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W© can summarize all the efects that we shall now discuss by remarking that
they have to do with the effects OŸ rmouing sources. We no longer assume that
the source is localized, with all its motion beïng at a relatively low speed near a
fñxed point.
We recall that the fundamental laws of electrodynamics say that, at large
distances from a moving charge, the electric field is given by the formula
q d2e R-
J— _.nn (34.1)
'The second derivative of the unit vector ep; which points in the apparent direction
of the charge, is the determining feature of the electric fñeld. 'This unit vector
does not point toward the presen£ position of the charge, of course, but rather in
the direction that the charge would seem to be, if the information travels only at
the fñnite speed c from the charge to the observer.
--- Trang 589 ---
Associated with the electric feld is a magnetic feld, always at right angles
to the electric fñeld and at right angles to the apparent direction of the source,
given by the formula
b= —€©hP:X Hực. (34.2)
Until now we have considered only the case in which motions are nonrelativistic
in speed, so that there is no appreciable motion in the direction of the source
to be considered. Now we shall be more general and study the case where the
motion is at an arbitrary velocity, and see what different efects may be expected
in those cireumstances. We shall let the motion be at an arbitrary speed, but of
course we shall still assumne that the detector is very far from the source.
y_ (xứ)
ep! z(r)
P — 6A x
Fig. 34-1. The path of a moving charge. The true position at the
time 7 Is at Ï, but the retarded position Is at A.
W© already know from our discussion in Chapter 28 that the only things that
count in d2ep/đ‡2 are the changes in the đirection of en;. Let the coordinates
of the charge be (z,,2z), with z measured along the direction of observation
(Fig. 34-1). At a given moment in tỉme, say the moment 7, the three components
of the position are #(7), (7), and z(7). The distance ?? is very nearly equal
to fr) = eo + z(r). Ñow the direction of the vector eq; depends mainly on #
and ø, but hardly at all upon z: the transverse components of the unit vector are
z/R and /R, and when we diferentiate these components we get things like #2
in the denominator:
d(4/R) — dz/dL dz +
dc hR dị R2 `
So, when we are far enough away the only terms we have to worry about are the
--- Trang 590 ---
variations of z and ø. Thus we take out the factor Ïọ and get
m—....,
4mcoc2Ro_ di2
_ _ 4megc2Rg dị2 ` (343)
where #ọ is the distance, more or less, to g; let us take it as the distance @?P to the
origin of the coordinates (z,,z). Thus the electric field is a constant multiplied
by a very simple thing, the second derivatives oŸ the z- and -coordinates. (We
could put it more mathematically by calling z and the fransuerse components
of the position vector ? of the charge, but this would not add to the clarity.)
Of course, we realize that the coordinates must be measured at the retarded
tỉme. Here we find that z(7) does afect the retardation. What tỉme is the
retarded time? Tf the time of observation is called ¿ (the time at P) then the
tỉme 7 to which this corresponds at A is not the time ý, but ¡is delayed by the
total distance that the light has to go, divided by the speed of light. In the
frst approximation, this delay is ffo/c, a constant (an uninteresting feature),
but in the next approximation we must include the efects of the position in the
z-direction at the time 7, because 1Ý g is a little farther back, there is a little
more retardation. 'This is an efect that we have neglected before, and ït is the
only change needed in order to make our results valid for all speeds.
What we must now do is to choose a certain value of £ and calculate the value
of r from it, and thus fnd out where z and ø are at that 7. These are then the
retarded z and ø, which we call z“ and z⁄, whose second derivatives debermine
the fñeld. 'Thus 7 is determined by
t=T+ Ro + zữ)
z() =z(). — w()=(). (31.4)
Now these are complicated equations, but it is easy enough to make a geometrical
picture to describe theïr solution. 'Phis picture will give us a good qualitative
feeling for how things work, but ít still takes a lot of detailed mathematics to
deduce the precise results of a complicated problem.
--- Trang 591 ---
x x'(t)
¬— c TÊN
'TO OBSERVER 0
Fig. 34-2. A geometrical solution of Eq. (34.5) to find x/(£).
34-2 Einding the “apparent” motion
The above equation has an interesting simplifcation. If we disregard the
uninteresting constant delay ?2o/c, which just means that we must change the
origin of ý by a constant, then i% says that
cÈ = œT + Z(T), z' = #(T), ự =9(1). (34.5)
NÑow we need to fnd zˆ and ø as functions of f, not 7, and we can do this in
the following way: Ed. (34.5) says that we should take the actual motion and
add a constant (the speed of light) times 7. What that turns out to mean is
shown in Fig. 34-2. We take the actual motion of the charge (shown at left) and
imagine that as it is going around it is being swept away from the point ? at
the speed e (there are no contractions from relativity or anything like that; this
is Just a mathematical addition of the cr). In this way we get a new motion,
in which the line-ofsight coordinate is cý, as shown at the right. (The figure
shows the result for a rather complicated motion ¡in a plane, but of course the
motion may not be in one plane—it may be even more complicated than motion
in a plane.) The point is that the horizontal (¡.e., line-of-sight) distance now is
no longer the old z, but is z + cr, and therefore is cứ. Thus we have found a
picture of the curve, #“ (and #') against ¿l All we have to do to fnd the field
1s to look at the acceleration of this curve, 1.e., to diferentiate I% twice. So the
fnal answer is: in order to find the electric feld for a moving charge, take the
motion of the charge and translate it back at the speed e to “open it out”; then
the curve, so drawn, is a curve of the #” and z positions of the function of¿. The
acceleration of this curve gives the electric field as a function of ý. Ôr, If we wish,
we can now imagine that this whole “rigid” curve moves forward at the speed