- Poisson's equation
In
mathematics , Poisson's equation is apartial differential equation with broad utility inelectrostatics ,mechanical engineering andtheoretical physics . It is named after the Frenchmathematician ,geometer andphysicist Siméon-Denis Poisson. The Poisson equation is:Deltavarphi=f
where Delta is the
Laplace operator , and "f" and φ are real or complex-valued functions on amanifold . When the manifold isEuclidean space , the Laplace operator is often denoted as abla}^2 and so Poisson's equation is frequently written as:abla}^2 varphi = f.
In three-dimensional
Cartesian coordinate s, it takes the form:left( frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2} + frac{partial^2}{partial z^2} ight)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 thescreened Poisson equation . There are various methods for numerical solution. Therelaxation method , an iterative algorithm, is one example.Electrostatics
One of the 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 theelectric potential for a given charge distribution.The derivation of Poisson's equation in electrostatics follows.
SI units are used and Euclidean space is assumed.Starting with
Gauss' law for electricity (also part ofMaxwell's equations ) in a differential control volume, we have::mathbf{ abla} cdot mathbf{D} = ho::mathbf{ abla} cdot means to take the
divergence .::mathbf{D} is theelectric displacement field .::ho is thecharge density .Assuming the medium is linear, isotropic, and homogeneous (see
polarization density ), then::mathbf{D} = varepsilon mathbf{E}::varepsilon is the
permittivity of the medium.::mathbf{E} is theelectric field .By substitution and division, we have:
:mathbf{ abla} cdot mathbf{E} = frac{ ho}{varepsilon}
In the absence of a changing magnetic field, mathbf{B},
Faraday's law of induction gives::abla imes mathbf{E} = -dfrac{partial mathbf{B {partial t} = 0::imes is the
cross product .::t is time.Since the curl of the electric field is zero, it is defined by a scalar electric potential field, varphi (see
Helmholtz decomposition ).:mathbf{E} = - abla varphi
Eliminating mathbf{E} by substitution, we have a form of the Poisson equation:
:abla cdot abla varphi = { abla}^2 varphi = -frac{ ho}{varepsilon}.
Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then
Laplace's equation results. If the charge density follows aBoltzmann distribution , then thePoisson-Boltzmann equation results. The Poisson-Boltzmann equation plays a role in the development of the Debye-Hückel theory of dilute electrolyte solutions.(Note: Although the above discussion assumes that the magnetic field not varying in time, the same Poisson equation arises even if it does vary in time, as long as the
Coulomb gauge is used. However, in this more general context, computing varphi is no longer sufficient to calculate mathbf{E}, since the latter also depends on themagnetic vector potential , which must be independently computed.)Potential of a Gaussian charge density
If there is a spherically symmetric Gaussian charge density ho(r) :
:ho(r) = frac{Q}{sigma^3sqrt{2pi}^3},e^{-r^2/(2sigma^2)},
where "Q" is the total charge, then the solution φ ("r") of Poisson's equation,
:abla}^2 varphi = - { ho over varepsilon } ,
is given byFact|date=September 2007
:varphi(r) = { 1 over 4 pi varepsilon } frac{Q}{r},mbox{erf}left(frac{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 varphi.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 varepsilon_0 } {Q over r} , as one would expect. Furthermore the erf function approaches 1 extremely fast as its argument increase; in practice for r > 3σ the relative error is smaller than 1/1000.ee also
*
Discrete Poisson equation References
* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde302.pdf 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 1-58488-299-9External links
* [http://planetmath.org/encyclopedia/PoissonsEquation.html Poisson's equation] on
PlanetMath .
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