# Charge conservation

1. Jul 15, 2010

### hunt_mat

I have been working with someone on plasma physics. We have a simple model of a laser hitting a charge neutral plasma. The laser promotes the electrons in the plasma into a high energy electron beam. We have been looking at the problem in 1D using the Lorentz force law, the conservation of number density and Ampere's law, these form a set of first order hyperbolic PDEs.

My colleague is concerned with conservation of charge, he says that as the plasma was initially charge neutral then the total charge (included in the electron beam) must remain zero. I thought that all we had to show was that:
$$\nabla\cdot\mathbf{J}+\frac{\partial\rho}{\partial t}=0$$
He calculated
$$Q=\int_{0}^{\infty}\rho dx$$
He wanted this to be zero. Is this right?

2. Jul 15, 2010

### Antiphon

The first equation is local charge conservation and is strictly obeyed. The second one is global charge conservation. Using the second equation would allow you to create a positive charge on one place and a negative charge someplace else. Since simultaneity is observer-dependent for widely separated points, charge conservation would appear to be violated in some reference frames.

3. Jul 15, 2010

### hunt_mat

He managed to creates a delta function of +ve charge at x=0, so the whole thing turns out to be zero, was this correct?

4. Jul 15, 2010

### Antiphon

Is is not physical. Such charge creation will lead to unphysical effects like non-transverse radiation fields.

5. Jul 15, 2010

### hunt_mat

As we're not injecting any electron into the system the global charge should remain zero (this was his argument), If we calculate the charge in the electron beam produced my the laser then we get a non-zero answer.

What should we be doing?

Thanks for your help by the way.

6. Jul 16, 2010

### Born2bwire

Maybe you are forgetting the ions that are left behind when you strip off the electrons. Have you looked at say Francis Chen's textbook? He has a very good book on introductory plasma physics that starts with basic classical first principles. It may give you the examples that you are looking for in terms of how to apply the Lorentz force and charge conservation when it comes to a plasma.

7. Jul 16, 2010

### hunt_mat

We're only looking at a simple 1D model to start with, how would we incorporate the positive ions in our 1D model. The equations we're using for the 1D equations are the Lorentz equation:
$$\gamma^{3}(v)\Bigg(\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}\Bigg)=-\frac{e}{m}E$$
The continuity equation
$$\frac{\partial n}{\partial t}+\frac{\partial}{\partial x}(nv)=0$$
And Ampere's law
$$\frac{\partial E}{\partial t}+a_{1}E+a_{2}nv=0$$
Where the a_{i} are some constants defining the current in the plasma. Is there a way of dealing with the positive ions in this simple mode or do we have to give with the fact that this model is just too simplistic?

Mat