# 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.
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.