Electrostatics - vector calculus, gauss's law, laplace and poisson's equation
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partial differential equation of elliptic type with broad utility in mechanical engineeringand theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the Frenchmathematician, geometer, and physicistSiméon Denis Poisson.[1]
Poisson's equation is
{\displaystyle \Delta \varphi =f}
where {\displaystyle \Delta } is the Laplace operator, and {\displaystyle f} and {\displaystyle \varphi
are real or complex-valued functions on a manifold. Usually, {\displaystyle f} is given and {\displaystyle \varphi } is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as
{\displaystyle \nabla ^{2}\varphi =f.}
In three-dimensional Cartesian coordinates, it takes the form
{\displaystyle \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).}
When {\displaystyle f=0 identically we obtain Laplace's equation.
Poisson's equation may be solved using a Green's function:
{\displaystyle \varphi (\mathbf {r} )=-\iiint {\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\!r',}
where the integral is over all of space. A general exposition of the Green's function for Poisson's equation is given in the article on the screened Poisson equation. There are various methods for numerical solution, such as the relaxation method, an iterative algorithm.
Poisson's equation is
{\displaystyle \Delta \varphi =f}
where {\displaystyle \Delta } is the Laplace operator, and {\displaystyle f} and {\displaystyle \varphi
are real or complex-valued functions on a manifold. Usually, {\displaystyle f} is given and {\displaystyle \varphi } is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as
{\displaystyle \nabla ^{2}\varphi =f.}
In three-dimensional Cartesian coordinates, it takes the form
{\displaystyle \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).}
When {\displaystyle f=0 identically we obtain Laplace's equation.
Poisson's equation may be solved using a Green's function:
{\displaystyle \varphi (\mathbf {r} )=-\iiint {\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\!r',}
where the integral is over all of space. A general exposition of the Green's function for Poisson's equation is given in the article on the screened Poisson equation. There are various methods for numerical solution, such as the relaxation method, an iterative algorithm.
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