A line charge of length 2d with a total charge Q has a linear charge density varying as (d2 – z2), where z is the distance from the midpoint. A grounded, conducting, spherical shell of inner radius b > d is centered at the midpoint of the line charge.
(a) Find the potential everywhere inside the spherical shell as an expansion in Legendre polynomials.
(b) Calculate the surface-charge density induced on the shell.
(c) Discuss your answers to parts a and b in the limit that d << b.