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the only thing that can affect the fñield at a given place and time is the behavior |
of the charges in the øasứ. How ƒár in the past? "The time delay, or refarded |
time, so-called, is the time it takes, at speed e, to get from the charge to the field |
point ?P. The delay is ?//e. |
So to allow for this time delay, we put a littÌe prime on r, meaning how far |
away 1% uas when the information now arriving at left g. Just for a moment |
suppose that the charge carried a light, and that the light could only come |
to at the speed c. Then when we look at g, we would not see where ït 1s |
now, of course, but where it Ͽs at some earlier time. What appears in our |
formula is the apparen‡ direction ezz—the direction it used to be—the so-called |
retarded direction——and at the retarded distance r7. That would be easy enough to |
understand, too, but it is also wrong. The whole thing is much more complicated. |
There are several more terms. The next term is as though nature were trying |
to allow for the fact that the efect is retarded, ¡f we might put it very crudely. It |
suggests that we should calculate the delayed Coulomb field and add a correction |
to it, which is its rate of change times the time delay that we use. Nature seems |
to be attempting to guess what the field at the present time is going to be, by |
taking the rate of change and multiplying by the time that ¡is delayed. But we |
are not yet through. 'Phere is a third term——the second derivative, with respect |
to ứ, of the unit vector in the direction of the charge. Now the formula ¡s finished, |
and that is all there is to the electric ñeld from an arbitrarily moving charge. |
'The magnetic field is given by |
B=-e,: x Eực. (28.4) |
We have written these down only for the purpose of showing the beauty of nature |
or, in a way, the power of mathematics. We do not pretend to understand :ø0hø it |
is possible to write so much in such a small space, but (28.3) and (28.4) contain |
the machinery by which electric generators work, how light operates, all the |
phenomena of electricity and magnetism. Of course, to complete the story we |
also need to know something about the behavior of the materials involved——the |
properties of matter——which are not described properly by (28.3). |
To fñnish with our description of the world of the 19th century we must |
mention one other great synthesis which occurred in that century, one with which |
Maxwell had a great deal to do also, and that was the synthesis of the phenomena, |
of heat and mechanics. We shall study that subject soon. |
'What had to be added in the 20th century was that the dynamical laws of |
Newton were found to be all wrong, and quantum mechanies had to be introduced |
--- Trang 489 --- |
to correct them. Newton'”s laws are approximately valid when the scale of things |
1s sufficiently large. These quantum-mechanical laws, combined with the laws of |
electricity, have only recently been combined to form a set of laws called guantwm |
clectrodunøam#cs. In addition, there were discovered a number of new phenomena, |
of which the first was radioactivity, discovered by Becquerel in 1898——he just |
sneaked 1t in under the 19th century. 'PThis phenomenon of radioactivity was |
followed up to produce our knowledge of nuclei and new kinds of forces that are |
not gravitational and not electrical, but new particles with diferent interactions, |
a subJect which has still not been unravelled. |
Eor those purists who know more (the professors who happen to be reading |
this), we should add that when we say that (28.3) is a complete expression of the |
knowledge of electrodynamies, we are not being entirely accurate. There was a |
problem that was not quite solved at the end of the 19th century. When we try |
to calculate the ñeld from all the charges ?ncluding the charge itselƒ that tue tuanứ |
the ficld to ac† on, we get into trouble trying to fnd the distance, for example, of |
a charge om itself, and dividing something by that distance, which is zero. The |
problem of how to handle the part of this fñeld which ¡is generated by the very |
charge on which we want the field to act is not yet solved today. So we leave 1t |
there; we do not have a complete solution to that puzzle yet, and so we shall |
avoid the puzzle for as long as we can. |
28-2 Radiation |
That, then, is a summary of the world picture. Now let us use it to discuss |
the phenomena called radiation. To discuss these phenomena, we must select |
from Eq. (28.3) only that piece which varies inversely as the distance and not as |
the square of the distance. lt turns out that when we fñnally do fnd that piece, it |
1s so simple in its form that it is legitimate to study optics and electrodynamics |
in an elementary way by taking it as “the law” of the electric ñeld produced by a |
moving charge far away. We shall take it temporarily as a given law which we |
will learn about in detail next year. |
Of the terms appearing in (28.3), the first one evidentÌy goes inversely as |
the square of the distance, and the second is only a correction for delay, so 1E |
1s easy to show that both of them vary inversely as the square of the distance. |
All of the efects we are interested in come from the third term, which is not |
very complicated, after all. What this term says 1s: look at the charge and note |
the direction of the unit vector (we can project the end of it onto the surface of |
--- Trang 490 --- |
a unit sphere). As the charge moves around, the unit vector wiggles, and fhe |
acceleration oƒ that ni 0ector is that tục are looking or. Phat is all. Thus |
q d2e„› |
⁄= 4mcạc2 d2. (285) |
1s a statement of the laws of radiation, because that is the only important term |
when we get far enough away that the fñelds are varying inversely as the distance. |
(The parts that go as the square have fallen off so mụuch that we are not interested |
in them.) |
NÑow we can go a little bít further in studying (28.5) to see what it means. |
Suppose a charge is moving in any manner whatsoever, and we are observing it |
from a distance. We imagine for a moment that in a sense it is “li up” (although |
1t is light that we are trying to explain); we imagine it as a little white dot. Then |
we would see this white dot running around. But we don” see ezøcfg how it is |
running around right =øu, because of the delay that we have been talking about. |
What counts is how iÿ was moving earler. The unit vector e„; is pointed toward |
the apparent position of the charge. Of course, the end of ez:; goes on a slipght |
curve, so that its acceleration has two components. One is the transverse piece, |
because the end of it goes up and down, and the other is a radial piece because |
1t stays on a sphere. Ït is easy to demonstrate that the latter is much smaller |
and varies as the inverse square of? when r7 is very great. This is easy to see, Íor |
when we imagine that we move a given source farther and farther away, then the |
wigplings of ez; look smaller and smaller, inversely as the distance, but the radial |
component of acceleration is varying much more rapidly than inversely as the |
distance. So for practical purposes all we have to do is project the motion on a |
plane at unit distance. 'Therefore we fñnd the following rule: Imagine that we look |
at the moving charge and that everything we see is delayed——like a painter trying |
to paint a scene on a screen at a unit distance. Á real painter, oŸ course, does |
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