I have tried to derive, in a particular form, an equation for the electric field of light propagating in a dielectric. Unfortunately, for my intended purpose, for the case of electromagnetic wave propagation, in a linear, isotropic dielectric, I appear to have a ‘grad div E’ term in the equation.
If you consider plane waves propagating in the z-direction with an electric vector in the x-direction, then the ‘grad div E’ term becomes zero.
My question is, ’What is the best way to handle the ‘grad div E’ term?’
See the attachment for more detail and references.
Also, how should I derive the propagation equations for light in a magnetic medium, with relative permeability μ≠1?
I'm afraid your document opened without revealing any of the equations. They were just blank spaces. Can you make a PDF of it?
PDF attached.
NB: I now think I know how to handle the magnetic media case I mentioned. Chapter 36 of reference 1 was illuminating, concerning what currents might appear in Maxwell's equations.
1) R.P Feynman, R.B. Leighton, M. Sands, Volume II, The Feynman Lecture On Physics, Addison-Wesley, 1964.
For some reason, in answer I gave to your question did not get printed.
That answer was "div E=0, if there is no free charge in the medium."
If \mu is constant, I don't see any new problem even if \mu doesn't equal one.
In electromagnetic radiation, E is transverse to the polar origin, which means that div E will be equal to zero. The polar origin lies at the centre of each of Maxwell's tiny vortices that fill all of space. In electromagnetic radiation, E is not a radial force arising from Coulomb's law.
If I use the fact that the polarisation, P, is proportional to E, in EQ (1) of either attachment to previous posts of this topic, then I obtain that div E = 0. Hence the offending term, grad div E, drops out.
I think that P being proportional to E is what is meant by a linear dielectric, which you specified when you defined your problem. The fact that it is homogeneous too means div E is zero, as there are no boundaries or gradients where there will be polarization charges which give non-zero div E. The linear, homogeneous condition is probably equivalent to specifying div E = 0. In the magnetic case it is probably equivalent to div H = 0.
I did not make full use of the linearity of the dielectric, in setting up the equations.
EQ (1) should be altered to “div E = 0” (ish)
NB: With light passing through a dielectric you have a time varying electric field, of the light, polarizing the medium. So, at any given time, P will vary from place to place within the dielectric.
The traversal of light would also make the magnetization of a magnetic material vary from point to point at a given time, because of its magnetic field.
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