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A short Word document dated 9.19.13, marked by Phil as his working document for updating this section, which is now installed in the Chapter 1 notes. It outlines solving the wave equations for A and φ using a retarded Green's function, with source values evaluated at time t - R/v, and compares the result to the electrostatic Poisson solution. It explains why the approach is not used: with loss and frequency-dependent μ and ε, the frequency domain is preferable.
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New Retarded path not taken PhL 9.19.13
I think this is my working doc for updating this section. It is now installed.
1.4 Retarded Solutions: a path not taken
In a medium with σ = 0 and in which μ and ε are time-independent, equations (1.3.7) and (1.3.8) apply,
2 - με ∂t2)A = - μJT (1.3.7)
(2 - με ∂t2)φ = - (1/ε)ρT (1.3.8)
One approach to solving these equations for A and φ is the method of retarded solutions. Although we shall not use this approach, we outline the method here. We seek to solve an equation of this form
(2 - με ∂t2) u = - f (1.4.1)
where for example in (1.3.8) we have u = φ and f = ρT/ε. Since με = 1/v2 where v is the wave velocity in the medium, write (1.4.1) as
(∂t2 - v22) u = v2f
or
u = f where ≡ ∂t2 - 2 . (1.4.2)
This last equation is similar to (A.7.2) of Appendix A and we solve it in the same manner. Define a Green's function g as the solution of
v2 g(x,t; x',t') = δ(x-x')δ(t-t') . (1.4.3)
As Appendix A section 7 shows, the solution is given by
v2 g(x,t; x',t') = (1/4πR)δ(t-t'-R/v) with R = |x-x'| . (1.4.4)
Jackson (6.41) and (6.44) uses G(+) = 4πv2g with v = c and refers to the solution as a "retarded Green function". See also Stakgold references in Appendix A. Then the solution to (1.4.1) is
u(x,t) = ∫d3x' ∫dt' v2 g(x,t; x',t') f(x',t')
as can be verified by applying to both sides and making use of (1.4.3). Inserting (1.4.4) then gives
u(x,t) = ∫d3x' ∫dt' (1/4πR)δ(t-t'-R/v) f(x',t')
= ∫d3x' (1/4πR) f(x',t-R/v)
= ∫d3x' (1.4.5)
Thus, the solutions to (1.3.7) and (1.3.8) are
A(x,t) = ∫d3x' (1.4.6)
φ(x,t) = ∫d3x' (1.4.7)
The potentials at time t are generated by the values the sources had at time t - R/v since the influence of the sources travels at finite velocity v through the medium. Compare (1.4.7) to (A.0.2) which is the solution to the electrostatic Poisson equation.
With the medium σ loss term present, and with frequency dependent (hence time dependent) μ and ε, it is better to work in the frequency domain instead of the time domain. Thus, although they are interesting, we shall not use these retarded solutions.