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Old retarded potential section 1_4 REVIEWED
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Short note dated 9.19.13 by Phil, saved as the original version of section 1.4 of the Transmission Lines chapter 1 notes and marked as discardable. It defines a time-domain Green's function for the wave equation with a delta source, gives the retarded solution G = (1/R) δ(t' + R/v - t), and writes the potentials φ and A from it. It explains why the frequency domain is preferred when loss and frequency-dependent μ and ε are present.
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Old retarded potential section 1.4 PhL 9.19.13
This is just storage of the original section 1.4, can be thrown eventually.
1.4 The Retarded Potentials: a path not taken
In simple situations it is possible to solve the above wave equations for φ and A directly in the time-domain. For example, if σ = 0 ( so Ja = JT) and μ and ε are not time dependent one can define a Green's Function G by
( 2 - με ∂t2) G(x,t;x',t') = - 4πδ(3)(x-x') δ(t-t') (1.4.1)
[ In light of our previous note, this is ∂2G(xμ; x'μ) = - 4π δ(4)(xμ-x'μ).] The solution of (1.4.1) is
G(x,t;x',t') = (1/R) δ( t' + (R/v) - t ) (1.4.2)
where R = |x-x'|, and v = c/= the phase velocity in the medium . This lets one write down the solution to either potential equation as follows:
φx,t = (1.4.3)
A(x,t) = (1.4.4)
These expressions for the potentials are typically used in radiation problems to find the fields of accelerating particles or antennas. Notice that the time in the sources is "retarded" by R/v.
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 potentials.