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| The '''uniqueness theorem''' for [[Poisson's equation]] states that the equation has a unique [[gradient]] of the solution for a large class of [[boundary condition]]s. In the case of [[electrostatics]], this means that if an [[electric field]] satisfying the boundary conditions is found, then it is the complete electric field.
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| __TOC__
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| ==Proof==
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| In [[Gaussian units]], the general expression for [[Poisson's equation]] in [[electrostatics]] is
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| :<math>\mathbf{\nabla}\cdot(\epsilon \mathbf{\nabla}\varphi)= -4 \pi \rho_{f}</math>
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| Here <math>\varphi</math> is the [[electric potential]] and <math>\mathbf{E}=-\mathbf{\nabla}\varphi</math> is the [[electric field]].
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| The uniqueness of the gradient of the solution (the uniqueness of the electric field) can be proven for a large class of boundary conditions in the following way.
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| Suppose that there are two solutions <math>\varphi_{1}</math> and <math>\varphi_{2}</math>. One can then define <math>\phi=\varphi_{2}-\varphi_{1}</math> which is the difference of the two solutions. Given that both <math>\varphi_{1}</math> and <math>\varphi_{2}</math> satisfy [[Poisson's Equation]], <math>\phi</math> must satisfy
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| :<math>\mathbf{\nabla}\cdot(\epsilon \mathbf{\nabla}\phi)= 0</math> | |
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| Using the identity
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| :<math>\nabla \cdot (\phi \epsilon \, \nabla \phi )=\epsilon \, (\nabla \phi )^2 + \phi \nabla \cdot (\epsilon \, \nabla \phi )</math>
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| And noticing that the second term is zero one can rewrite this as
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| :<math>\mathbf{\nabla}\cdot(\phi\epsilon \mathbf{\nabla}\phi)= \epsilon (\mathbf{\nabla}\phi)^2</math>
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| Taking the volume integral over all space specified by the boundary conditions gives
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| :<math>\int_V \mathbf{\nabla}\cdot(\phi\epsilon \mathbf{\nabla}\phi) d^3 \mathbf{r}= \int_V \epsilon (\mathbf{\nabla}\phi)^2 \, d^3 \mathbf{r}</math>
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| Applying the [[divergence theorem]], the expression can be rewritten as
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| :<math>\sum_i \int_{S_i} (\phi\epsilon \mathbf{\nabla}\phi) \cdot \mathbf{dS}= \int_V \epsilon (\mathbf{\nabla}\phi)^2 \, d^3 \mathbf{r}</math>
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| Where <math>S_i</math> are boundary surfaces specified by boundary conditions.
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| Since <math>\epsilon > 0</math> and <math>(\mathbf{\nabla}\phi)^2 \ge 0</math>, then <math>\mathbf{\nabla}\phi</math> must be zero everywhere (and so <math>\mathbf{\nabla}\varphi_{1} = \mathbf{\nabla}\varphi_{2}</math>) when the surface integral vanishes.
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| This means that the gradient of the solution is unique when
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| :<math>\sum_i \int_{S_i} (\phi\epsilon \, \mathbf{\nabla}\phi) \cdot \mathbf{dS} =
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| 0</math>
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| The boundary conditions for which the above is true are:
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| # [[Dirichlet boundary condition]]: <math>\varphi</math> is well defined at all of the boundary surfaces. As such <math>\varphi_1=\varphi_2</math> so at the boundary <math>\phi = 0</math> and correspondingly the surface integral vanishes.
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| # [[Neumann boundary condition]]: <math>\mathbf{\nabla}\varphi</math> is well defined at all of the boundary surfaces. As such <math>\mathbf{\nabla}\varphi_1=\mathbf{\nabla}\varphi_2</math> so at the boundary <math>\mathbf{\nabla}\phi=0</math> and correspondingly the surface integral vanishes.
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| # Modified [[Neumann boundary condition]] (also called [[Robin boundary condition]] - conditions where boundaries are specified as conductors with known charges): <math>\mathbf{\nabla}\varphi</math> is also well defined by applying locally [[Gauss's Law]]. As such, the surface integral also vanishes.
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| # Mixed boundary conditions (a combination of Dirichlet, Neumann, and modified Neumann boundary conditions): the uniqueness theorem will still hold.
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| ==See also==
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| *[[Poisson's equation]]
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| *[[Gauss's law]]
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| *[[Coulomb's law]]
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| *[[Method of images]]
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| *[[Green's function]]
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| *[[Uniqueness theorem]]
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| *[[Spherical harmonics]]
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| ==References==
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| *{{cite book
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| |author=L.D. Landau, E.M. Lifshitz
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| |year=1975
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| |title=The Classical Theory of Fields
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| |edition=4th |volume=Vol. 2
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| |publisher=[[Butterworth–Heinemann]]
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| |isbn=978-0-7506-2768-9
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| }}
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| *{{cite book
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| |author=J. D. Jackson
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| |year=1998
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| |title=Classical Electrodynamics
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| |edition=3rd
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| |publisher=[[John Wiley & Sons]]
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| |isbn=978-0-471-30932-1
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| }}
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| {{DEFAULTSORT:Uniqueness Theorem}}
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| [[Category:Electrostatics]]
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| [[Category:Vector calculus]]
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Nice to satisfy you, my name is Numbers Held although I don't truly like becoming known as like that. For many years he's been living in North Dakota and his family members loves it. Bookkeeping is her day job now. One of the very very best issues in the world for me is to do aerobics and I've been doing it for quite a while.
my web-site; dns125.dolzer.at