Average field on/in a sphere **

(a) A point charge q is located at an arbitrary position inside a sphere (just an imaginary sphere in space) with radius R. Show that the average electric field over the surface of the sphere is zero. Hint: Use an argument involving Newton's third law, along with what you know about spherical shells.

(b) If the point charge q is instead located outside the sphere, a distance r from the center, show that the average electric field over the surface of the sphere has magnitude q/4πE0r.

(c) Return to the case where the point charge q is located inside the sphere of radius R. Let the distance from the center be r. Use the above results to show that the average electric field over the entire volume of the sphere of radius R has magnitude qr/4πE0R3 and points toward the center (if q is positive).