Poisson’s Equation in Plasma

Intuition

Poisson’s equation links charge density to electric potential. In plasma physics, it closes the fluid/kinetic description by relating particle densities to the self-consistent electrostatic field.

Formal Definition

\[\\nabla^2 \\phi = -\\frac{\\rho}{\\varepsilon_0}\]

where \(\\rho = e\\Bigl(\\sum_s Z_s n_{is} - n_{es}\\Bigr)\) is the net charge density.

Mathematical Formulation

For a single species with Boltzmann response:

\[\\nabla^2 \\phi - \\frac{1}{\\lambda_D^2}\\phi = -\\frac{e}{\\varepsilon_0}\\bigl(Z n_{i0} - n_{e0}\\bigr)\]

The homogeneous solution decays as \(\\exp(-r/\\lambda_D)/r\).

Derivation

Start from Gauss’s law \(\\nabla\\cdot\\mathbf{E} = \\rho/\\varepsilon_0\) and \(\\mathbf{E} = -\\nabla\\phi\). The screened Poisson equation follows by substituting the linearized Boltzmann relation for electrons.

Worked Example

For a point charge \(+Ze\) in an electron-ion plasma with fixed ions, the screened potential is

\[\\phi(r) = \\frac{Ze}{4\\pi\\varepsilon_0 r} e^{-r/\\lambda_D}\]

At \(r = \\lambda_D\), the potential has dropped to \(1/e\) of its vacuum value.

Common Mistakes

Forgetting the sign: positive charge gives \(\\nabla^2\\phi < 0\).
Using vacuum Green’s function when \(\\lambda_D\) is finite.

Quiz Questions

What happens to Poisson’s equation when \(\\lambda_D \\to 0\)?
Why does the Yukawa potential reduce to the Coulomb potential at \(r \\ll \\lambda_D\)?