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argument against the corpuscular theory; it was used by Huygens. If light were
like a lot of arrows shooting along, how could other arrows go through them so
easily? Such philosophical arguments are not of mụch weight. One could always
say that light is made up of arrows which go through each otherl
26-2 Reflection and refraction
The discussion above gives enough of the basic iđeø of geometrical optics—now
we have to go a little further into the quantitative features. Thus far we have light
going only in straight lines bebween two points; now let us study the behavior
of light when it hits various materials. The simplest object is a mirror, and the
law for a mirror is that when the light hits the mirror, it does not continue in a
straight line, but bounces of the mirror into a new straight line, which changes
when we change the inclination of the mirror. 'Phe question for the aneients
was, what is the relation between the two angles involved? This is a very simple
relation, discovered long ago. 'Phe light striking a mirror travels in such a way
that the two angles, between each beam and the mirror, are equal. For some
reason iÈ is customary to measure the angles from the normal to the mirror
surface. Thus the so-called law of refection 1s
0; = Ú,. (26.1)
'That is a simple enough proposition, but a more dificult problem is encoun-
tered when light goes from one medium into another, for example from air into
water; here also, we see that it does not go in a straight line. In the water the
ray is a% an inclination to its path in the air; if we change the angle Ø; so that
1t comes down more nearly vertically, then the angle of “breakage” is not as
\ø, ro, Ế
Fig. 26-1. The angle of incidence ¡s equal to the angle of reflection.
--- Trang 455 ---
Fig. 26-2. A light ray Is refracted when It posses from one medium
Into another.
Table 26-1 Table 26-2
Anglein air Angle in water Anglein air Angle in water
10° 8° 10° 7-1/2°
207 15-1/2 207 15°
30 22-1/2° 30 22”
400 29 40 29°
502 35° 502 35°
60° 40-1/2° 60° 40-1/2°
702 45-1/2° 702 45”
80 502 802 48”
great. But iŸ we tilt the beam of light at quite an angle, then the deviation angle
1s very large. The question is, what is the relation of one angle to the other?
This also puzzled the ancients for a long time, and here they never found the
answerl It is, however, one of the few places in all of Greek physics that one
may fñnd any experimental results listed. Claudius Ptolemy made a list of the
angle in water for each of a number of diferent angles in air. Table 26-1 shows
the angles In the air, in degrees, and the corresponding angle as measured In
the water. (Ordinarily it is said that Greek scientists never did any experiments.
But it would be impossible to obtain this table of values without knowing the
ripht law, except by experiment. It should be noted, however, that these do
not represent independent careful measurements for each angle but only some
numbers interpolated from a few measurements, for they all ft perfectly on a
parabola.)
--- Trang 456 ---
'This, then, is one of the important steps in the development of physical law:
frst we observe an efect, then we measure it and list it in a table; then we try
to fnd the ruie by which one thing can be connected with another. The above
numerical table was made in 140 A.D., but ít was not until 1621 that someone
finally found the rule connecting the two anglesl The rule, found by Willebrord
Snell, a Dutch mathematician, is as follows: if Ø; is the angle in air and Ø„ is the
angle in the water, then i% turns out that the sine of Ø; is equal to some constant
multiple of the sine of Ø,„:
sin Ø¿ = n0sin Ø„. (26.2)
For water the number ø is approximately 1.33. Equation (26.2) is called Šnells
la; 1 permits us to predict how the light is goïng to bend when it goes Írom air
into water. Table 26-2 shows the angles in air and in water according to Snells
law. Note the remarkable agreement with Ptolemy's list.
26-3 Eermat?s principle of least tỉme
Now in the further development of science, we want more than just a formula.
Pirst we have an observation, then we have numbers that we measure, then we
have a law which summarizes all the numbers. But the real gior of science 1s
that te can fnd a U0ay öƒ thinkứng súch that the law 1s cuident.
The fñrst way of thinking that made the law about the behavior of light evident
was discovered by Fermat in about 1650, and it is called ¿he pr¿inciple oƒ least
time, or Ferma†s principle. His idea 1s thịs: that out of all possible paths that it
might take to get from one point to another, light takes the path which requires
the shortest từmc.
Let us first show that this is true for the case of the mirror, that this simple
prineiple contains both the law of straight-line propagation and the law for the
mirror. So, we are growing in our understandingl Let us try to ñnd the solution
to the following problem. In Eig. 26-3 are shown two points, A and Ö, and a
plane mirror, 1ƒ“. What is the way to get rom A to in the shortest time?
The answer is to go straight from 4 to Bƒ But if we add the extra rule that the
light has to sfứrike the mưrror and come back in the shortest time, the answer is
not so easy. QÔne way would be to go as quickly as possible to the mirror and
then go to Ö, on the path AJD2B. Of course, we then have a long path 2Ö. If we
move over a little to the right, to 2, we slipghtly increase the first distance, but
we greatly decrease the second one, and so the total path length, and therefore
--- Trang 457 ---
A __===—— Z
ụ _+Z :
Ẫ ` ZZ ⁄
LÀ<z ~
M—*^ _< E—M
XI ÔNG Ộ
Fig. 26-3. lllustration of the principle of least time.
the travel time, is less. How can we find the point Πfor which the time is the
shortest? We can fñnd it very nicely by a geometrical trick.
W© construct on the other side of MƒÄ⁄” an artifcial point ”, which is the
same distance below the plane ăĔ as the point Ö is above the plane. Then
we draw the line #Z/. Now because ÖƑ'ÄM is a right angle and ÐF'= F'B', ⁄B
is equal to #'. Therefore the sum of the bwo distances, A4 + EB, which is