Plasmas, Random Walks and Applications: A first taste

1) Debye screening

Let us examine the behavior of a point charge inside a plasma.

The electric potential of a point charge Q in vacuum (or any linear dielectric) is , according to the known Coulomb law.

Suppose we insert a positive point charge Q inside an electron-ion plasma, where the ions are too heavy to move (they produce a homogeneous positive ion background). The free electrons inside the plasma will have the tendency to cluster around the positive charge. However, as the region around Q gets crowded with electrons, they will start to repel each other. So there is some equilibrium situation, where the electrons are attracted by Q and repelled by each other!

If we want to be quantitative, the potential generated away from the point charge will be

Clearly, we see that there is an exponential suppressing the potential. The factor can be interpreted as the point charge minus the electrons that gathered around it! This is exactly the concept of shielding. The constant λ_D is called Debye length, and is a function of the temperature and density of the plasma. It gives an estimate for the distance scale, beyond which the point charge can be characterized as unimportant!

If we graphically present the charge density in the plasma around the point charge (setting every constant to 1) we get

We can thus see that a couple of Debye lengths (set to 1) away from the charge, the charge density returns practically to 1 (its “normal” value), which means that after this distance the plasma is unaffected by the existence of the point charge!

So, if we are “far enough” from the point charge, what we see is the initial charge, shielded (partially cancelled) by the electrons that clustered around it. This is called Debye Shielding, and is a characteristic of plasmas! In general, static (i.e. constant in time) electric fields are suppressed inside a plasma!

We should keep that in mind, because now the only thing we have left to manipulate plasmas, are static MAGNETIC fields, and of course electromagnetic waves (i.e. fields that evolve in time). We will talk about them in subsequent sections.