Question

4. Using the formula V(r) 1 /TTda , calculate the potential inside and outside a uniformly charged spherical shell of total charge q and radius R A useful formula fo VR-2Rz cos& =[1 Virw-mcosor sine'de'

Answer 1

For spherical shill sigma = q/4pi R^2 = q/4pi R^2 V(pi) = 1/4pi epsilon_0 integral sigma(r)/ Vertical-bar r - r' Vertical-bar 'da' da = R^2 sin theta d theta d phi V(R) = 1/4pi epsilon_0 integral^pi_0 integral^2pi_0 R^2 sin theta d theta d theta/squareroot R^2 + Z^2 - 2RZ cos theta = sigma/4pi epsilon_0 2piR^2 integral^pi_0 sin theta d theta/squareroot R^2 + Z^2 - 2RZ cos theta V(R) = sigma/4pi epsilon_0 (2pi R^2) [1/RZ squareroot R^2 + Z^2 - 2RZ cos theta]^pi_0 = q/8pi epsilon_0[1/RZ [(R + Z) - (R - Z)] = q/8pi epsilon_0[1/RZ (2Z)] V(R) = q/8pi epsilon_0 R (inside) \r\n sine outside q = 0 so V(R) = 0 (outside)