Test charge in an electron plasma: Yukawa potential φ ∝ e^{-r/λ_D}/r, λ_D = √(ε₀k_BT/n_e e²), 2D map and λ_D vs T, n.
About this model
In a plasma, mobile electrons respond to a localized electric potential and rearrange to partially cancel external fields — Debye shielding. For a test point charge Q immersed in a quasineutral electron plasma with density n_e and temperature T_e, linearized Poisson–Boltzmann theory gives the Yukawa potential φ(r) = (Q/4πε₀r) exp(−r/λ_D), where the Debye length λ_D = √(ε₀ k_B T_e / (n_e e²)) sets the exponential decay scale. At r ≫ λ_D the potential is strongly suppressed compared with the bare Coulomb 1/r law; at r ≪ λ_D the field resembles vacuum Coulomb behavior. The screening parameter κ = 1/λ_D grows with density and falls with temperature. This simulator plots screened versus bare φ(r), the radial electric field magnitude, a two-dimensional color map of φ around the charge with a circle at r = λ_D, and curves of λ_D versus T (fixed n) and versus n (fixed T). It uses SI units with sliders for T_e, log₁₀ n_e, and test charge Q. Ion dynamics, magnetic fields, and nonlinear shielding at very large potentials are omitted.
Who it's for: Undergraduate plasma physics, space physics, or electromagnetism students after Coulomb's law and before kinetic theory or Langmuir waves.
Key terms
Debye shielding
Debye length
Yukawa potential
Plasma quasineutrality
Screening
Poisson–Boltzmann
Electron temperature
How it works
Debye shielding of a test charge in a plasma: Yukawa potential φ ∝ e^{-r/λ_D}/r, λ_D from T and n_e, 2D map and λ_D scaling curves.
Frequently asked questions
What is the physical meaning of λ_D?
It is the characteristic distance over which mobile charges can rearrange to cancel an imposed electric field. Beyond a few λ_D, the plasma is nearly neutral on average.
Why does φ(r) look like (1/r) exp(−r/λ_D)?
The Laplacian of a spherically symmetric Yukawa potential solves Poisson's equation with an exponential screening term from linearized charge response, giving exponential decay on top of the Coulomb geometry.
How do T and n affect shielding?
Higher n_e means more electrons available to screen → smaller λ_D. Higher T_e means less willingness to stay in the potential well → larger λ_D.
When does this linear model break down?
If eφ ≳ k_B T_e the response is nonlinear; very low density or collisionless kinetic effects also modify the profile. This page is the standard introductory Yukawa picture.