Uniqueness theorem for Poisson's equation: Difference between revisions

Revision as of 19:49, 4 January 2021

Revision as of 17:02, 6 January 2021

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==Proof==

TtheThe general expression for [[Poisson's equation]] in [[electrostatics]] is

:<math>\mathbf{\nabla}^2 \varphi = -\frac{\rho_f}{\epsilon_0},</math>

where <math>\varphi</math> is the [[electric potential]] and <math>\rho_f</math> is the [[charge density|charge distribution]] over some region.

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Suppose that we claim to have two solutions of Poisson's equation. Let us call these two solutions <math>\varphi_1</math> and <math>\varphi_2</math>. Then

:<math>\mathbf{\nabla}^2 \varphi_1 = - \frac{\rho_f}{\epsilon_0}</math>, and

:<math>\mathbf{\nabla}^2 \varphi_2 = - \frac{\rho_f}{\epsilon_0}</math>.

It follows that <math>\varphi=\varphi_2-\varphi_1</math> is a solution of [[Laplace's equation]] (a special case of [[Poisson's equation]] which equals <math>0</math>) because subtracting the two solutions above gives