consider no magnetic sources
gauss's law for magnetic fields indicates mu*H is divergenceless, and thus can be written as the curl of a vector field Ae
faraday's law implies Ee + j * w * Ae is curlless, and thus can be written in terms of the gradient of a scalar field phi_e
thus Ee can be written in terms of potentials
the above equations impose no restrictions on the divergence of Ae (think helmholtz decomposition, but i need to make this more clear in my notes)
so impose the lorenz gauge. this is arbitrary, but makes the math simple as follows
rewrite rho_e in terms of div(A_e), plug this into the expression for Ee, and plug this into ampere's law to derive a vector PDE
rewrite div(A_e) in terms of rho, plug this into the expression for Ee, and plug this into gauss's law for electricity to get a scalar PDE
the result is a vector and a scalar inhomogeneous helmholtz equation. if we do math in rectangular coordinates, we see that we really have 4 versions of the same scalar helmholtz equation. this is why the lorenz gauge is chosen
this can be done for no electric sources to, and you can define the potentials Am and pho_m
>>12765469thanks, and i agree. the EEs got the sign backwards and should have used i.
