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For a system of charges and currents varying in time,
we cam handle each Fourier component separately.
Consider the potentials, fields, and radiation from a
localized system of charges and currents that vary
sinusoidally in time:
It was shown in Chapter 6 that the solution for the
vector potential A(x, t) in the Lorenz gauge is
provided no boundary surfaces are present. With the
sinusoidal time dependence (9.1), the solution for A
becomes
where k = ω/c is the wave number, and a sinusoidal
time dependence is understood.
The magnetic field is given by
while, outside the source, the electric field is
where
is the impedance of free space.
Given a current distribution J(x’), the field can, in
principle at least, be determined by calculating the
integral in (9.3). But in general, properties of the fields
in the limit that the source of current is confined to a
small region, very small in fact compared to a
wavelength.
If the source dimensions are of order d and the
wavelength is
and if d << λ, then there are
three spatial regions of interest:
We will see that the fields have very different
properties in the different zones.
For the near zone where r << λ the exponential in (9.3)
can be replaced by unity. The inverse distance can be
expanded using (3.70), with the result,
This shows that the near fields are quasi-stationary,
oscillating harmonically as e-iωt, but otherwise static in
character.
In the far zone (kr >> 1) the exponential in (9.3)
oscillates rapidly and determines the behavior of the
vector potential. It is sufficient to approximate
where n is a unit vector in the direction of x. The
inverse distance in (9.3) can be replaced by r. Then the
vector potential is
This demonstrates that in the far zone the vector
potential behaves as an outgoing spherical wave with
an angular dependent coefficient.
It’s easy to show that the fields calculated from (9.4)
and (9.5) are transverse to the radius vector and fall off
as r-1. If the source dimensions are small compared to
a wave length it is appropriate to expand the integral
in (9.8) in powers of k:
The magnitude of the nth term is given by
Since the order of magnitude of x’ is d and kd is small
compared to unity by assumption, the successive terms
in the expansion of A evidently fall off rapidly with n.
In the intermediate or induction zone the two
alternative approximations leading to (9.6) and (9.8)
cannot be made; all powers of kr must be retained.
The key result is the exact expansion (9.98) for the
Green function appearing in (9.3). For points outside
the source (9.3) then becomes
If the source dimensions are small compared to a
wavelength, jl(kr’) can be approximated by (9.88).
Then the result for A is (9.6) with the replacement
The right-hand side shows the transition from staticzone result (9.6) for kr << 1 to the radiation-zone form
(9.9) for kr >> 1.
The analog of (9.2) for the scalar potential is
The electric monopole contribution is obtained by
replacing |x – x’| → |x| ≡ r under the integral. The
result is
where q(t) is the total charge of the source. Since
charge is conserved and it’s a localized source, the
total charge q is independent of time. Thus the electric
monopole part of the potential (and fields) of a
localized source is of necessity static.
If only the first term in (9.9) is kept, the vector
potential is
Examination of (9.11) and (9.12) shows that (9.13) is
the l = 0 part of the series and that it is valid
everywhere outside the source, not just in the far zone.
The integral can be put in more familiar terms by an
integration by parts:
since from the continuity equation,
Thus the vector potential is
where
is the electric dipole moment.
The electric dipole fields from (9.4) and (9.5) are
In the radiation zone the fields take on limiting forms,
showing the typical behavior of radiation fields.
In the near zone, on the other hand, the fields approach
The electric field, apart from its oscillations in time, is
just the static electric dipole field (4.13).
The fields in the near zone are dominantly electric in
nature.
The time-averaged power radiated per unit solid angle
by the oscillating dipole moment p is
where E and H are given by (9.19). Thus we find
If the components of p all have the same phase, the
angular distribution is a typical dipole pattern,
where the angle θ is measured from direction of p.
The total power radiated, independent of the relative
phases of the components of p, is
A simple example of an electric dipole radiator is a
center-fed, linear antenna, as shown in Fig. 9.1.
The current is in the same direction in each half of the
antenna, having a value I0 at the gap and falling
approximately linearly to zero at the ends:
From the continuity equation the linear-charge density
ρ’ (charge per unit length) is constant along each arm
of the antenna, with the valu,
the upper (lower) sign being appropriate for positive
(negative) values of z.
The dipole moment (9.17) is parallel to the z axis and
has the magnitude
The angular distribution of radiated power is
while the total power radiated is
We see that for a fixed input current the power
radiated increase as the square of the frequency, at
least in the long-wavelength domain where kd <<1
The customary basic situation is for a plane
monochromatic wave to be incident on a scatterer. For
simplicity the surrounding medium is taken to have
If the incident direction is defined by the unit vector n0,
and the incident polarization vector is ε0, the incident
fields are
where k = ω/c and a time-dependence e-iωt is
understood.
Far away from the scatterer, the scattered (radiated)
fields are found from (9.19) and (9.36) to be
where n is a unit vector in the direction of observation
and r is the distance away from scatterer.
The power radiated in the direction n with polarization
ε, per unit solid angle, per unit incident flux (power
per unit area) in the direction n0 with polarization ε0, is
called the differential scattering cross section:
With (10.1) and (10.2), the differential cross section
can be written
The dependence on frequency is called Rayleigh’s law.
We say that the laws of mechanics are invariant under
Galilean transformations. For two reference frames K
and K’ with coordinates (x, y, z, t) and (x’, y’, z’, t),
respectively, and moving with relative velocity v, the
space and time coordinates in the two frames are
related according to Galilean relative by
As an example of a mechanical system, consider a
group of particles interacting via two-body central
potentials. In an obvious notation the equation of
motion of the ith particle in the reference frame K’ is
From the connections (11.1) between the two
coordinates it is evident that
Thus (11.2) can be transformed into
Suppose that a field ψ(x’, t’) satisfies the wave
equation in the reference frame K’.
By straightforward use of (11.1), in terms of the
coordinates in the reference frame K the wave
equation becomes:
No kinematic transformation of ψ can restore to (11.5)
the appearance of (11.4).
Einstein began to think about these matters there
existed several possibilities:
1.The Maxwell equations were incorrect.
2.Galilean relativity applied to electromagnetism in a
preferred reference frame, in which the luminiferous
ether was at rest.
3.There existed relativity principle for both mechanics
and electromagnetism, but it was not Galilean
relativity.
The second was accepted by most physicists of time.
FitzGerald-Lorentz contraction hypothesis (1892)
where by objects moving at a velocity v through the
ether are contracted in the direction of motion
according to the formula
This contraction held for moving charge densities.
Some experiments convinced Einstein of the
unacceptability of the hypothesis of an ether. He chose
the third alternative above and sought principles of
relativity that would encompass classical mechanics,
electrodynamics, and indeed all natural phenomena.
Einstein’s special theory of relativity is based on two
postulates:
1.Postulate of relativity.
2.Postulate of the constancy of the speed of light.
Because special relativity applies to everything, not
just light, it is desirable to express the second postulate
in terms that convey its generality.
2’.Postulate of a universal limiting speed
If there is a plane electromagnetic wave in vacuum its
phase as observed in the inertial frames K and K’,
connected by the Galilean coordinate transformation
(11.1), is
If t and x are expressed in terms of t’ and x’ from
(11.1), we obtain
Since this equality must hold for all t’ and x’, we
therefore find
These are the standard Doppler shift formulas of
Galilean relativity.
The unit wave normal n is seen to be an invariant in
all inertial frames. However, the direction of energy
flow changes from frame to frame. Consider the
segments of a plane wave sketched in Fig. 11.1.
At t = t’ = 0 the center of the segment is at the point A
in both K and K’. The direction of motion of the wave
packet, assumed to be the direction of energy flow, is
thus not parallel to n in K’, but along a unit vector m
shown in Fig. 11.1 and specified by
It’s convenient to have (11.8) expressed in terms of the
m appropriate to the laboratory rather than n. It’s
sufficient to have n in terms of m correct to first order
in v/c. We find
Where v0 is the velocity of the laboratory relative to
the ether rest frame
Consider now a plane wave whose frequency is ω in
the ether rest frame, ω0 in the laboratory, and ω1 in an
inertial frame K1 moving with a velocity v1 = u1 + v0
relative to the ether rest frame. From (11.8) the
observed frequencies are
If ω1 is expressed in terms of the laboratory frequency
ω0 correct to order v2/c2, is easily shown to be
From (11.11) the difference in frequency between
emitter and absorber is
If the emitter and absorber
are located on the opposite
ends of a rod of length 2R
that is rotated about its center
with angular velocity Ω, as
indicated in Fig.11.2
Then (u2-u1)˙m = 0 and the fractional frequency
difference is
Extraterrestrial light sources (sun or other stars) and
light from binary stars as establishing the second
postulate and ruling out Ritz’s theory. Unfortunately, it
seems clear that most of the early evidence is invalid
because of the interaction of the radiation with the
matter through which it passes before detection.
(CERN in 1964) The speed of 6 GeV photons
produced in the decay of very energetic neutral pions
was measured by time of flight over paths up to 80
meters. The pions were produced by bombardment of
a beryllium target by 19.2 GeV and had speeds of
0.99975c
One can ask whether there is any evidence for a
frequency dependence of the speed of electromagnetic
waves in vacuum. One possible source of variation is
attributable to a photon mass. The group velocity in
this case is
where the photon rest energy is ħω0.
Another source of frequency variation in the speed of
light is dispersion of the vacuum. The discovery of
pulsars make it possible to test this idea with high
precision. Pulsar observation cover at least 13 decades
of frequency. Variation on the speed of light for two
frequencies ω1 and ω2 is:
Where Δt is the pulse duration and D is the distance
from the source to observer. For the Crab pulsar Np
0532, Δt ≈ 0.003 s and D ≈ 6000 light-years so that
(c Δt/D ) ≈ 1.7 × 10-14.