Relativistic Ohms Law: Modeling 1D Hot Electron Beam in Plasma

[hunt_mat]

A colleague and I are looking at modelling a hot electron beam hitting a initially charge neutral plasma. Initially we're looking at the 1D problem, the equations we're using are:
<br /> \begin{array}{rcl}<br /> \gamma^{3}(v/c)\left(\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}\right) &amp; = &amp; -\frac{e}{m}E \\<br /> \frac{\partial n}{\partial t}+\frac{\partial }{\partial x}(nv) &amp; = &amp; 0 \\<br /> \frac{\partial E}{\partial t}+\frac{E}{\eta\epsilon_{0}} &amp; = &amp; \frac{nev}{\epsilon_{0}}<br /> \end{array}<br />
where e is the charge on the electron, m is the mass of the electron, \eta is the resistivity of the charge neutral plasma, v is the speed of the electrons, n is the number density of the electrons and E is the electric field produced by the electron beam. It was brought to our attention that Ohms law is not relativitically invariant but it is possible to make it so.

So my question is this, "Are the equations we have correct for a simple 1D model or do we need to change something?"

 

[Phrak]

in the sum on the left hand side of your first equation the units don't match.

 

[hunt_mat]

Yes they do. \gamma has no units, the \partial_{t}y} and v\partial_{x}v both have units of acceleration. The first is LT^(-2), and the other is LT-1(LT^-1/L)=LT^(-2).
The units on the RHS are Newtons (eE and m is mass), so that side is acceleration also.