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down, depending on whether the fñeld was in the positive or negative -direction.
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Although we shall not at the present time give the correct law of force between
charges moving in an arbitrary manner, one relative to the other, because iE is
too complicated, we shall give one aspect of it: the complete law of the forces
£J the ields are knoumn. "The force on a charged object depends upon its motion;
1ƒ, when the objJect is standing still at a given place, there is some force, this
1s taken to be proportional to the charge, the coefficient being what we call
the electric field. When the object moves the force may be different, and the
correction, the new “piece” of force, turns out to be dependent exactly lineariu
on the 0elocitu, but at right angles to 9ø and to another vector quantity which we
call the magnetic induction ÐB. Tf the components of the electric fñeld # and the
magnetic induction Ö are, respectively, (E„, Ey, #;) and (B„, By, B,), and if the
velocity ø has the components (0z, 0y, 0„), then the total electric and magnetic
force on a moving charge g has the components
Ty = q(E„ +uyB, — 0„BỤ),
Tụ = q(Ey +u„B„ — 0y„B,), (12.11)
1; =q(E„ + uy„Bụ — 0y).
Tí, for instance, the only component of the magnetic feld were Ö„ and the only
component of the velocity were „, then the only term left in the magnetic force
would be a force in the z-direction, at right angles to both Ö and 0.
12-ã Pseudo forces
The next kind of force we shall discuss might be called a pseudo force. In
Chapter I1 we discussed the relationship between two people, Joe and Moe, who
use diferent coordinate systems. Let us suppose that the positions of a particle
as measured by Joe are z and by Moe are #; then the laws are as follows:
z—=#+$, U—=, z=#Z,
where s is the displacement of Moeˆs system relative to Joe's. If we suppose that
the laws of motion are correct for Joe, how do they look for Moe? We fnd frst,
d+/dt = da" (dt + ds/di.
Previously, we considered the case where s was constant, and we found that s
made no diference in the laws of motion, since đs/dt = 0; ultimately, therefore,
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the laws of physics were the same in both systems. But another case we can
take is that s — œ‡, where w is a uniform velocity In a straight line. “Then
ø is nob constant, and đs/đf is not zero, but is u, a constant. However, the
acceleration đ2z/đi2 is still the same as đˆz/đf”, because du/d‡ = 0. Thịis proves
the law that we used in Chapter 10, namely, that if we move in a straight line
with uniform velocity the laws of physics will look the same to us as when we are
standing still. 'Phat is the Galilean transformation. But we wish to discuss the
interesting case where s is still more complicated, say s = af2/2. Then ds/df = at
and đ2s/đi2 = aø, a uniform acceleration; or in a still more complicated case, the
acceleration might be a function of time. Thịs means that although the laws of
motion from the point of view of Joe would look like
m Tên đụ,
the laws of motion as looked upon by Moe would appear as
m = Hạ — h„ — ma.
'That is, since Moe”s coordinate system is accelerating with respect to Joe”s, the
extra term mø comes in, and Moe will have to correct his forces by that amount
in order to get Newton's laws to work. In other words, here is an apparent,
mmysterious new force of unknown origin which arises, of course, because Moe
has the wrong coordinate system. 'This is an example of a pseudo force; other
examples occur in coordinate systems that are rofating.
Another example of pseudo force is what is ofben called “centrifugal force.”
An observer in a rotating coordinate system, e.g., in a rotating box, will ñnd
mmysterious forces, not accounted for by any known origin oŸ force, throwing
things outward toward the walls. Thhese forces are due merely to the fact that
the observer does not have NÑewton's coordinate system, which is the simplest
coordinate system.
Pseudo force can be ïllustrated by an interesting experiment in which we
push a jar of water along a table, with acceleration. Gravity, of course, acts
downward on the water, but because of the horizontal acceleration there is also a
pseudo force acting horizontally and in a direction opposite to the acceleration.
'The resultant of gravity and pseudo force makes an angle with the vertical, and
during the acceleration the surface of the water will be perpendicular to the
--- Trang 248 ---
resultant force, ¡.e., inclined at an angle with the table, with the water standing
higher in the rearward side of the jar. When the push on the jar stops and the
jar decelerates because of friction, the pseudo force is reversed, and the water
stands higher in the forward side of the jar (Eig. 12-4).
_> ———————> -^=—
Fig. 12-4. lllustration of a pseudo force.
One very important feature of pseudo forces 1s that they are always Dropor-
tional to the masses; the same is true of gravity. The possibility exists, therefore,
that grauift ?selƒ ¡s a pseudo ƒorce. Ïs it not possible that perhaps gravitation is
due simply to the fact that we do not have the right coordinate system? After
all, we can always get a force proportional to the mass if we imagine that a
body is accelerating. Eor instance, a man shut up in a box that is standing
still on the earth ñnds himself held to the ñoor of the box with a certain force
that is proportional to his mass. But ïf there were no earth at all and the box
were sianding still, the man inside would foat in space. Ôn the other hand, If
there were no earth at all and something were puilmg the box along with an
acceleration ø, then the man in the box, analyzing physics, would ñnd a pseudo
force which would pull him to the foor, just as gravity does.
Binstein put forward the famous hypothesis that accelerations give an imitation
OoŸ gravitation, that the forces of acceleration (the pseudo forces) cœwnot be
địstinguished from those oŸ gravity; 1 is not possible to tell how much of a given
force is gravity and how much is pseudo force.
Tt might seem all right to consider gravity to be a pseudo force, to say that we
are all held down because we are accelerating upward, but how about the people
in Madagascar, on the other side of the earth—are they accelerating too? Einstein
found that gravity could be considered a pseudo force only at one point at a time,
and was led by his considerations to suggest that the geometrU oj the tuuorld 1s
more complicated than ordinary Euclidean geometry. The present discussion is
only qualitative, and does not pretend to convey anything more than the general
idea. To give a rough idea of how gravitation could be the result of pseudo fÍorces,
we present an ïllustration which is purely geometrical and does not represent the
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real situation. Suppose that we all lived in two dimensions, and knew nothing of
a thid. We think we are on a plane, but suppose we are really on the surface of a