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

In summary, a colleague and I are considering a 1D model of a hot electron beam interacting with a charge neutral plasma. The equations we are using involve the speed of the electrons, number density, and electric field. However, it was brought to our attention that Ohm's law is not relativistically invariant, but there is a way to make it so. We are wondering if our equations need to be adjusted for this.
  • #1
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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:
[tex]
\begin{array}{rcl}
\gamma^{3}(v/c)\left(\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}\right) & = & -\frac{e}{m}E \\
\frac{\partial n}{\partial t}+\frac{\partial }{\partial x}(nv) & = & 0 \\
\frac{\partial E}{\partial t}+\frac{E}{\eta\epsilon_{0}} & = & \frac{nev}{\epsilon_{0}}
\end{array}
[/tex]
where e is the charge on the electron, m is the mass of the electron, [itex]\eta[/itex] 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?"
 
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  • #2
in the sum on the left hand side of your first equation the units don't match.
 
  • #3
Yes they do. [itex]\gamma[/itex] has no units, the [itex]\partial_{t}y}[/itex] and [itex]v\partial_{x}v[/itex] 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.
 

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

1. What is Relativistic Ohm's Law?

Relativistic Ohm's Law is a mathematical equation that describes the relationship between electric current, electromagnetic fields, and the properties of a plasma. It is used to model the behavior of a 1D hot electron beam in plasma.

2. Why is the modeling of 1D hot electron beams in plasma important?

Understanding the behavior of hot electron beams in plasma is crucial in many areas of science, such as fusion research and space physics. The modeling of these beams can help us predict and control their behavior, leading to potential advancements in technology and energy production.

3. How does Relativistic Ohm's Law take into account the relativistic effects?

In the relativistic regime, the velocity of particles approaches the speed of light, and the traditional form of Ohm's Law is no longer applicable. Relativistic Ohm's Law takes into account the effects of special relativity, such as length contraction and time dilation, to accurately model the behavior of hot electron beams in plasma.

4. What are some applications of Relativistic Ohm's Law in research?

Relativistic Ohm's Law is used in various research areas, including plasma physics, astrophysics, and fusion energy. It has been applied to study the behavior of plasma in laboratory experiments, to understand the dynamics of astrophysical jets, and to improve the efficiency of fusion energy production.

5. Are there any limitations to Relativistic Ohm's Law?

Like any scientific model, there are limitations to Relativistic Ohm's Law. It assumes certain simplifications, such as a 1D electron beam and a uniform plasma, which may not always accurately reflect real-world conditions. Additionally, it may not be applicable in extreme relativistic regimes, where quantum effects become significant.

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