# Two field form of the Maxwell equations: Why are effects due to polarisation ignored.

I have a couple of questions regarding the following section in my notes and would like some help in resolving some confusions.

1)why is εr = 1 in the plasma?

2) If µr = 1 it means we switched off magnetic effects so the conduction current density is not 0 hence it is included in the fourth Maxwell equation. Furthermore magnetisation current density is 0 because magnetic effects are ignored but i do not get why polarisation current density is 0. For a similar reason I also do not get why in the first Maxwell equation polarisation charge density is equal to zero.

• $\varepsilon_{r} \neq 1$ in a plasma unless the frequency of oscillations is well above the electron plasma frequency. That is, only for $\omega > \omega_{pe}$ are electromagnetic waves free modes. Meaning, they can propagate as if in vacuum (ignoring Faraday rotation). – honeste_vivere Apr 25 '18 at 13:17

both 1) and 2) are essentially high frequency approximation. certainly not true in general. For instance, a common expression for the plasma permittivity is $$\epsilon = 1 - \omega_p^2/\omega^2$$ You can see what happens when the $\omega$ is much larger than the plasma frequency. But even this expression already has assumed that the frequency is much larger than the inverse of collisional relaxation time. This causes $\epsilon(\omega)$ to be real, which violates causality. If you want to see the full derivation in both time-domain and frequency-domain check out this paper on linear response laws and electrodynamics. A number of examples are worked out in detail. BTW, I prefer to work in CGS where $\epsilon$ and $\mu$ are dimensionless and $D$ and $E$ have the same units. Much more natural.

added: Here is a nice plasma frequency calculator. It is not hard to work near the plasma frequency of many materials by choosing the right EM frequency. This can lead to very interesting ENZ (epsilon near zero) situations. Not what you asked, but cool nonetheless.

You are showing the 4 Maxwell equations for homogeneous isotropic linear media and you are assuming that there are no magnetic materials, i.e. $\mu_r=1$.

ad (1): In plasmas very often $\epsilon_r =1$ is assumed which is a good approximation for gas plasmas of low density. If you consider a plasma in a solid, like a metal or semiconductor, you have to include $\epsilon_r \gt 1$ for the displacement current and in the plasma frequency formula.

ad (2): The polarization current density $$\frac {\partial \vec P}{\partial t}=\epsilon_0\chi\frac {\partial \vec E}{\partial t}=\epsilon_0(\epsilon_r-1)\frac {\partial \vec E}{\partial t}$$ is included in your 4th Maxwell equation, it is not zero! It is part of the total displacement current density $$\epsilon_0\epsilon_r\frac {\partial \vec E}{\partial t}=\epsilon_0\chi\frac {\partial \vec E}{\partial t}=\epsilon_0(\epsilon_r-1)\frac {\partial \vec E}{\partial t}+\epsilon_0\frac {\partial \vec E}{\partial t}$$ Likewise the polarization charge $\rho_b=-\nabla \vec P$ is not zero, it is also included by using $\epsilon_r \gt 1$ in the first Maxwell equation $$\nabla·\epsilon_0 \epsilon_r \vec E=\nabla·\epsilon_0 (1+\chi) \vec E=\nabla·\epsilon_0 \vec E+\nabla·\epsilon_0 \chi \vec E=\nabla·\epsilon_0 \vec E+\nabla·\vec P=\nabla·\epsilon_0 \vec E-\rho_b=\rho_f$$ Thus there is no missing polarization current and polarization charge in these equations.