Prove Green’s Reciprocation Theorem | Quantum Science Philippines

## 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:

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