Debye Shielding

Let us consider the simplest possible example. Suppose that a quasi-neutral plasma is sufficiently close to thermal equilibrium that the number densities of its two species are distributed according to the Maxwell-Boltzmann law (Reif 1965),

(1.11) |

where is the electrostatic potential, and and are constant. From , it is clear that quasi-neutrality requires the equilibrium potential to be zero. Suppose that the equilibrium potential is perturbed, by an amount , as a consequence of a small, localized, perturbing charge density, . The total perturbed charge density is written

(1.12) |

Thus, Poisson's equation yields

(1.13) |

which reduces to

(1.14) |

If the perturbing charge density actually consists of a point charge , located at the origin, so that , then the solution to the previous equation is written

(1.15) |

This expression implies that the Coulomb potential of the perturbing point charge is shielded over distances longer than the Debye length by a shielding cloud of approximate radius that consists of charge of the opposite sign.

By treating as a continuous function, the previous analysis implicitly assumes that there are many particles in the shielding cloud. Actually, Debye shielding remains statistically significant, and physical, in the opposite limit in which the cloud is barely populated. In the latter case, it is the probability of observing charged particles within a Debye length of the perturbing charge that is modified (Hazeltine and Waelbroeck 2004).