Question

Show that Laplace's equation $\nabla^2V=0$ minimizes the energy (in electrostatics).

Answer 1

Answer:

where {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}is the Laplace operator,[note 1] {\displaystyle \nabla \cdot }\nabla \cdot is the divergence operator (also symbolized "div"), {\displaystyle \nabla }\nabla is the gradient operator (also symbolized "grad"), and {\displaystyle f(x,y,z)}{\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

If the right-hand side is specified as a given function, {\displaystyle h(x,y,z)}{\displaystyle h(x,y,z)}, we have

{\displaystyle \Delta f=h.}{\displaystyle \Delta f=h.}

This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.

The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions,[1] which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation.[2] In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.

Explanation:

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