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Recall the following morphs of Maxwell's equations, this time with
the sources and expressed in terms of potentials by means of the
homogeneous equations. Gauss's Law for magnetism is:
|
(11.3) |
This is an identity if we define
:
|
(11.4) |
Similarly, Faraday's Law is
and is satisfied as an identity by a scalar potential such that:
Now we look at the inhomogeneous equations in terms of the potentials.
Ampere's Law:
|
|
|
(11.10) |
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(11.11) |
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|
(11.12) |
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|
(11.13) |
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(11.14) |
Similarly Gauss's Law for the electric field becomes:
In the the Lorentz gauge,
|
(11.18) |
the potentials satisfy the following inhomogeneous wave equations:
where and
are the charge density and current density
distributions, respectively. For the time being we will stick with
the Lorentz gauge, although the Coulomb gauge:
|
(11.21) |
is more convenient for certain problems. It is probably worth reminding
y'all that the Lorentz gauge condition itself is really just one out of
a whole family of choices.
Recall that (or more properly, observe that in its role in these wave
equations)
|
(11.22) |
where is the speed of light in the medium. For the time being,
let's just simplify life a bit and agree to work in a vacuum:
|
(11.23) |
so that:
If/when we look at wave sources embedded in a dielectric medium, we can
always change back as the general formalism will not be any different.
Next: Green's Functions for the
Up: Maxwell's Equations, Yet Again
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Robert G. Brown
2007-12-28