Use Green's Reciprocation Theorem to show the G.S to the Electrostatic Potential

jhosamelly

1. The problem statement, all variables and given/known data

Use Green's Reciprocation Theorem to show the G.S to the Electrostatic Potential


2. Relevant equations

Green's Reciprocation Theorem

[itex]\int_{v} \rho^{'} \Phi d^{3}x + \int_{s} \sigma^{'} \Phi da = \int_{v} \rho \Phi^{'} d^{3}x + \int_{s} \sigma \Phi^{'} da [/itex]

The General Solution to the Electrostatic Potential

[itex] \Phi (x,y,z) = \frac{1}{4 ∏ \epsilon_{0}} \int^{+∞}_{-∞} d x^{'} \int^{+∞}_{-∞} d y^{'} \int^{+∞}_{-∞} [ \frac{\rho (x',y',z')}{\sqrt{(x-x')^{2} + (y-y')^{2} + (z-z')^{2}}} - \frac{\rho (x',y',z')}{\sqrt{(x-x')^{2} + (y-y')^{2} + (z+z')^{2}}}]d z^{'} + \frac{1}{4 ∏} \int^{+∞}_{-∞} d x^{'} \int^{+∞}_{-∞} d y^{'} [ \frac{2 z' V (x',y')}{[{(x-x')^{2} + (y-y')^{2} + z'^{2}]^{3/2}}}]d z^{'}[/itex]

3. The attempt at a solution

So, I should be able to derive the General Solution to the Electrostatic Potential from the Green's Reciprocation Theorem

I have the following to substitute.

[itex] \rho ^{'} (\vec{x'}) = 4 ∏ \delta (\vec{x} - \vec{x'}) [/itex]

[itex] \Phi ^{'} (x) = G_(D) (\vec{x} ; \vec{x'}) [/itex]

[itex] \sigma ^{'} (\vec{x'}) = \epsilon_{0} \frac{∂ G_{D}}{∂{n'}} [/itex]

but i don't know how to continue. All of these are just the ones with ' . What would I substitute to those without ' . What should I do? Please help.