I Do electron density waves accompany EM waves in coaxial cables?

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Maxwell's equations suggest a wave equation for the electric field that can be divided into transverse and longitudinal components. The transverse part describes how electromagnetic (EM) waves propagate, while the longitudinal part involves electron density waves in conductors. By applying the continuity equation, a wave equation for electron density is derived, indicating that these density waves travel at the speed of light. The discussion concludes that in coaxial cables, electron "waves of compression" accompany the transverse EM waves. This phenomenon occurs whenever the applied voltage changes across any conductor.
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Maxwell’s equations imply the following wave equation for the electric field
$$\nabla^2\mathbf{E}-\frac{1}{c^2}\frac{\partial^2\mathbf{E}}{\partial t^2}
= \frac{1}{\varepsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf J}{\partial t}.\tag{1}$$
I wonder if eqn.##(1)## can be split into the following transverse part
$$\nabla^2\mathbf{E}_T-\frac{1}{c^2}\frac{\partial^2\mathbf{E}_T}{\partial t^2}
= \mu_0\frac{\partial\mathbf{J}_T}{\partial t}\tag{2}$$
and longitudinal part
$$\frac{1}{\varepsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf{J}_L}{\partial t}=0.\tag{3}$$
Taking the divergence of eqn.##(3)## and substituting in the continuity equation ##\nabla \cdot \mathbf{J}_L=-\partial\rho/\partial t## we obtain a wave equation
$$\nabla^2\rho-\frac{1}{c^2}\frac{\partial^2\rho}{\partial t^2}=0.\tag{4}$$
Do these equations describe how transverse EM waves ##(2)## travel down the dielectric in a coaxial cable accompanied by electron density waves ##(4)## in the conductors?
 
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Typo in line 4. That wave is moving at the speed of light.
 
As far as I can see, I think you do get an electron "wave of compression" travelling along a transmission line in addition to the TEM wave. The longitudinal E-field is developed across the inductance-per-unit-length of the line. This applies to any conductor when the applied voltage varies.
 
DaveE said:
Typo in line 4. That wave is moving at the speed of light.
I guess I should use ##\epsilon##, ##\mu## and ##c## appropriate to the material in the coaxial cable.
 
I think so. I believe the wave has the same propagation constants and Zo as the TEM wave, so you don't notice it. It is of small magnitude in a coaxial cable but is important for widely spaced conductors.
 
Bob44 said:
I guess I should use ##\epsilon##, ##\mu## and ##c## appropriate to the material in the coaxial cable.
The velocity must match that of the EM wave which is ##\frac{1}{\sqrt{\epsilon \mu}}##. If your derivation doesn't produce that then I think it's wrong.
 
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