Poisson's equation

In mathematics, Poisson's equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. It is named after the French mathematician, geometer and physicist Siméon-Denis Poisson.

The Poisson equation is

\Delta\varphi=f

where $\Delta$ is the Laplace operator, and f and φ are real or complex-valued functions on a manifold. When the manifold is Euclidean space, the Laplace operator is often denoted as ${\nabla}^2$ and so Poisson's equation is frequently written as

{\nabla}^2 \varphi = f

In three-dimensional Cartesian coordinates, it takes the form

\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right)\varphi(x,y,z) = f(x,y,z).

For vanishing f, this equation becomes Laplace's equation

\Delta \varphi = 0. \!

The Poisson equation may be solved using a Green's function; a general exposition of the Green's function for the Poisson equation is given in the article on the screened Poisson equation. There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example.

Electrostatics

One of the principal cornerstones of electrostatics is the posing and solving of problems that are described by the Poisson equation. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. In SI units:

{\nabla}^2 \Phi = - {\rho \over \epsilon_0}

where $\Phi \!$ is the electric potential (in volts), $\rho \!$ is the charge density (in coulombs per cubic meter), and $\epsilon_0 \!$ is the permittivity of free space (in farads per meter).

In a region of space where there is no unpaired charge density, we have

\rho = 0, \,

and the equation for the potential becomes Laplace's equation:

{\nabla}^2 \Phi = 0.

Potential of a Gaussian charge density

If there is a tridimensional spherically symmetric Gaussian charge density $\rho(r)$ :

\rho(r) = \frac{Q}{\sigma^3\sqrt{2\pi}^3}\,e^{-r^2/(2\sigma^2)},

where Q is the total charge, then the solution Φ (r) of the Poisson's equation:

{\nabla}^2 \Phi = - { \rho \over \epsilon_0 }

is given by:

\Phi(r) = { 1 \over 4 \pi \epsilon_0 } \frac{Q}{r}\,\mbox{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)

where erf(x) is the error function. This solution can be checked explicitly by a careful manual evaluation of ${\nabla}^2 \Phi$ . Note that, for r much greater than σ, erf(x) approaches unity and the potential Φ (r) approaches the point charge potential ${ 1 \over 4 \pi \epsilon_0 } {Q \over r}$ , as one would expect.

References

Poisson Equation at EqWorld: The World of Mathematical Equations.
L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

Partial differential equations | Electrostatics

Gourt Home::Gourt :: Poisson's equation

Electrostatics

Potential of a Gaussian charge density

See also

References