Suppose that you have two concentric spherical shells of radii a and b, with a<b. The innershell has a charge−Qand the outer shell has a charge Q; the charges are uniformly distributed over each shell.
a. Find the potential difference between the shells, going from the inner shell to the outershell.
b. Now suppose that the shells are nearly the same size in the sense that b=a+d where d<<a. Write the leading behaviour of the potential difference in the limit whered/a1. Express the answer in terms ofσ, the absolute value of the surface charge density on the innersphere, and d.
c. How does your answer compare with the potential difference between two infinite planes, of charge densities±σ, separated by a distance d?
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