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The Poisson equation is

:\Delta arphi=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 {
abla}^2 and so Poisson's equation is frequently written as

:{
abla}^2 arphi = f

In three-dimensional Cartesian Coordinate s, it takes the form

:
\left( rac{\partial^2}{\partial x^2} + rac{\partial^2}{\partial y^2} + rac{\partial^2}{\partial z^2} ight) arphi(x,y,z) = f(x,y,z).


For vanishing ''f'', this equation becomes Laplace's Equation

:\Delta arphi = 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 principle 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:

:{
abla}^2 \Phi = - { ho \over \epsilon_0}

where \Phi \! is the electric potential (in Volt s), ho \! is the Charge Density (in Coulomb s per cubic meter), and \epsilon_0 \! is the Permittivity of free space (in Farad s per meter).

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

: ho = 0, \,
and the equation for the potential becomes Laplace's Equation :

:{
abla}^2 \Phi = 0.


POTENTIAL OF A GAUSSIAN CHARGE DENSITY


If there is a tridimensional spherically symmetric Gaussian charge density
ho(r) :

: ho(r) = rac{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:

:{
abla}^2 \Phi = - { ho \over \epsilon_0 }

is given by:

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


where erf(''x'') is the Error Function .
This solution can be checked explicitly by a careful manual evaluation of {
abla}^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