Question

Problem 8 A positive charge is uniformly distributed through an insulating sphere of radius R. The point P that is located a distance r from the center of the sphere. (i) Determine the electric field when the point P is inside the sphere (r < R). (i) Determine the electric field when the point P is outside the sphere (r> R). (iii) Plot the magnitude of the electric field as a function of r.

Answer 1

r < R Gauss' law charge enclosed by sphere of radius r, q = p (4/3 pi r^3) rho: volume charge density, rho = theta/4/3 pi R^3 q = Q/(4/3) pi R^3 (4/3 pi r^3) = Qr^3/R^3 contourintegral E^vector middot ds^vector = q/epsilon_0 = Qr^3/R^3 epsilon_0 E(r) (4 pi r^2) = Qr^3/R^3 epsilon_0 E(r) = Q/4 pi epsilon_0 r/R^3 Total charge enclosed now = Q E (4 pi r^2) = Q/epsilon_0 E = Q/4 pi epsilon_0 r^2