## 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 is the potential due to a volume-charge density within a volume V and a surface charge density on the conducting surface S bounding the volume V, while is the potential due to another charge distribution and , then

*Solution:*

Using Green’s Theorem:

we can replace:

to and to

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

and

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

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

Plugging in:

a.) letting and

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

c.) Plugging in and

cancel out the we get Green’s reciprocation theorem: