonzero displacement? What about zero displacement but a nonzero distance? a. One-dimensional motion can have zero distance with a nonzero displacement. Displacement has both magnitude and direction, and it can also have zero displacement with nonzero distance because distance has only magnitude. b. One-dimensional motion can have zero distance with a nonzero displacement. Displacement has both magnitude and direction, but it cannot have zero displacement with nonzero distance because distance has only magnitude. c. One-dimensional motion cannot have zero distance with a nonzero displacement. Displacement has both magnitude and direction, but it can have zero displacement with nonzero distance because distance has only magnitude and any motion will be the distance it moves. d. One-dimensional motion cannot have zero distance with a nonzero displacement. Displacement has both magnitude and direction, and it cannot have zero displacement with nonzero distance because distance has only magnitude. 2. In which example would you be correct in describing an object in motion while your friend would also be correct in describing that same object as being at rest? a. You are driving a car toward the east and your friend drives past you in the opposite direction with the same speed. In your frame of reference, you will be in motion. In your friend’s frame of reference, you will be at rest. b. You are driving a car toward the east and your friend is standing at the bus stop. In your frame of reference, you will be in motion. In your friend’s frame of reference, you will be at rest. c. You are driving a car toward the east and your friend is standing at the bus stop. In your frame of reference, your friend will be moving toward the west. In your friend’s frame of reference, he will be at rest. d. You are driving a car toward the east and your friend is standing at the bus stop. In your frame of reference, your friend will be moving toward the east. In your friend’s frame of reference, he will be at rest. Access for free at openstax.org. 3. What does your car’s odometer record? a. displacement b. distance c. both distance and displacement d. the sum of distance and displacement 2.2 Speed and Velocity 4. In the definition of velocity, what physical quantity is changing over time? a. speed b. distance c. magnitude of displacement d. position vector 5. Which of the following best describes the relationship between instantaneous velocity and instantaneous speed? a. Both instantaneous speed and instantaneous b. velocity are the same, even when there is a change in direction. Instantaneous speed and instantaneous velocity cannot be the same even if there is no change in direction of motion. c. Magnitude of instantaneous velocity is equal to instantaneous speed. d. Magnitude of instantaneous velocity is always greater than instantaneous speed. 2.3 Position vs. Time Graphs 6. Use the graph to describe what the runner’s motion looks like. How are average velocity for only the first four seconds and instantaneous velocity related? What is the runner's net displacement over the time shown? a. The net displacement is 12 m and the average velocity is equal to the instantaneous velocity. b. The net displacement is 12 m and the average velocity is two times the instantaneous velocity. c. The net displacement is 10 + 12 = 22 m and the average velocity is equal to the instantaneous velocity. d. The net displacement is 10 + 12 = 22 m and the average velocity is two times the instantaneous velocity. 7. A position vs. time graph of a frog swimming across a . If each section lasts pond has two distinct straight-line sections. The slope of . The slope of the second section the first section is is , then what is the frog’s total average velocity? a. b. c. d. 2.4 Velocity vs. Time Graphs 8. A graph of velocity vs. time of a ship coming into a harbor is shown. Chapter 2 • Chapter Review 83 Describe the acceleration of the ship based on the graph. a. The ship is moving in the forward direction at a steady rate. Then it accelerates in the forward direction and then decelerates. b. The ship is moving in the forward direction at a steady rate. Then it turns around and starts decelerating, while traveling in the reverse direction. It then accelerates, but slowly. c. The ship is moving in the forward direction at a steady rate. Then it decelerates in the forward direction, and then continues to slow down in the forward direction, but with more deceleration. d. The ship is moving in the forward direction at a steady rate. Then it decelerates in the forward direction, and then continues to slow down in the forward direction, but with less deceleration. Critical Thinking Items 2.1 Relative Motion, Distance, and Displacement 9. Boat A and Boat B are traveling at a constant speed in opposite directions when they pass each other. If a person in each boat describes motion based on the boat’s own reference frame, will the description by a person in Boat A of Boat B’s motion be the same as the description by a person in Boat B of Boat A’s motion? a. Yes, both persons will describe the same motion because motion is independent of the frame of reference. b. Yes, both persons will describe the same motion because they will perceive the other as moving in the backward direction. c. No, the motion described by each of them will be different because motion is a relative term. d. No, the motion described by each of them will be different because the motion perceived by each will be opposite to each other. 10. Passenger A sits inside a moving train and throws a ball vertically upward. How would the motion of the ball be described by a fellow train passenger B and an observer C who is standing on the platform outside the train? a. Passenger B sees that the ball has vertical, but no horizontal, motion. Observer C sees the ball has vertical as well as horizontal motion. b. Passenger B sees the ball has vertical as well as horizontal motion. Observer C sees the ball has the vertical, but no horizontal, motion. c. Passenger B sees the ball has horizontal but no vertical motion. Observer C sees the ball has vertical as well as horizontal motion. d. Passenger B sees the ball has vertical as well as horizontal motion. Observer C sees the ball has horizontalbut no vertical motion. 2.2 Speed and Velocity 11. Is it possible to determine a car’s instantaneous velocity from just the speedometer reading? a. No, it reflects speed but not the direction. b. No, it reflects the average speed of the car. c. Yes, it sometimes reflects instantaneous velocity of the car. d. Yes, it always reflects the instantaneous velocity of the car. 12. Terri, Aaron, and Jamal all walked along straight paths. 84 Chapter 2 • Chapter Review Terri walked 3.95 km north in 48 min. Aaron walked 2.65 km west in 31 min. Jamal walked 6.50 km south in 81 min. Which of the following correctly ranks the three boys in order from lowest to highest average speed? Jamal, Terri, Aaron a. Jamal, Aaron, Terri b. c. Terri, Jamal, Aaron d. Aaron, Terri, Jamal 13. Rhianna and Logan start at the same point and walk due Identify the time (ta, tb, tc, td, or te) at which at which the instantaneous velocity is greatest, the time at which it is zero, and the time at which it is negative. 2.4 Velocity vs. Time Graphs 15. Identify the time, or times, at which the instantaneous velocity is greatest, and the time, or times, at which it is negative. A sketch of velocity vs. time derived from the figure will aid in arriving at the correct answers. north. Rhianna walks with an average velocity Logan walks three times the distance in twice the time as Rhianna. Which of the following expresses Logan’s average velocity in terms of a. Logan’s average velocity = ? . . b. Logan’s average velocity = c. Logan’s average velocity = d. Logan’s average velocity = . . . 2.3 Position vs. Time Graphs 14. Explain how you can use the graph of position vs. time to describe the change in velocity over time. a. The instantaneous velocity is greatest at f, and it is negative at d, h, I, j, and k. b. The instantaneous velocity is greatest at e, and it is negative at a, b, and f. c. The instantaneous velocity is greatest at f, and it is negative at d, h, I, j, and k d. The instantaneous velocity is greatest at d, and it is negative at a, b, and f. Problems 2.1 Relative Motion, Distance, and Displacement Up is the positive direction. What are the total displacement of the ball and the total distance traveled by the ball? a. The displacement is equal to -4 m and the distance 16. In a coordinate system in which the direction to the is equal to 4 m. right is positive, what are the distance and displacement of a person who walks to the right, and then a. Distance is b. Distance is c. Distance is d. Distance is to the left, to the left? and displacement is and displacement is and displacement is and displacement is . . . . 17. Billy drops a ball from a height of 1 m. The ball bounces back to a height of 0.8 m, then bounces again to a height of 0.5 m, and bounces once more to a height of 0.2 m. Access for free at openstax.org. b. The displacement is equal to 4 m and the distance is equal to 1 m. c. The displacement is equal to 4 m and the distance is equal to 1 m. d. The displacement is equal to -1 m and the distance is equal to 4 m. 2.2 Speed and Velocity 18. You sit in a car that is moving at an average speed of 86.4 km/h. During the 3.3 s that you glance out the window, how far has the car traveled? 7.27 m a. 79 m b. c. 285 km 1026 m d. 2.3 Position vs. Time Graphs 19. Using the |
graph, what is the average velocity for the whole Chapter 2 • Chapter Review 85 moments in time. What is the minimum number of data points you would need to estimate the average acceleration of the object? a. 1 b. 2 3 c. d. 4 10 seconds? 22. Which option best describes the average acceleration from 40 to 70 s? a. The total average velocity is 0 m/s. b. The total average velocity is 1.2 m/s. c. The total average velocity is 1.5 m/s. d. The total average velocity is 3.0 m/s. 20. A train starts from rest and speeds up for 15 minutes until it reaches a constant velocity of 100 miles/hour. It stays at this speed for half an hour. Then it slows down for another 15 minutes until it is still. Which of the following correctly describes the position vs time graph of the train’s journey? a. The first 15 minutes is a curve that is concave upward, the middle portion is a straight line with slope 100 miles/hour, and the last portion is a concave downward curve. b. The first 15 minutes is a curve that is concave downward, the middle portion is a straight line with slope 100 miles/hour, and the last portion is a concave upward curve. c. The first 15 minutes is a curve that is concave upward, the middle portion is a straight line with slope zero, and the last portion is a concave downward curve. d. The first 15 minutes is a curve that is concave downward, the middle portion is a straight line with slope zero, and the last portion is a concave upward curve. 2.4 Velocity vs. Time Graphs 21. You are characterizing the motion of an object by measuring the location of the object at discrete a. b. c. d. It is negative and smaller in magnitude than the initial acceleration. It is negative and larger in magnitude than the initial acceleration. It is positive and smaller in magnitude than the initial acceleration. It is positive and larger in magnitude than the initial acceleration. 23. The graph shows velocity vs. time. Calculate the net displacement using seven different divisions. Calculate it again using two divisions: 0 → 40 s 86 Chapter 2 • Test Prep and 40 → 70 s . Compare. Using both, calculate the average velocity. a. Displacement and average velocity using seven divisions are 14,312.5 m and 204.5 m/s while with two divisions are 15,500 m and 221.4 m/s respectively. b. Displacement and average velocity using seven divisions are 15,500 m and 221.4 m/s while with two divisions are 14,312.5 m and 204.5 m/s respectively. c. Displacement and average velocity using seven divisions are 15,500 m and 204.5 m/s while with two divisions are 14,312.5 m and 221.4 m/s respectively. d. Displacement and average velocity using seven divisions are 14,312.5 m and 221.4 m/s while with two divisions are 15,500 m and 204.5 m/s respectively. Performance Task 2.4 Velocity vs. Time Graphs 24. The National Mall in Washington, DC, is a national park containing most of the highly treasured memorials and museums of the United States. However, the National Mall also hosts many events and concerts. The map in shows the area for a benefit concert during which the president will speak. The concert stage is near the Lincoln Memorial. The seats and standing room for the crowd will stretch from the stage east to near the Washington Monument, as shown on the map. You are planning the logistics for the concert. Use the map scale to measure any distances needed to make the calculations below. The park has three new long-distance speakers. They would like to use these speakers to broadcast the concert audio to other parts of the National Mall. The speakers can project sound up to 0.35 miles away but they must be connected to one of the power supplies within the concert area. What is the minimum amount of wire needed for each speaker, in miles, in order to project the audio to the following areas? Assume that wire cannot be placed over buildings or any memorials. Part A. The center of the Jefferson Memorial using power supply 1 (This will involve an elevated wire that can travel over water.) Part B. The center of the Ellipse using power supply 3 (This wire cannot travel over water.) Part C. The president’s motorcade will be traveling to the concert from the White House. To avoid concert traffic, the motorcade travels from the White House west down E Street and then turns south on 23rd Street to reach the Lincoln memorial. What minimum speed, in miles per hour to the nearest tenth, would the motorcade have to travel to make the trip in 5 minutes? Part D. The president could also simply fly from the White House to the Lincoln Memorial using the presidential helicopter, Marine 1. How long would it take Marine 1, traveling slowly at 30 mph, to travel from directly above the White House landing zone (LZ) to directly above the Lincoln Memorial LZ? Disregard liftoff and landing times and report the travel time in minutes to the nearest minute. TEST PREP Multiple Choice 2.1 Relative Motion, Distance, and Displacement 25. Why should you specify a reference frame when describing motion? a. a description of motion depends on the reference frame Access for free at openstax.org. b. motion appears the same in all reference frames c. reference frames affect the motion of an object d. you can see motion better from certain reference frames 26. Which of the following is true for the displacement of an object? a. It is always equal to the distance the object moved b. c. d. between its initial and final positions. It is both the straight line distance the object moved as well as the direction of its motion. It is the direction the object moved between its initial and final positions. It is the straight line distance the object moved between its initial and final positions. 27. If a biker rides west for 50 miles from his starting position, then turns and bikes back east 80 miles. What is his net displacement? a. b. c. d. Cannot be determined from the information given 130 miles 30 miles east 30 miles west 28. Suppose a train is moving along a track. Is there a single, correct reference frame from which to describe the train’s motion? a. Yes, there is a single, correct frame of reference Chapter 2 • Test Prep 87 d. 1,500 m south 32. A bicyclist covers the first leg of a journey that is long in , at a speed of , , at a in . If his average speed is equal to the , then which of the following is and and the second leg of speed of average of true? a. b. c. d. 33. A car is moving on a straight road at a constant speed in a single direction. Which of the following statements is true? a. Average velocity is zero. b. The magnitude of average velocity is equal to the average speed. c. The magnitude of average velocity is greater than because motion is a relative term. the average speed. b. Yes, there is a single, correct frame of reference d. The magnitude of average velocity is less than the which is in terms of Earth’s position. average speed. c. No, there is not a single, correct frame of reference because motion is a relative term. 2.3 Position vs. Time Graphs d. No, there is not a single, correct frame of reference because motion is independent of frame of reference. 29. If a space shuttle orbits Earth once, what is the shuttle’s distance traveled and displacement? a. Distance and displacement both are zero. b. Distance is circumference of the circular orbit while displacement is zero. c. Distance is zero while the displacement is circumference of the circular orbit. d. Distance and displacement both are equal to circumference of the circular orbit. 2.2 Speed and Velocity 34. What is the slope of a straight line graph of position vs. time? a. Velocity b. Displacement c. Distance d. Acceleration 35. Using the graph, what is the runner’s velocity from 4 to 10 s? 30. Four bicyclists travel different distances and times along a straight path. Which cyclist traveled with the greatest average speed? a. Cyclist 1 travels b. Cyclist 2 travels c. Cyclist 3 travels d. Cyclist 4 travels in in in in . . . . 31. A car travels with an average velocity of 23 m/s for 82 s. Which of the following could NOT have been the car's displacement? a. b. c. 1,700 m east 1,900 m west 1,600 m north a. –3 m/s b. 0 m/s c. d. 1.2 m/s 3 m/s 88 Chapter 2 • Test Prep 2.4 Velocity vs. Time Graphs 36. What does the area under a velocity vs. time graph line represent? a. acceleration b. displacement c. distance d. instantaneous velocity 37. An object is moving along a straight path with constant Short Answer 2.1 Relative Motion, Distance, and Displacement 38. While standing on a sidewalk facing the road, you see a bicyclist passing by toward your right. In the reference frame of the bicyclist, in which direction are you moving? a. b. c. d. in the same direction of motion as the bicyclist in the direction opposite the motion of the bicyclist stationary with respect to the bicyclist in the direction of velocity of the bicyclist 39. Maud sends her bowling ball straight down the center of the lane, getting a strike. The ball is brought back to the holder mechanically. What are the ball’s net displacement and distance traveled? a. Displacement of the ball is twice the length of the lane, while the distance is zero. b. Displacement of the ball is zero, while the distance is twice the length of the lane. c. Both the displacement and distance for the ball are equal to zero. d. Both the displacement and distance for the ball are twice the length of the lane. 40. A fly buzzes four and a half times around Kit Yan’s head. The fly ends up on the opposite side from where it started. If the diameter of his head is the total distance the fly travels and its total displacement? a. The distance is with a displacement of , what is zero. b. The distance is c. The distance is with a displacement of zero. with a displacement of . d. The distance is with a displacement of . 2.2 Speed and Velocity 41. Rob drove to the nearest hospital with an average speed of v m/s in t seconds. In terms of t, if he drives home on the same path, but with an average sp |
eed of 3v m/s, how Access for free at openstax.org. and . acceleration. A velocity vs. time graph starts at ends at , stretching over a time-span of What is the object’s net displacement? a. b. c. d. cannot be determined from the information given long is the return trip home? t/6 a. t/3 b. 3t c. d. 6t 42. What can you infer from the statement, Velocity of an object is zero? a. Object is in linear motion with constant velocity. b. Object is moving at a constant speed. c. Object is either at rest or it returns to the initial point. d. Object is moving in a straight line without changing its direction. 43. An object has an average speed of 7.4 km/h. Which of the following describes two ways you could increase the average speed of the object to 14.8 km/h? a. Reduce the distance that the object travels by half, keeping the time constant, or keep the distance constant and double the time. b. Double the distance that the object travels, keeping the time constant, or keep the distance constant and reduce the time by half. c. Reduce the distance that the object travels to onefourth, keeping the time constant, or keep the distance constant and increase the time by fourfold. Increase the distance by fourfold, keeping the time constant, or keep the distance constant and reduce the time by one-fourth. d. 44. Swimming one lap in a pool is defined as going across a pool and back again. If a swimmer swims 3 laps in 9 minutes, how can his average velocity be zero? a. His average velocity is zero because his total distance is zero. b. His average velocity is zero because his total displacement is zero. c. His average velocity is zero because the number of laps completed is an odd number. d. His average velocity is zero because the velocity of each successive lap is equal and opposite. 2.3 Position vs. Time Graphs 45. A hockey puck is shot down the arena in a straight line. Assume it does not slow until it is stopped by an opposing player who sends it back in the direction it came. The players are 20 m apart and it takes 1 s for the puck to go there and back. Which of the following describes the graph of the displacement over time? Consider the initial direction of the puck to be positive. a. The graph is an upward opening V. b. The graph is a downward opening V. c. The graph is an upward opening U. d. The graph is downward opening U. 46. A defensive player kicks a soccer ball 20 m back to her own goalie. It stops just as it reaches her. She sends it back to the player. Without knowing the time it takes, draw a rough sketch of the displacement over time. Does this graph look similar to the graph of the hockey puck from the previous question? a. Yes, the graph is similar to the graph of the hockey puck. b. No, the graph is not similar to the graph of the hockey puck. c. The graphs cannot be compared without knowing the time the soccer ball was rolling. 47. What are the net displacement, total distance traveled, and total average velocity in the previous two problems? a. net displacement = 0 m, total distance = 20 m, total average velocity = 20 m/s b. net displacement = 0 m, total distance = 40 m, total average velocity = 20 m/s c. net displacement = 0 m, total distance = 20 m, total average velocity = 0 m/s d. net displacement = 0 m, total distance = 40 m, total average velocity = 0 m/s 48. A bee flies straight at someone and then back to its hive along the same path. Assuming it takes no time for the bee to speed up or slow down, except at the moment it changes direction, how would the graph of position vs time look? Consider the initial direction to be positive. a. The graph will look like a downward opening V shape. b. The graph will look like an upward opening V shape. c. The graph will look like a downward opening parabola. Chapter 2 • Test Prep 89 a. b. c. d. It is a straight line with negative slope. It is a straight line with positive slope. It is a horizontal line at some negative value. It is a horizontal line at some positive value. 50. Which statement correctly describes the object’s speed, as well as what a graph of acceleration vs. time would look like? a. The object is not speeding up, and the acceleration vs. time graph is a horizontal line at some negative value. b. The object is not speeding up, and the acceleration vs. time graph is a horizontal line at some positive value. c. The object is speeding up, and the acceleration vs. time graph is a horizontal line at some negative value. d. The object is speeding up, and the acceleration vs. time graph is a horizontal line at some positive value. d. The graph will look like an upward opening 51. Calculate that object’s net displacement over the time parabola. shown. 2.4 Velocity vs. Time Graphs 49. What would the velocity vs. time graph of the object whose position is shown in the graph look like? 90 Chapter 2 • Test Prep a. 540 m b. 2,520 m c. 2,790 m 5,040 m d. a. 18 m/s b. 84 m/s c. 93 m/s 168 m/s d. 52. What is the object’s average velocity? Extended Response 2.1 Relative Motion, Distance, and Displacement 53. Find the distance traveled from the starting point for each path. Which path has the maximum distance? a. The distance for Path A is 6 m, Path B is 4 m, Path C is 12 m and for Path D is 7 m. The net displacement for Path A is 7 m, Path B is –4m, Path C is 8 m and for Path D is –5m. Path C has maximum distance and it is equal to 12 meters. b. The distance for Path A is 6 m, Path B is 4 m, Path C is 8 m and for Path D is 7 m. The net displacement for Path A is 6 m, Path B is –4m, Path C is 12 m and for Path D is –5 m. Path A has maximum distance and it is equal to 6 meters. c. The distance for Path A is 6 m, Path B is 4 m, Path C is 12 m and for Path D is 7 m. The net displacement for Path A is 6 m, Path B is –4 m, Path C is 8 m and for Path D is –5 m. Path C has maximum distance Access for free at openstax.org. and it is equal to 12 meters. d. The distance for Path A is 6 m, Path B is –4 m, Path C is 12 m and for Path D is –5 m. The net displacement for Path A is 7 m, Path B is 4 m, Path C is 8 m and for Path D is 7 m. Path A has maximum distance and it is equal to 6 m. 54. Alan starts from his home and walks 1.3 km east to the library. He walks an additional 0.68 km east to a music store. From there, he walks 1.1 km north to a friend’s house and an additional 0.42 km north to a grocery store before he finally returns home along the same path. What is his final displacement and total distance traveled? a. Displacement is 0 km and distance is 7 km. b. Displacement is 0 km and distance is 3.5 km. c. Displacement is 7 km towards west and distance is 7 km. d. Displacement is 3.5 km towards east and distance is 3.5 km. 2.2 Speed and Velocity 55. Two runners start at the same point and jog at a constant speed along a straight path. Runner A starts at time t = 0 s, and Runner B starts at time t = 2.5 s. The runners both reach a distance 64 m from the starting point at time t = 25 s. If the runners continue at the same speeds, how far from the starting point will each be at time t = 45 s? a. Runner A will be m away and Runner B will be m away from the starting point. b. Runner A will be will be c. Runner A will be will be d. Runner A will be will be m away and runner B m away from the starting point. away and Runner B away from the starting point. away and Runner B away from the starting point. 56. A father and his daughter go to the bus stop that is located 75 m from their front door. The father walks in a straight line while his daughter runs along a varied path. Despite the different paths, they both end up at the bus stop at the same time. The father’s average speed is 2.2 m/s, and his daughter’s average speed is 3.5 m/s. (a) How long does it take the father and daughter to reach the bus stop? (b) What was the daughter’s total distance traveled? (c) If the daughter maintained her same average speed and traveled in a straight line like her father, how far beyond the bus stop would she have traveled? a. b. c. d. (a) 21.43 s (b) 75 m (c) 0 m (a) 21.43 s (b) 119 m (c) 44 m (a) 34 s (b) 75 m (c) 0 m (a) 34 s (b) 119 m (c) 44 m 2.3 Position vs. Time Graphs 57. What kind of motion would create a position graph like the one shown? a. uniform motion b. any motion that accelerates c. motion that stops and then starts d. motion that has constant velocity 58. What is the average velocity for the whole time period shown in the graph? Chapter 2 • Test Prep 91 a. b. c. d. 2.4 Velocity vs. Time Graphs 59. Consider the motion of the object whose velocity is charted in the graph. During which points is the object slowing down and speeding up? a. It is slowing down between dand e. It is speeding up between aand dand eand h It is slowing down between aand dand eand h. It is speeding up between dand eand then after i. It is slowing down between dand eand then after h. It is speeding up between aand dand eand h. It is slowing down between aand dand eand h. It is speeding up between dand eand then after i. b. c. d. 60. Divide the graph into approximate sections, and use those sections to graph the velocity vs. time of the object. 92 Chapter 2 • Test Prep Then calculate the acceleration during each section, and calculate the approximate average velocity. a. Acceleration is zero and average velocity is 1.25 m/s. b. Acceleration is constant with some positive value and average velocity is 1.25 m/s. c. Acceleration is zero and average velocity is 0.25 m/s. d. Acceleration is constant with some positive value and average velocity is 0.25 m/s. Access for free at openstax.org. CHAPTER 3 Acceleration Figure 3.1 A plane slows down as it comes in for landing in St. Maarten. Its acceleration is in the opposite direction of its velocity. (Steve Conry, Flickr) Chapter Outline 3.1 Acceleration 3.2 Representing Acceleration with Equations and Graphs You may have heard the term accelerator, referring to the gas pedal in a car. When the gas pedal is pushed INTRODUCTION down, the flow of gasoline to |
the engine increases, which increases the car’s velocity. Pushing on the gas pedal results in acceleration because the velocity of the car increases, and acceleration is defined as a change in velocity. You need two quantities to define velocity: a speed and a direction. Changing either of these quantities, or both together, changes the velocity. You may be surprised to learn that pushing on the brake pedal or turning the steering wheel also causes acceleration. The first reduces the speedand so changes the velocity, and the second changes the directionand also changes the velocity. In fact, any change in velocity—whether positive, negative, directional, or any combination of these—is called an acceleration in physics. The plane in the picture is said to be accelerating because its velocity is decreasing as it prepares to land. To begin our study of acceleration, we need to have a clear understanding of what acceleration means. 3.1 Acceleration Section Learning Objectives By the end of this section, you will be able to do the following: • Explain acceleration and determine the direction and magnitude of acceleration in one dimension • Analyze motion in one dimension using kinematic equations and graphic representations 94 Chapter 3 • Acceleration Section Key Terms average acceleration instantaneous acceleration negative acceleration Defining Acceleration Throughout this chapter we will use the following terms: time, displacement, velocity, and acceleration. Recall that each of these terms has a designated variable and SI unit of measurement as follows: • Time: t, measured in seconds (s) • Displacement: Δd, measured in meters (m) • Velocity: v, measured in meters per second (m/s) • Acceleration: a, measured in meters per second per second (m/s2, also called meters per second squared) • Also note the following: ◦ Δ means change in ◦ The subscript 0 refers to an initial value; sometimes subscript i is instead used to refer to initial value. ◦ The subscript f refers to final value ◦ A bar over a symbol, such as , means average Acceleration is the change in velocity divided by a period of time during which the change occurs. The SI units of velocity are m/s and the SI units for time are s, so the SI units for acceleration are m/s2. Average acceleration is given by Average acceleration is distinguished from instantaneous acceleration, which is acceleration at a specific instant in time. The magnitude of acceleration is often not constant over time. For example, runners in a race accelerate at a greater rate in the first second of a race than during the following seconds. You do not need to know all the instantaneous accelerations at all times to calculate average acceleration. All you need to know is the change in velocity (i.e., the final velocity minus the initial velocity) and the change in time (i.e., the final time minus the initial time), as shown in the formula. Note that the average acceleration can be positive, negative, or zero. A negative acceleration is simply an acceleration in the negative direction. Keep in mind that although acceleration points in the same direction as the changein velocity, it is not always in the direction of the velocity itself. When an object slows down, its acceleration is opposite to the direction of its velocity. In everyday language, this is called deceleration; but in physics, it is acceleration—whose direction happens to be opposite that of the velocity. For now, let us assume that motion to the right along the x-axis is positiveand motion to the left is negative. Figure 3.2 shows a car with positive acceleration in (a) and negative acceleration in (b). The arrows represent vectors showing both direction and magnitude of velocity and acceleration. Figure 3.2 The car is speeding up in (a) and slowing down in (b). Access for free at openstax.org. 3.1 • Acceleration 95 Velocity and acceleration are both vector quantities. Recall that vectors have both magnitude and direction. An object traveling at a constant velocity—therefore having no acceleration—does accelerate if it changes direction. So, turning the steering wheel of a moving car makes the car accelerate because the velocity changes direction. Virtual Physics The Moving Man With this animation in , you can produce both variations of acceleration and velocity shown in Figure 3.2, plus a few more variations. Vary the velocity and acceleration by sliding the red and green markers along the scales. Keeping the velocity marker near zero will make the effect of acceleration more obvious. Try changing acceleration from positive to negative while the man is moving. We will come back to this animation and look at the Chartsview when we study graphical representation of motion. Click to view content (https://archive.cnx.org/specials/e2ca52af-8c6b-450e-ac2f-9300b38e8739/moving-man/) GRASP CHECK Figure 3.3 Which part, (a) or (b), is represented when the velocity vector is on the positive side of the scale and the acceleration vector is set on the negative side of the scale? What does the car’s motion look like for the given scenario? a. Part (a). The car is slowing down because the acceleration and the velocity vectors are acting in the opposite direction. b. Part (a). The car is speeding up because the acceleration and the velocity vectors are acting in the same direction. c. Part (b). The car is slowing down because the acceleration and velocity vectors are acting in the opposite directions. d. Part (b). The car is speeding up because the acceleration and the velocity vectors are acting in the same direction. Calculating Average Acceleration Look back at the equation for average acceleration. You can see that the calculation of average acceleration involves three values: change in time, (Δt); change in velocity, (Δv); and acceleration (a). Change in time is often stated as a time interval, and change in velocity can often be calculated by subtracting the initial velocity from the final velocity. Average acceleration is then simply change in velocity divided by change in time. Before you begin calculating, be sure that all distances and times have been converted to meters and seconds. Look at these examples of acceleration of a subway train. 96 Chapter 3 • Acceleration WORKED EXAMPLE An Accelerating Subway Train A subway train accelerates from rest to 30.0 km/h in 20.0 s. What is the average acceleration during that time interval? Strategy Start by making a simple sketch. Figure 3.4 This problem involves four steps: 1. Convert to units of meters and seconds. 2. Determine the change in velocity. 3. Determine the change in time. 4. Use these values to calculate the average acceleration. Solution 1. Identify the knowns. Be sure to read the problem for given information, which may not looklike numbers. When the problem states that the train starts from rest, you can write down that the initial velocity is 0 m/s. Therefore, v0 = 0; vf = 30.0 km/h; and Δt= 20.0 s. 2. Convert the units. 3. Calculate change in velocity, velocity is to the right. 3.1 where the plus sign means the change in 4. We know Δt, so all we have to do is insert the known values into the formula for average acceleration. Discussion The plus sign in the answer means that acceleration is to the right. This is a reasonable conclusion because the train starts from rest and ends up with a velocity directed to the right (i.e., positive). So, acceleration is in the same direction as the changein velocity, as it should be. 3.2 WORKED EXAMPLE An Accelerating Subway Train Now, suppose that at the end of its trip, the train slows to a stop in 8.00 s from a speed of 30.0 km/h. What is its average acceleration during this time? Strategy Again, make a simple sketch. Access for free at openstax.org. 3.1 • Acceleration 97 Figure 3.5 In this case, the train is decelerating and its acceleration is negative because it is pointing to the left. As in the previous example, we must find the change in velocity and change in time, then solve for acceleration. Identify the knowns: v0 = 30.0 km/h; vf = 0; and Δt= 8.00 s. Solution 1. 2. Convert the units. From the first problem, we know that 30.0 km/h = 8.333 m/s. 3. Calculate change in velocity, change in velocity points to the left. where the minus sign means that the 4. We know Δt= 8.00 s, so all we have to do is insert the known values into the equation for average acceleration. 3.3 Discussion The minus sign indicates that acceleration is to the left. This is reasonable because the train initially has a positive velocity in this problem, and a negative acceleration would reduce the velocity. Again, acceleration is in the same direction as the changein velocity, which is negative in this case. This acceleration can be called a deceleration because it has a direction opposite to the velocity. TIPS FOR SUCCESS • • It is easier to get plus and minus signs correct if you always assume that motion is away from zero and toward positive values on the x-axis. This way valways starts off being positive and points to the right. If speed is increasing, then acceleration is positive and also points to the right. If speed is decreasing, then acceleration is negative and points to the left. It is a good idea to carry two extra significant figures from step-to-step when making calculations. Do not round off with each step. When you arrive at the final answer, apply the rules of significant figures for the operations you carried out and round to the correct number of digits. Sometimes this will make your answer slightly more accurate. Practice Problems 1. A cheetah can accelerate from rest to a speed of in . What is its acceleration? a. b. c. d. 2. A women backs her car out of her garage with an acceleration of . How long does it take her to reach a speed of ? a. b. c. d. 98 Chapter 3 • Acceleration WATCH PHYSICS Acceleration This video shows the basic calculation of acceleration and some useful unit conversions. Click to view content (https://www.khanaca |
demy.org/embed_video?v=FOkQszg1-j8) GRASP CHECK Why is acceleration a vector quantity? a. b. c. d. It is a vector quantity because it has magnitude as well as direction. It is a vector quantity because it has magnitude but no direction. It is a vector quantity because it is calculated from distance and time. It is a vector quantity because it is calculated from speed and time. GRASP CHECK What will be the change in velocity each second if acceleration is 10 m/s/s? a. An acceleration of b. An acceleration of c. An acceleration of d. An acceleration of means that every second, the velocity increases by means that every second, the velocity decreases by means that every means that every , the velocity increases by , the velocity decreases by . . . . Snap Lab Measure the Acceleration of a Bicycle on a Slope In this lab you will take measurements to determine if the acceleration of a moving bicycle is constant. If the acceleration is constant, then the following relationships hold: , then and If You will work in pairs to measure and record data for a bicycle coasting down an incline on a smooth, gentle slope. The data will consist of distances traveled and elapsed times. • Find an open area to minimize the risk of injury during this lab. stopwatch • • measuring tape • bicycle 1. Find a gentle, paved slope, such as an incline on a bike path. The more gentle the slope, the more accurate your data will likely be. 2. Mark uniform distances along the slope, such as 5 m, 10 m, etc. 3. Determine the following roles: the bike rider, the timer, and the recorder. The recorder should create a data table to collect the distance and time data. 4. Have the rider at the starting point at rest on the bike. When the timer calls Start, the timer starts the stopwatch and the rider begins coasting down the slope on the bike without pedaling. 5. Have the timer call out the elapsed times as the bike passes each marked point. The recorder should record the times in the data table. It may be necessary to repeat the process to practice roles and make necessary adjustments. 6. Once acceptable data has been recorded, switch roles. Repeat Steps 3–5 to collect a second set of data. 7. Switch roles again to collect a third set of data. 8. Calculate average acceleration for each set of distance-time data. If your result for is not the same for different pairs of Δvand Δt, then acceleration is not constant. Interpret your results. 9. Access for free at openstax.org. 3.2 • Representing Acceleration with Equations and Graphs 99 GRASP CHECK If you graph the average velocity (y-axis) vs. the elapsed time (x-axis), what would the graph look like if acceleration is uniform? a. a horizontal line on the graph b. a diagonal line on the graph c. an upward-facing parabola on the graph d. a downward-facing parabola on the graph Check Your Understanding 3. What are three ways an object can accelerate? a. By speeding up, maintaining constant velocity, or changing direction b. By speeding up, slowing down, or changing direction c. By maintaining constant velocity, slowing down, or changing direction d. By speeding up, slowing down, or maintaining constant velocity 4. What is the difference between average acceleration and instantaneous acceleration? a. Average acceleration is the change in displacement divided by the elapsed time; instantaneous acceleration is the acceleration at a given point in time. b. Average acceleration is acceleration at a given point in time; instantaneous acceleration is the change in displacement divided by the elapsed time. c. Average acceleration is the change in velocity divided by the elapsed time; instantaneous acceleration is acceleration at a given point in time. d. Average acceleration is acceleration at a given point in time; instantaneous acceleration is the change in velocity divided by the elapsed time. 5. What is the rate of change of velocity called? a. Time b. Displacement c. Velocity d. Acceleration 3.2 Representing Acceleration with Equations and Graphs Section Learning Objectives By the end of this section, you will be able to do the following: • Explain the kinematic equations related to acceleration and illustrate them with graphs • Apply the kinematic equations and related graphs to problems involving acceleration Section Key Terms acceleration due to gravity kinematic equations uniform acceleration How the Kinematic Equations are Related to Acceleration We are studying concepts related to motion: time, displacement, velocity, and especially acceleration. We are only concerned with motion in one dimension. The kinematic equations apply to conditions of constant acceleration and show how these concepts are related. Constant acceleration is acceleration that does not change over time. The first kinematic equation relates displacement d, average velocity , and time t. 3.4 The initial displacement is often 0, in which case the equation can be written as 100 Chapter 3 • Acceleration This equation says that average velocity is displacement per unit time. We will express velocity in meters per second. If we graph displacement versus time, as in Figure 3.6, the slope will be the velocity. Whenever a rate, such as velocity, is represented graphically, time is usually taken to be the independent variable and is plotted along the xaxis. Figure 3.6 The slope of displacement versus time is velocity. The second kinematic equation, another expression for average velocity divided by two. is simply the initial velocity plus the final velocity Now we come to our main focus of this chapter; namely, the kinematic equations that describe motion with constant acceleration. In the third kinematic equation, acceleration is the rate at which velocity increases, so velocity at any point equals initial velocity plus acceleration multiplied by time 3.6 Note that this third kinematic equation does not have displacement in it. Therefore, if you do not know the displacement and are not trying to solve for a displacement, this equation might be a good one to use. The third kinematic equation is also represented by the graph in Figure 3.7. 3.5 The fourth kinematic equation shows how displacement is related to acceleration Figure 3.7 The slope of velocity versus time is acceleration. When starting at the origin, and, when starting from rest, , in which case the equation can be written as 3.7 Access for free at openstax.org. 3.2 • Representing Acceleration with Equations and Graphs 101 This equation tells us that, for constant acceleration, the slope of a plot of 2dversus t2 is acceleration, as shown in Figure 3.8. Figure 3.8 When acceleration is constant, the slope of 2dversus t2 gives the acceleration. The fifth kinematic equation relates velocity, acceleration, and displacement 3.8 This equation is useful for when we do not know, or do not need to know, the time. When starting from rest, the fifth equation simplifies to According to this equation, a graph of velocity squared versus twice the displacement will have a slope equal to acceleration. Figure 3.9 Note that, in reality, knowns and unknowns will vary. Sometimes you will want to rearrange a kinematic equation so that the knowns are the values on the axes and the unknown is the slope. Sometimes the intercept will not be at the origin (0,0). This will happen when v0 or d0 is not zero. This will be the case when the object of interest is already in motion, or the motion begins at some point other than at the origin of the coordinate system. Virtual Physics The Moving Man (Part 2) Look at the Moving Man simulation again and this time use the Chartsview. Again, vary the velocity and acceleration by sliding the red and green markers along the scales. Keeping the velocity marker near zero will make the effect of acceleration more obvious. Observe how the graphs of position, velocity, and acceleration vary with time. Note which are linear plots and which are not. Click to view content (https://archive.cnx.org/specials/e2ca52af-8c6b-450e-ac2f-9300b38e8739/moving-man/) GRASP CHECK On a velocity versus time plot, what does the slope represent? a. Acceleration 102 Chapter 3 • Acceleration b. Displacement c. Distance covered d. Instantaneous velocity GRASP CHECK On a position versus time plot, what does the slope represent? a. Acceleration b. Displacement c. Distance covered d. Instantaneous velocity The kinematic equations are applicable when you have constant acceleration. , or , or 1. 2. 3. 4. 5. when d0 = 0 when v0 = 0 , or , or when d0 = 0 and v0 = 0 when d0 = 0 and v0 = 0 Applying Kinematic Equations to Situations of Constant Acceleration Problem-solving skills are essential to success in a science and life in general. The ability to apply broad physical principles, which are often represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Essential analytical skills will be developed by solving problems in this text and will be useful for understanding physics and science in general throughout your life. Problem-Solving Steps While no single step-by-step method works for every problem, the following general procedures facilitate problem solving and make the answers more meaningful. A certain amount of creativity and insight are required as well. 1. Examine the situation to determine which physical principles are involved. It is vital to draw a simple sketch at the outset. Decide which direction is positive and note that on your sketch. 2. Identify the knowns: Make a list of what information is given or can be inferred from the problem statement.Remember, not all given information will be in the form of a number with units in the problem. If something starts from rest, we know the initial velocity is zero. If something stops, we know the fin |
al velocity is zero. 3. Identify the unknowns: Decide exactly what needs to be determined in the problem. 4. Find an equation or set of equations that can help you solve the problem.Your list of knowns and unknowns can help here. For example, if time is not needed or not given, then the fifth kinematic equation, which does not include time, could be useful. 5. Insert the knowns along with their units into the appropriate equation and obtain numerical solutions complete with units.This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. 6. Check the answer to see if it is reasonable: Does it make sense?This final step is extremely important because the goal of physics is to accurately describe nature. To see if the answer is reasonable, check its magnitude, its sign, and its units. Are the significant figures correct? Summary of Problem Solving • Determine the knowns and unknowns. • Find an equation that expresses the unknown in terms of the knowns. More than one unknown means more than one equation is needed. • Solve the equation or equations. Access for free at openstax.org. 3.2 • Representing Acceleration with Equations and Graphs 103 • Be sure units and significant figures are correct. • Check whether the answer is reasonable. FUN IN PHYSICS Drag Racing Figure 3.10 Smoke rises from the tires of a dragster at the beginning of a drag race. (Lt. Col. William Thurmond. Photo courtesy of U.S. Army.) The object of the sport of drag racing is acceleration. Period! The races take place from a standing start on a straight onequarter-mile (402 m) track. Usually two cars race side by side, and the winner is the driver who gets the car past the quarter-mile point first. At the finish line, the cars may be going more than 300 miles per hour (134 m/s). The driver then deploys a parachute to bring the car to a stop because it is unsafe to brake at such high speeds. The cars, called dragsters, are capable of accelerating at 26 m/s2. By comparison, a typical sports car that is available to the general public can accelerate at about 5 m/s2. Several measurements are taken during each drag race: • Reaction time is the time between the starting signal and when the front of the car crosses the starting line. • Elapsed time is the time from when the vehicle crosses the starting line to when it crosses the finish line. The record is a little over 3 s. • Speed is the average speed during the last 20 m before the finish line. The record is a little under 400 mph. The video shows a race between two dragsters powered by jet engines. The actual race lasts about four seconds and is near the end of the video (https://openstax.org/l/28dragsters) . GRASP CHECK A dragster crosses the finish line with a velocity of start to finish, what was its average velocity for the race? a. b. c. d. . Assuming the vehicle maintained a constant acceleration from WORKED EXAMPLE Acceleration of a Dragster The time it takes for a dragster to cross the finish line is unknown. The dragster accelerates from rest at 26 m/s2 for a quarter mile (0.250 mi). What is the final velocity of the dragster? Strategy The equation displacement, without involving the time. is ideally suited to this task because it gives the velocity from acceleration and 104 Chapter 3 • Acceleration Solution 1. Convert miles to meters. 3.9 2. 3. Identify the known values. We know that v0 = 0 since the dragster starts from rest, and we know that the distance traveled, d− d0 is 402 m. Finally, the acceleration is constant at a= 26.0 m/s2. Insert the knowns into the equation and solve for v. 3.10 Taking the square root gives us Discussion 145 m/s is about 522 km/hour or about 324 mi/h, but even this breakneck speed is short of the record for the quarter mile. Also, note that a square root has two values. We took the positive value because we know that the velocity must be in the same direction as the acceleration for the answer to make physical sense. An examination of the equation physical quantities: can produce further insights into the general relationships among • The final velocity depends on the magnitude of the acceleration and the distance over which it applies. • For a given acceleration, a car that is going twice as fast does not stop in twice the distance—it goes much further before it stops. This is why, for example, we have reduced speed zones near schools. Practice Problems 6. Dragsters can reach a top speed of 145 m/s in only 4.45 s. Calculate the average acceleration for such a dragster. a. −32.6 m/s2 b. 0 m/s2 c. d. 32.6 m/s2 145 m/s2 7. An Olympic-class sprinter starts a race with an acceleration of 4.50 m/s2. Assuming she can maintain that acceleration, what is her speed 2.40 s later? a. 4.50 m/s 10.8 m/s b. c. 19.6 m/s d. 44.1 m/s Constant Acceleration In many cases, acceleration is not uniform because the force acting on the accelerating object is not constant over time. A situation that gives constant acceleration is the acceleration of falling objects. When air resistance is not a factor, all objects near Earth’s surface fall with an acceleration of about 9.80 m/s2. Although this value decreases slightly with increasing altitude, it may be assumed to be essentially constant. The value of 9.80 m/s2 is labeled gand is referred to as acceleration due to gravity. Gravity is the force that causes nonsupported objects to accelerate downward—or, more precisely, toward the center of Earth. The magnitude of this force is called the weight of the object and is given by mgwhere mis the mass of the object (in kg). In places other than on Earth, such as the Moon or on other planets, gis not 9.80 m/s2, but takes on other values. When using gfor the acceleration ain a kinematic equation, it is usually given a negative sign because the acceleration due to gravity is downward. Access for free at openstax.org. 3.2 • Representing Acceleration with Equations and Graphs 105 WORK IN PHYSICS Effects of Rapid Acceleration Figure 3.11 Astronauts train using G Force Simulators. (NASA) When in a vehicle that accelerates rapidly, you experience a force on your entire body that accelerates your body. You feel this force in automobiles and slightly more on amusement park rides. For example, when you ride in a car that turns, the car applies a force on your body to make you accelerate in the direction in which the car is turning. If enough force is applied, you will accelerate at 9.80 m/s2. This is the same as the acceleration due to gravity, so this force is called one G. One G is the force required to accelerate an object at the acceleration due to gravity at Earth’s surface. Thus, one G for a paper cup is much less than one G for an elephant, because the elephant is much more massive and requires a greater force to make it accelerate at 9.80 m/s2. For a person, a G of about 4 is so strong that his or her face will distort as the bones accelerate forward through the loose flesh. Other symptoms at extremely high Gs include changes in vision, loss of consciousness, and even death. The space shuttle produces about 3 Gs during takeoff and reentry. Some roller coasters and dragsters produce forces of around 4 Gs for their occupants. A fighter jet can produce up to 12 Gs during a sharp turn. Astronauts and fighter pilots must undergo G-force training in simulators. The video (https://www.youtube.com/ watch?v=n-8QHOUWECU) shows the experience of several people undergoing this training. People, such as astronauts, who work with G forces must also be trained to experience zero G—also called free fall or weightlessness—which can cause queasiness. NASA has an aircraft that allows it occupants to experience about 25 s of free fall. The aircraft is nicknamed the Vomit Comet. GRASP CHECK A common way to describe acceleration is to express it in multiples of g, Earth's gravitational acceleration. If a dragster accelerates at a rate of 39.2 m/s2, how many g's does the driver experience? a. 1.5 g b. 4.0 g c. 10.5 g d. 24.5 g WORKED EXAMPLE Falling Objects A person standing on the edge of a high cliff throws a rock straight up with an initial velocity v0 of 13 m/s. (a) Calculate the position and velocity of the rock at 1.00, 2.00, and 3.00 seconds after it is thrown. Ignore the effect of air resistance. Strategy Sketch the initial velocity and acceleration vectors and the axes. 106 Chapter 3 • Acceleration Figure 3.12 Initial conditions for rock thrown straight up. List the knowns: time t= 1.00 s, 2.00 s, and 3.00 s; initial velocity v0 = 13 m/s; acceleration a= g= –9.80 m/s2; and position y0 = 0 m List the unknowns: y1, y2, and y3; v1, v2, and v3—where 1, 2, 3 refer to times 1.00 s, 2.00 s, and 3.00 s Choose the equations. 3.11 3.12 These equations describe the unknowns in terms of knowns only. Solution Discussion The first two positive values for y show that the rock is still above the edge of the cliff, and the third negative y value shows that it has passed the starting point and is below the cliff. Remember that we set upto be positive. Any position with a positive value is above the cliff, and any velocity with a positive value is an upward velocity. The first value for v is positive, so the rock is still on the way up. The second and third values for v are negative, so the rock is on its way down. (b) Make graphs of position versus time, velocity versus time, and acceleration versus time. Use increments of 0.5 s in your graphs. Strategy Time is customarily plotted on the x-axis because it is the independent variable. Position versus time will not be linear, so calculate points for 0.50 s, 1.50 s, and 2.50 s. This will give a curve closer to the true, smooth shape. Solution The three graphs are shown in Figure 3.13. Access for free at openstax.org. 3.2 • Representing Acceleration with Equations and Graphs 107 Figure 3.13 Discussion • yvs. tdoes notrepresent the two-dimensional parabolic path |
of a trajectory. The path of the rock is straight up and straight down. The slope of a line tangent to the curve at any point on the curve equals the velocity at that point—i.e., the instantaneous velocity. 108 Chapter 3 • Acceleration • Note that the vvs. tline crosses the vertical axis at the initial velocity and crosses the horizontal axis at the time when the rock changes direction and begins to fall back to Earth. This plot is linear because acceleration is constant. • The avs. tplot also shows that acceleration is constant; that is, it does not change with time. Practice Problems 8. A cliff diver pushes off horizontally from a cliff and lands in the ocean later. How fast was he going when he entered the water? a. b. c. d. 9. A girl drops a pebble from a high cliff into a lake far below. She sees the splash of the pebble hitting the water later. How fast was the pebble going when it hit the water? a. b. c. d. Check Your Understanding 10. Identify the four variables found in the kinematic equations. a. Displacement, Force, Mass, and Time b. Acceleration, Displacement, Time, and Velocity c. Final Velocity, Force, Initial Velocity, and Mass d. Acceleration, Final Velocity, Force, and Initial Velocity 11. Which of the following steps is always required to solve a kinematics problem? a. Find the force acting on the body. b. Find the acceleration of a body. c. Find the initial velocity of a body. d. Find a suitable kinematic equation and then solve for the unknown quantity. 12. Which of the following provides a correct answer for a problem that can be solved using the kinematic equations? for for . The body’s final velocity is . The body’s final velocity is . . a. A body starts from rest and accelerates at b. A body starts from rest and accelerates at c. A body with a mass of d. A body with a mass of is acted upon by a force of is acted upon by a force of . The acceleration of the body is . The acceleration of the body is . . Access for free at openstax.org. Chapter 3 • Key Terms 109 KEY TERMS acceleration due to gravity acceleration of an object that is subject only to the force of gravity; near Earth’s surface this acceleration is 9.80 m/s2 average acceleration change in velocity divided by the time interval over which it changed instantaneous acceleration rate of change of velocity at a specific instant in time kinematic equations the five equations that describe motion in terms of time, displacement, velocity, and acceleration constant acceleration acceleration that does not change negative acceleration acceleration in the negative direction with respect to time SECTION SUMMARY 3.1 Acceleration • Acceleration is the rate of change of velocity and may be negative or positive. • Average acceleration is expressed in m/s2 and, in one dimension, can be calculated using 3.2 Representing Acceleration with Equations and Graphs • The kinematic equations show how time, displacement, KEY EQUATIONS 3.1 Acceleration Average acceleration • velocity, and acceleration are related for objects in motion. In general, kinematic problems can be solved by identifying the kinematic equation that expresses the unknown in terms of the knowns. • Displacement, velocity, and acceleration may be displayed graphically versus time. Average velocity Velocity , or when v0 = 0 3.2 Representing Acceleration with Equations and Graphs Average velocity , or when d0 = 0 Displacement when d0 = 0 and v0 = 0 Acceleration when d0 = 0 and v0 = 0 , or , or CHAPTER REVIEW Concept Items 3.1 Acceleration 1. How can you use the definition of acceleration to explain the units in which acceleration is measured? a. Acceleration is the rate of change of velocity. Therefore, its unit is m/s2. b. Acceleration is the rate of change of displacement. Therefore, its unit is m/s. c. Acceleration is the rate of change of velocity. Therefore, its unit is m2/s. d. Acceleration is the rate of change of displacement. Therefore, its unit is m2/s. 2. What are the SI units of acceleration? a. b. c. d. 3. Which of the following statements explains why a racecar going around a curve is accelerating, even if the speed is constant? a. The car is accelerating because the magnitude as well as the direction of velocity is changing. b. The car is accelerating because the magnitude of velocity is changing. c. The car is accelerating because the direction of velocity is changing. 110 Chapter 3 • Chapter Review d. The car is accelerating because neither the magnitude nor the direction of velocity is changing. 3.2 Representing Acceleration with Equations and Graphs 4. A student calculated the final velocity of a train that decelerated from 30.5 m/s and got an answer of −43.34 m/ s. Which of the following might indicate that he made a mistake in his calculation? a. The sign of the final velocity is wrong. b. The magnitude of the answer is too small. c. There are too few significant digits in the answer. d. The units in the initial velocity are incorrect. 5. Create your own kinematics problem. Then, create a flow chart showing the steps someone would need to take to solve the problem. a. Acceleration b. Distance c. Displacement d. Force 6. Which kinematic equation would you use to find the after she jumps from a plane velocity of a skydiver and before she opens her parachute? Assume the positive direction is downward. a. b. c. d. Critical Thinking Items 3.1 Acceleration 7. Imagine that a car is traveling from your left to your right at a constant velocity. Which two actions could the driver take that may be represented as (a) a velocity vector and an acceleration vector both pointing to the right and then (b) changing so the velocity vector points to the right and the acceleration vector points to the left? a. (a) Push down on the accelerator and then (b) push down again on the accelerator a second time. (a) Push down on the accelerator and then (b) push down on the brakes. (a) Push down on the brakes and then (b) push down on the brakes a second time. (a) Push down on the brakes and then (b) push down on the accelerator. b. c. d. 8. A motorcycle moving at a constant velocity suddenly accelerates at a rate of to a speed of in . What was the initial speed of the motorcycle? a. b. c. d. 3.2 Representing Acceleration with Equations and Graphs 9. A student is asked to solve a problem: An object falls from a height for 2.0 s, at which point it is still 60 m above the ground. What will be the velocity of the object when it hits the ground? Which of the following provides the correct order of kinematic equations that can be used to solve the problem? a. First use then use Access for free at openstax.org. b. First use then use c. First use then use d. First use then use 10. Skydivers are affected by acceleration due to gravity and by air resistance. Eventually, they reach a speed where the force of gravity is almost equal to the force of air resistance. As they approach that point, their acceleration decreases in magnitude to near zero. Part A. Describe the shape of the graph of the magnitude of the acceleration versus time for a falling skydiver. Part B. Describe the shape of the graph of the magnitude of the velocity versus time for a falling skydiver. Part C. Describe the shape of the graph of the magnitude of the displacement versus time for a falling skydiver. a. Part A. Begins with a nonzero y-intercept with a downward slope that levels off at zero; Part B. Begins at zero with an upward slope that decreases in magnitude until the curve levels off; Part C. Begins at zero with an upward slope that increases in magnitude until it becomes a positive constant b. Part A. Begins with a nonzero y-intercept with an upward slope that levels off at zero; Part B. Begins at zero with an upward slope that decreases in magnitude until the curve levels off; Part C. Begins at zero with an upward slope that increases in magnitude until it becomes a positive constant c. Part A. Begins with a nonzero y-intercept with a downward slope that levels off at zero; Part B. Begins at zero with a downward slope that Chapter 3 • Chapter Review 111 decreases in magnitude until the curve levels off; Part C. Begins at zero with an upward slope that increases in magnitude until it becomes a positive constant d. Part A. Begins with a nonzero y-intercept with an upward slope that levels off at zero; Part B. Begins at zero with a downward slope that decreases in magnitude until the curve levels off; Part C. Begins at zero with an upward slope that increases in magnitude until it becomes a positive constant 11. Which graph in the previous problem has a positive slope? a. Displacement versus time only b. Acceleration versus time and velocity versus time c. Velocity versus time and displacement versus time d. Acceleration versus time and displacement versus time Problems 3.1 Acceleration 12. The driver of a sports car traveling at 10.0 m/s steps down hard on the accelerator for 5.0 s and the velocity increases to 30.0 m/s. What was the average acceleration of the car during the 5.0 s time interval? a. –1.0 × 102 m/s2 b. –4.0 m/s2 c. 4.0 m/s2 d. 1.0 × 102 m/s2 13. A girl rolls a basketball across a basketball court. The ball slowly decelerates at a rate of −0.20 m/s2. If the initial velocity was 2.0 m/s and the ball rolled to a stop at 5.0 sec after 12:00 p.m., at what time did she start the ball rolling? a. 0.1 seconds before noon b. 0.1 seconds after noon 5 seconds before noon c. 5 seconds after noon d. Performance Task 3.2 Representing Acceleration with Equations and Graphs 16. Design an experiment to measure displacement and elapsed time. Use the data to calculate final velocity, average velocity, acceleration, and acceleration. Materials • a small marble or ball bearing • a garden hose • a measuring tape • a stopwatch or stopwatch software download • a sloping driveway or lawn as long as the garden 3.2 Representing Acceleration with Equations and Graphs 14. A swan on a lake gets airborne by flapping its wings and running on |
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