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cach point the fñeld is determined by the acceleration of the charge at an earlier |
time, the amount earlier being the delay r/c. The field at farther and farther |
points is determined by the acceleration at earlier and earlier times. So the curve |
in Eig. 29-3 is really, in a sense, a “reversed” plot of the acceleration as a function |
of time; the distanece is related to time by a constant scale factor c, which we |
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often take as unity. This is easily seen by considering the mathematical behavior |
of ø(£ — r/c). Evidently, if we add a little time Af, we get the same value for |
a(È — r/c) as we would have if we had subtracted a little distance: Ar = —c Ai. |
Stated another way: if we add a little time A£, we can restore œ(£ — r/e) to |
1ts former value by adding a little distance Az = cAf. Thhat is, as tỉme goes on |
the fteld mmoues qs a U0aue outuUard jrom the source. Thhat is the reason why we |
sometimes say light is propagated as waves. I% is equivalent to saying that the |
field is delayed, or to saying that the electric feld is moving outward as time |
ØO©s OH. |
An interesting special case is that where the charge g is moving up and down |
in an oscillatory manner. The case which we studied experimentally in the last |
chapter was one in which the displacement ø at any time ý was equal to a certain |
constant zọ, the magnitude of the oscillation, times cos/. 'Then the acceleration |
d = —02#0 COS UÉ — đọ COS UÉ, (29.2) |
where ứo is the maximum acceleration, —œ2#o. Putting this formula into (29.1), |
we fñnd (t— r/ |
. Œọ COS(U(È — rc |
1 =-qsin8 _.~ (29.3) |
Now, ignoring the angle Ø and the constant factors, let us see what that looks |
like as a function of position or as a function of time. |
29-2 Energy of radiation |
First of all, at any particular moment or in any particular place, the strength |
of the field varies inversely as the distance r, as we mentioned previously. NÑow |
we must point out that the energu content of a wave, or the energy efects that |
such an electrie field can have, are proportional to the sợuare of the field, because |
1Ý, for instance, we have some kind of a charge or an oscillator in the electric field, |
then I1f we let the field act on the oscillator, it makes it move. lf this is a linear |
oscillator, the acceleration, velocity, and displacement produced by the electric |
fñeld acting on the charge are all proportional to the feld. So the kinetic energy |
which is developed in the charge is proportional to the square of the fñield. Š5o we |
shall take it that the energy that a field can deliver to a system is proportional |
somehow to the square of the field. |
This means that the energy that the source can deliver decreases as we gøet |
farther away; in fact, 1t varles ?nuersclU as the square oƒ the đistance. But that |
--- Trang 500 --- |
Fig. 29-4. The energy flowing within the cone ABC D is independent |
of the distance r at which ït is measured. |
has a very simple interpretation: If we wanted to pick up all the energy we could |
from the wave in a certain cone at a distance ?¡ (Eig. 29-4), and we do the same |
at another distance r›, we ñnd that the amount of energy per unit area at any |
one place øoes inversely as the square of r, but the area of the surface intercepted |
by the cone goes đ/recfu as the square of z. So the energy that we can take out |
of the wave within a given conical angle is the same, no matter how far away |
we arel In particular, the total energy that we could take out of the whole wave |
by putting absorbing oscillators all around is a certain fñxed amount. So the |
fact that the amplitude of E varies as 1/7 is the same as saying that there is an |
energy fux which is never lost, an energy which goes on and on, spreading over a |
greater and greater effective area. Thus we see that after a charge has oscillated, |
1t has lost some energy which it can never recover; the energy keeps going farther |
and farther away without diminution. So ïÝ we are far enough away that our |
basic approximation is good enouph, the charge cannot recover the energy which |
has been, as we say, radiated away. Of course the energy still exists somewhere, |
and is available to be picked up by other systems. We shall study this energy |
“loss” further in Chapter 32. |
Let us now consider more carefully how the wave (29.3) varies as a function |
of time at a given place, and as a function of position at a given time. Again we |
ignore the 1/r variation and the constants. |
29-3 Sinusoidal waves |
Pirst let us ñx the position r, and watch the field as a function of time. Ït is |
oscillatory at the angular frequency œ. The angular frequency œ can be defned |
--- Trang 501 --- |
as the ra£© oƒ chưnge öoƒ phase tuïth từme (radians per second). We have already |
studied such a thing, so it should be quite familiar to us by now. 'Phe per?od is |
the time needed for one oscillation, one complete cycle, and we have worked that |
out too; it is 2#/œ, because œ times the period is one cycle of the cosine. |
Now we introduce a new quantity which is used a great deal in physics. This |
has to do with the opposite situation, in which we fix £ and look at the wave |
as a function of distance r. Of course we notice that, as a function of r, the |
wave (29.3) is also oscillatory. That is, aside from 1/r, which we are ignoring, we |
see that # oscillates as we change the position. So, in analogy with œ, we can |
defne a quantity called the 0œøe rruwmber, symbolized as k. 'This is deñned as ứhe |
rake oƒ change oƒ phase tuïth distœnce (radians per meter). 'That 1s, as we move |
in space at a fñxed time, the phase changes. |
'There is another quantity that corresponds to the period, and we might call |
1t the period in space, but it is usually called the wavelength, symbolized À. The |
wavelength is the distance occupied by one complete cycle. Ït is easy tO see, |
then, that the wavelength is 2Z/&k, because & times the wavelength would be the |
number of radians that the whole thing changes, being the product o£ the rate of |
change of the radians per meter, times the number of meters, and we must make |
a 27 change for one cycle. So &À = 27 is exactly analogous to ¿fọ = 27. |
Now in our particular wave there is a definite relationship between the fre- |
quency and the wavelength, but the above definitions of k and œ are actually |
quite general. 'Phat is, the wavelength and the frequency may not be related in |
the same way in other physical circumstances. However, in our circumstance |
the rate of change of phase with distance is easily determined, because if we call |
Ó = w(t — r/c) the phase, and diferentiate (partially) with respect to distance r7, |
the rate of change, Øj/Ôr, is |
lg|=£=Š: (29.4) |
There are many ways to represent the same thing, such as |
À =cío (29.5) À/=ec (29.7) |
œ = €k (29.6) œÀ = 27c (29.8) |
'Why is the wavelength equal to e times the period? 'Phat”s very easy, Of course, |
because if we sit still and wait for one period to elapse, the waves, travelling at |
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