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Mathematics for Physics

76 Poisson Equation

A partial differential equation relating the Laplacian of the potential to charge density that is used to determine electrostatic potentials from sources, where the Laplacian is the divergence of the gradient.

Poisson’s equation. Combining \(\mathbf{E} = -\nabla V\) with Gauss’s law yields Poisson’s equation. This principle is used to compute the potential of a known charge density.

Poisson’s equation is

\[ \nabla^{2}V = -\dfrac{\rho}{\epsilon_{0}} \]

where

  • \(\nabla^{2}\) is the Laplacian.
  • \(V\) is the electric potential.
  • \(\rho\) is the charge density.
  • \(\epsilon_{0}\) is the permittivity of free space.

Its reduction to Laplace’s equation. When the charge density vanishes, Poisson’s equation reduces to Laplace’s equation. This principle is used to treat empty regions as a special case of the sourced problem.

Laplace’s equation is

\[ \nabla^{2}V = 0 \]

where

  • \(\nabla^{2}\) is the Laplacian.
  • \(V\) is the electric potential.

The source as curvature of \(V\). The charge density sets the curvature of the potential. This principle is used to solve for \(V\) inside a uniformly charged region subject to boundary values.

Note: Also written \(\nabla^{2}V=-\rho/\epsilon_{0}\).