# Green's reciprocation theorem

Please see the attach file, The Green reciprocation theorem also attached. it take from " Classical Electrodynamic 3rd by Jackson"

Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use the reciprocation theorem of Green to prove that the total induced charge on one of the planes is equal to (-q) times the fractional perpendicular distance of the point charge from the other plane.

(Hint: As your comparison electrostatic problem with the same surfaces choose one whose charge densities and potential are known and simple.)

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#### Solution Preview

Green Reciprocation theorem states that if is a potential due to some volume charge density in volume V and surface charge density on a conducting surface S bounding V, while is the potential due to a different charge densities and then

(1.1)

Our system is that of two parallel infinite conducting planes.

The reciprocal system that we can use (since we know all about it) is that of a parallel plate capacitor with no charge between the plates.

We hold the bottom plane at potential 0 while the top plane at potential V.

We know that ...

#### Solution Summary

The solution shows how to find a reciprocal system that when applied to the Green's reciprocal theorem simplifies our calculations dramatically.