A conducting sphere of radius c is grounded and placed in a uniform electric field that
Chapter 14, Problem 12(choose chapter or problem)
A conducting sphere of radius c is grounded and placed in a uniform electric field that has intensity E in the z-direction. The potential u(r, 6) outside the sphere is determined from the boundary-value problem a2u + au + _!_ aiu + cot 8 au = 0 r > c. 0 < 8 < 1f ;;r2 r ar r2 ae2 r2 a8 ' u(c, 8) = o. 0 < 8 < 'IT r-+oo limu(r,6) = -Ez = -Ercos8. Show that c' u(r. 6) = -Er cos(}+ E 2 cos 8. r nonneg [Hint: Explain why f: cos 8 P,.(cos 8) sin 6 dlJ = 0 for all ative integers except n = 1. See (24) of Section 12.6.) In 13 and 14, you are asked to find a product solution u(r. 8, (
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