Consider a point charge q between two infinite parallel conducting planes held at zero potential. Let the planes be located at z = 0 and z = L in a cylindrical coordinate system, with the charge on the z axis at z = z₀, 0 0

(a) Show that the amount of induced charge on the plate at z = L inside a circle of radius a whose center is on the z axis is given by

Q_L(a) = ?q/V ?(z₀, 0)

Where ?(z₀, 0) is the potential is Problem 3.18 evaluate at z = z₀, ? = 0. Find the total charge induced on the upper plate. Compare with the solution (in method and answer) of Problem 1.13

(b) Show that the induced charge density on the upper plate on the upper plate can be written as

This integral can be expressed?as an infinite series involving the modified Bessel functions ?₀(n??/L), showing that at large radial distances the induced charge density falls off as (?)^-1/2e^-?p/L.

(c) Show that the charge density at p = 0 can be written as the series

sinh(kzo) dk sinh(kL) o(p) = kJ.(kp) 27 Jo