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

This demonstrates that in the far zone the vector potential behaves as an outgoing spherical wave with an angular dependent coefficient.

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

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 static-zone 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

Then (u2-u1)˙m = 0 and the fractional frequency difference is

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

(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

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.

where the photon rest energy is ħω0.

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 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.

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: