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Mathematics for Physics

64 Laplace Equation

A partial differential equation for the potential in charge-free regions that is used to solve electrostatic boundary-value problems, where a boundary-value problem is a differential equation whose solution is fixed by values on the edge of the region.

Laplace’s equation in charge-free regions. In a region with no charge the electrostatic potential satisfies Laplace’s equation. This principle is used to determine \(V\) between conductors from the values of \(V\) on the boundaries.

Laplace’s equation is

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

where

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

Harmonic functions. Solutions of Laplace’s equation are harmonic functions. A harmonic function is a function whose Laplacian vanishes. This principle is used to import the uniqueness and mean-value properties of potential theory into electrostatics.

The one-dimensional Laplace equation is

\[ \dfrac{d^{2}V}{dx^{2}} = 0 \]

where

  • \(V\) is the electric potential.
  • \(x\) is the Cartesian coordinate.

Uniqueness from boundary values. Uniqueness theorems fix the solution once the potential or its normal derivative is specified on the boundary. This principle is used to guarantee that a guessed potential that matches the boundaries is the physical potential.

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