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gets too large the body will be torn apart or crushed, depending on the kind of |
distortion. "The amount of force for which Hooke's law is valid depends upon |
the material; for instance, for dough or putty the force is very small, but for |
steel it is relatively large. Hookeˆs law can be nicely demonstrated with a long |
coil spring, made of steel and suspended vertically. A suitable weight hung on |
the lower end of the spring produces a tỉny twist throughout the length of the |
wire, which results in a small vertical deflection in each turn and adds up to a |
large displacement iŸ there are many turns. lf the total elongation produced, |
say, by a 100-gram weight, is measured, it is found that additional weights of |
100 grams will each produce an additional elongation that is very nearly equal |
to the stretch that was measured for the frst 100 grams. This constant ratio |
of force to displacement begins to change when the spring is overloaded, i.e., |
Hooke's law no longer holds. |
12-4 Eundamental forces. Fields |
We shall now discuss the only remaining forces that are fundamental. We |
call them fundamental in the sense that their laws are fundamentally simple. We |
shall first discuss electrical force. ObJects carry electrical charges which consist |
simply of electrons or protons. If any t£wo bodies are electrically charged, there |
1s an electrical force between them, and if the magnitudes of the charges are |
q¡ and qa, respectively, the force varies inversely as the square of the distance |
between the charges, or ' = (const)giqa/r2. Eor unlike charges, this law is like |
the law of gravitation, but for 2e charges the force is repulsive and the sign |
(direction) is reversed. The charges g¡ and qs can be intrinsically either positive |
or negative, and in any specifc application of the formula the direction of the |
force will come out right ïif the g's are given the proper plus or minus sign; the |
force is directed along the line between the two charges. The constant in the |
formula depends, of course, upon what units are used for the force, the charge, |
and the distance. In current practice the charge is measured in coulombs, the |
distance in meters, and the force in newtons. 'Then, in order to get the force |
--- Trang 241 --- |
to come out properly in newtons, the constant (which for historical reasons is |
written 1/4zeo) takes the numerical value |
cọ = 8.854 x 10~12 coul”/newton - m? |
1/4meo = 8.99 x 109 N - m2/coulŸ. |
Thus the force law for static charges is |
F. = qiqar/4aegrẺ. (12.2) |
In nature, the most important charge of all is the charge on a single elec- |
tron, which is 1.60 x 10~†! coulomb. In working with electrical forces between |
fundamental particles rather than with large charges, many people prefer the |
combination (qei)Ÿ/4zeo, in which qe is deñned as the charge on an electron. This |
combination occurs frequently, and to simplify calculations it has been defned |
by the symbol eŸ; its numerical value in the mks system of units turns out to |
be (1.52 x 10~14)2, The advantage of using the constant in this form is that the |
force between two electrons in newtons can then be written simply as e2/z?, with |
r in meters, without all the individual constants. Electrical forces are much more |
complicated than this simple formula indicates, since the formula. gives the Íorce |
between two objects only when the objects are standing still. We shall consider |
the more general case shortÌy. |
In the analysis oŸ forces oŸ the more fundamental kinds (not such forces as |
friction, but the electrical force or the gravitational force), an interesting and very |
Important concept has been developed. Since at first sipht the Íorces are very |
much more complicated than ¡is indicated by the inverse-square laws and these |
laws hold true only when the interacting bodies are standing still, an improved |
method is needed to deal with the very complex forces that ensue when the bodies |
start to move in a complicated way. Experience has shown that an approach |
known as the concept of a “field” is of great utility for the analysis of forces of |
this type. To illustrate the idea for, say, electrical force, suppose we have two |
electrical charges, g¡ and qs, located at points ? and respectively. Then the |
force between the charges is given by |
F. = qiqar/4aegrẺ. (12.3) |
To analyze this force by means of the field concept, we say that the charge g |
at produces a “condition” at , such that when the charge ga is placed at ? |
--- Trang 242 --- |
1t “feels” the force. This is one way, strange perhaps, of describing 1t; we say |
that the force # on ga at Tỉ can be written in two parts. lt is g¿ multiplied by a |
quantity that would be there whether ga were there or not (provided we keep |
all the other charges in their right places). # is the “condition” produced by q, |
we say, and #' ¡is the response of ga to #. E/ is called an clectric feld, and ït 1s a |
vector. The formula for the electric fñeld # that is produced at by a charge g |
at P is the charge g¡ tỉimes the constant 1/4zeo divided by zŸ (z is the distance |
from to ?#?), and it is acting in the direction of the radius vector (the radius |
vector ? divided by its own length). The expression for # ¡is thus |
E = qir/4negrở. (12.4) |
We then write |
P=qsE, (12.5) |
which expresses the force, the field, and the charge in the field. What ¡is the point |
of all this? "The point ¡is to divide the analysis into two parts. One part says that |
something produces a field. 'Phe other part says that something is øcfed ơn by |
the fñeld. By allowing us to look at the two parts independently, this separation |
of the analysis simplifies the calculation of a problem in many situations. If many |
charges are present, we first work out the total electrie feld produced at ?# by all |
the charges, and then, knowing the charge that is placed at , we fñnd the force |
On I1. |
In the case of gravitation, we can do exactly the same thing. In this case, |
where the force #' = —Œmqmar/rỞ, we can make an analogous analysis, as |
follows: the force on a body in a gravitational fñeld is the mass of that body |
times the field Œ. The force on rn¿ is the mass ma times the field Œ produced |
by mị; that is, E! = mạ(C. Then the fñeld Œ produced by a body of mass rn |
is Œ = —Œm?/rỞ and it is direcbed radially, as in the electrical case. |
In spite of how it might at fñrst seem, this separation of one part from another |
1s not a triviality. It would be trivial, jus6 another way of writing the same |
thing, if the laws of force were simple, but the laws of force are so complicated |
that it turns out that the fields have a reality that is almost independent of |
the objects which create them. One can do something like shake a charge and |
produce an effect, a field, at a distance; if one then stops moving the charge, the |
field keeps track of all the past, because the interaction between two particles 1s |
not instantaneous. lt is desirable to have some way to remember what happened |
previously. If the force upon some charge depends upon where another charge |
--- Trang 243 --- |
was yesterday, which it does, then we need machinery to keep track of what went |
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