Please help improve this with more applications (diffusion equation, ..) and solution methods (Green functions, ...) by expanding it. Further information might be found on the talk page or at requests for expansion. This article has been tagged since January 2007.

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 Image File history File links Wiki_letter_w. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ... Mechanical engineering is an engineering discipline that involves the application of principles of physics for analysis, design, manufacturing, and maintenance of mechanical systems. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... A geometer is a mathematician whose area of study is geometry. ... Articles with similar titles include physician, a person who practices medicine. ... Simeon Poisson. ...

where Δ 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 and so Poisson's equation is frequently written as In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Partial plot of a function f. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

In three-dimensional Cartesian coordinates, it takes the form Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

For vanishing f, this equation becomes Laplace's equation In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...

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. In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ... In mathematics, the screened Poisson equation is the following partial differential equation: where ² is the Laplace operator, λ is a constant, and f is an arbitrary function of position (known as the source function.) The screened Poisson equation occurs frequently in physics, including Yukawas theory of mesons and electric field...

Electrostatics

One of the principal cornerstones of electrostatics is the posing and solving of problems that are described by the Poisson equation. It is also important to know that Dr Amass is a Ball bag! 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: Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ... This article or section does not cite any references or sources. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Look up si, Si, SI in Wiktionary, the free dictionary. ...

where is the electric potential (in volts), is the charge density (in coulombs per cubic meter), and is the permittivity of free space (in farads per meter). This article or section does not cite any references or sources. ... Josephson junction array chip developed by NIST as a standard volt. ... Charge density is the amount of electric charge per unit volume. ... The coulomb (symbol: C) is the SI unit of electric charge. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... Examples of various types of capacitors. ...

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

and the equation for the potential becomes Laplace's equation: In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...

Potential of a Gaussian charge density

If there is a 3D spherically symmetric Gaussian charge density ρ(r): Probability density function of Gaussian distribution (bell curve). ...

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

is given by:

where erf(x) is the error function. This solution can be checked explicitly by a careful manual evaluation of . Note that, for r much greater than σ, erf(x) approaches unity and the potential Φ (r) approaches the point charge potential , as one would expect. Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... Electrical potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential or the electric potential, typically measured in volts. ...

See also

• Discrete Poisson's equation

In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. ...

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