Green’s Reciprocation Theorem
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 54
9-8-2016
1628
Green’s Reciprocation Theorem
a) Prove Green’s reciprocation theorem: If ϕ is the potential due to a volume charge density ρ within a volume V and a surface charge density σ on the conducting surface S bounding the volume V, while ϕ' is the potential due to another charge distribution ρ' and σ' then
b) A point charge q is placed between two infinite grounded parallel conducting plates. If z0 is the distance between q and the lower plate, find the total charge induced on the upper plate in terms of q, z0, and l, where l is the distance between the plates (see Figure 1.1). Show your method clearly.
Figure 1.1
SOLUTION
a) We may prove the theorem by considering the volume integral of the following expression:
Integrating by parts in two ways, we have
(1)
Now, 
(where n points opposite to the directed area of the surface S) and 
so dividing (1) by 4π yields the desired result:
(2)
b) Let us introduce a second potential given by ϕ' = 2πσ' z, corresponding to a surface charge density on the upper plate of σ' and on the lower plate of 0 (see Figure 1.2). This introduced potential has no charge in the volume, and the real potential is zero on the plates so that the right-hand side of (2) is zero, yielding
where Qi is the induced charge. So on the upper plate,
Figure 1.2
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