Prove Green’s Reciprocation Theorem | Quantum Science Philippines
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Prove Green’s Reciprocation Theorem

Author: Kayrol Ann B. Vacalares

MS-Physics 1, MSU-Iligan Institute of Technology

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Prove Green’s Reciprocation Theorem:

If \Phi is the potential due to a volume-charge density \rho within a volume V and a surface charge density \sigma on the  conducting surface S bounding the volume V, while \Phi' is the potential due to another charge distribution \rho' and \sigma' , then

\int_v \rho \Phi' d^{3}x + \int_s \sigma \Phi' da = \int_v \rho' \Phi d^{3}x + \int_s \sigma' \Phi da

 

Solution:

Using Green’s Theorem:

\int_v (\phi \nabla^{2} \psi - \psi \nabla^{2} \phi) d^{3}x = \oint_s [\phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n}] da

we can replace:

\phi to \Phi  and \psi to \Phi'

and we can also use the Poisson’s Equation, where we have:

\nabla^{2} \Phi = - \frac{\rho}{\epsilon_0} and

\nabla^{2} \Phi' = - \frac{\rho'}{\epsilon_0}

and also the normal derivative of the potential derived from the boundary conditions to yield a surface charge density,

\sigma = \epsilon_0 \frac{\partial \Phi}{\partial n} \sigma' = \epsilon_0 \frac{\partial \Phi'}{\partial n}

We can use these equations and plug it in Green’s Theorem.

Plugging in:

a.) letting \psi = \Phi' and  \phi = \Phi

\int_v (\Phi \nabla^{2} \Phi') - \Phi' \nabla^{2} \Phi) d^{3}x = \oint_s [\Phi \frac{\partial \Phi'}{\partial n} - \Phi' \frac{\partial \Phi}{\partial n}] da \int_v (\Phi' \nabla^{2} \Phi d^{3}x + \oint_s \Phi' \frac{\partial \Phi}{\partial n} da = -\int_v \Phi \nabla^{2} \Phi' d^{3}x + \oint_s \Phi \frac{\partial \Phi'}{\partial n} da

 

b.) Plugging in Poisson’s Equation, we have:

\int_v \Phi (\frac{\rho}{\epsilon_0}) d^{3}x + \oint_s \Phi' \frac{\partial \Phi}{\partial n} da = \int_v \Phi (\frac{\rho'}{\epsilon_0} d^{3}x + \oint_s \Phi (\frac{\partial \Phi'}{\partial n} da

 

c.) Plugging in \sigma and \sigma'

\int_v \Phi' \frac{\rho}{\epsilon_0} d^{3}x + \oint_s \Phi' \frac{\sigma}{\epsilon_0} da = \int_v \Phi \frac{\rho'}{\epsilon_0} d^{3}x + \oint_s \Phi \frac{\sigma'}{\epsilon_0} da

cancel out the \epsilon_0 we get Green’s reciprocation theorem:

 

\int_v \rho \Phi' d^{3}x + \int_s \sigma \Phi' da = \int_v \rho' \Phi d^{3}x + \int_s \sigma' \Phi da

 

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