What is the Debye Shielding Problem?

[ggpwntthxbai]

Homework Statement

Consider a positive point charge +q, immersed in plasma. Show that the net charge in the Debye shielding cloud exactly cancels the test charge. Assume the ions are fixed and that e*phi <<< kTe.

Homework Equations

f(u) = A exp [-(1/2*m*u^2 + q*phi)/k_b*T_e]
u is the x component of velocity

The Attempt at a Solution

Really lost on this one. I think I could make a hand waivey argument with Gauss's Law but I don't see where to go on this one.

 

[Dickfore]

You should use the Boltzmann distribution:
$$n(\mathbf{x}) = n_{0} \, \exp\left(-\frac{U(\mathbf{x})}{k_{B} T_{e}}\right)$$
to give the distribution of electrons. Here, $n(\mathbf{x})$ is the number density of electrons around a point given by the position vector $\mathbf{x}$, $n_{0}$ is their equilibrium density, $U(\mathbf{x})$ is the potential energy of the electrons (the equilibrium density is reached at the part of space where $U = 0$ according to the above formula).

Think about what is the potential of charged particles in a polarized medium where electric fields might exist. After you answer this question, we can proceed further.