Question

Shown is a uniformly charged inner insulating sphere with radius a and with charge density given by ρ = ρ₀(r³/a³).

Outside of it is a conducting shell of inner radius b and outer radius c. This spherical shell also has double the charge of the inner non-conducting sphere. (So, if the inner sphere had charge “+Q”, the outer shell has charge “+2Q”.)

The space between the sphere and the shell is empty.

a) Describe/draw the charge distribution on the outer shell.

b) Show that the charge on the inner sphere is Q = ⅔πρ₀a³.

(For the rest of this problem, you can use the symbol Q if it is appropriate and if it makes the solution simpler to write.)                                        

c) Some of these can be written down “by inspection”. Some might need to be calculated. Derive/write down an expression for E for

1. r > c
2. b < r < c
3. a< r< b
4. r < a (derive this one)

d) If the electric potential is taken to be zero when r = ∞, write or derive expressions for V for

1. r > c
2. r = c
3. b < r < c
4. r = b (Be thoughtful about how potential is defined.)
5. a< r< b (derive this one)
6. r = a
7. r < a (derive this one)

Answer 1