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on yesterday, and that is the character of a fñeld. So when the forces get more
complicated, the feld becomes more and more real, and this technique becomes
less and less of an artificial separation.
In analyzing forces by the use of ñelds, we need two kinds of laws pertaining
to fields. “The first is the response to a field, and that gives the equations of
motion. For example, the law of response oŸ a mass to a gravitational feld is
that the force is equal to the mass times the gravitational fñeld; or, If there 1s
also a charge on the body, the response of the charge to the electric feld equals
the charge times the electric feld. 'Phe second part of the analysis of nature in
these situations is to formulate the laws which determine the strength of the
fñeld and how it is produced. 'These laws are sometimes called the feld cquafions.
W© shall learn more about them in due time, but shall write down a few things
about them now.
First, the most remarkable fact of all, which ¡is true exactly and which can be
easily understood, is that the total electric field produced by a number of sources
1s the vector sum of the electric felds produced by the first source, the second
Source, and so on. In other words, if we have numerous charges making a fñeld,
and ïf all by itself one of them would make the field #7, another would make the
feld 2z, and so on, then we merely add the vectors to get the total feld. 'This
prineciple can be expressed as
t=Ei+Ea+Es+--- (12.6)
or, in view of the defnition given above,
đG;T;¡
J— ——. 12.7
» 47corỷ ( )
Can the same methods be applied to gravitation? "The force between two
masses ?¡ and my was expressed by Newton as F' = -Gmạmar/rỶ. But
according to the field concept, we may say that ?mị creates a field Œ in all the
surrounding space, such that the Íorce on ?m¿ is given by
F'—=mạC. (12.8)
By complete analogy with the electrica] case,
--- Trang 244 ---
and the gravitational fñeld produced by several masses 1s
C =C+ạ+C¿+Ca+--- (12.10)
In Chapter 9, in working out a case of planetary motion, we used this principle
in essence. We simply added all the force vectors to get the resultant force on a
planet. If we divide out the mass oŸ the planet in question, we get Eq. (12.10).
Equations (12.6) and (12.10) express what is known as fhe principle oƒ
superposition of fields. 'Phis prineiple states that the total fñeld due to all the
sources is the sum of the fields due to each source. So far as we know today,
for electricity this is an absolutely guaranteed law, which is true even when
the force law is complicated because of the motions of the charges. There are
apparent violations, but more careful analysis has always shown these to be due
to the overlooking of certain moving charges. However, although the principle of
superposition applies exactly for electrical forces, it is not exact for gravity if the
fñeld is too strong, and NÑewton”s equation (12.10) is only approximate, according
to Binstein's gravitational theory.
Closely related to electrical force is another kind, called magnetic force, and
this too is analyzed in terms oŸ a field. Some of the qualitative relations bebween
electrical and magnetie forces can be ïllustrated by an experiment with an electron-
ray tube (Fig. 12-3). At one end of such a tube is a source that emits a stream
of electrons. Within the tube are arrangements for accelerating the electrons to
a high speed and sending some of them in a narrow beam to a Ñuorescent screen
at the other end of the tube. A spot of light glows in the center of the screen
where the electrons strike, and this enables us to trace the electron path. Ôn the
DI :M
+ __-——
NÓ, —4
I VN c— T—— Nị | 7
_ I”IL] J1
ELECTRON GUN ị É_-< Ự
HLCC TRƠN SoURCE V— — À~ 7 Ấ UORESCENT
Fig. 12-3. An electron-beam tube.
--- Trang 245 ---
way to the screen the electron beam passes through a narrow space between a
pair of parallel metal plates, which are arranged, say, horizontally. A voltage can
be applied across the plates, so that either plate can be made negative at will.
'When such a voltage is present, there is an electric fñeld between the plates.
The first part of the experiment is to apply a negative voltage to the lower
plate, which means that extra electrons have been placed on the lower plate.
Since like charges repel, the light spot on the screen instantly shifts upward.
(We could also say this in another way—that the electrons “felt” the ñeld, and
responded by deflecting upward.) We next reverse the voltage, making the upper
plate negative. The light spot on the screen now jumps below the center, showing
that the electrons in the beam were repelled by those in the plate above them.
(Or we could say again that the electrons had “responded” to the field, which is
now in the reverse direction.)
'The second part of the experiment is to disconnect the voltage from the plates
and test the efect ofa magnetic fñeld on the electron beam. 'This is done by means
of a horseshoe magnet, whose poles are far enough apart to more or less straddle
the tube. Suppose we hold the magnet below the tube in the same orientation
as the letter U, with its poles up and part of the tube in between. We note that
the light spot is deflected, say, upward, as the magnet approaches the tube from
below. So it appears that the magnet repels the electron beam. However, it is not
that simple, for If we invert the magnet without reversing the poles side-for-side,
and now approach the tube from above, the spot still moves øœrd, so the
electron beam is øøý repelled; instead, it appears to be attracted this time. Now
we start again, restoring the magnet to its original U orientation and holding
1t below the tube, as before. Yes, the spot is still defected upward; but now turn
the magnet 180 degrees around a vertical axis, so that ït is still in the Ù position
but the poles are reversed side-for-side. Behold, the spot now Jjumps downward,
and stays down, even if we invert the magnet and approach from above, as before.
'To understand this peculiar behavior, we have to have a new combination of
forces. We explain it thus: Across the magnet from one pole to the other there is a
magnetic field. Thịs fñeld has a direction which is always away from one particular
pole (which we could mark) and toward the other. Inverting the magnet did
not change the direction of the field, but reversing the poles side-for-side did
reverse is direction. For example, if the electron velocity were horizontal in the
z-direction and the magnetic field were also horizontal but in the ø-direction, the
magnetic force øn ‡he rnouïng clectrons would be in the z-direction, i.e., up or