Work to expand spherical shell to r3

Question

A capacitor is made out of two concentric spheres. The inner is a conducting solid sphere with an initial charge of Q, radius r1, and electric potential V. The outer is a conducting spherical shell of radius r2 and initial charge −Q. The outer shell is grounded throughout this problem. Calculate the work needed to expand the outer shell to radius r3 for the following two cases: (1) When the charges on the conductors Q and −Q are fixed. (2) When the electric potential difference between the conductors is fixed

Answer 1

The outer shell will lose its charge due to grounding.

So this might be a trick question to see whether you will realize that Q = 0.

Alternatively, it might be an attempt to say that the voltage V is

the potential difference between the two spheres.

I will assume the latter.

Fact 1:

The outer sphere experiences the same electrical attraction as if all the change +Q

were concentrated at the center of the spheres.

Fact 2:

The potential energy of the outer sphere when at radius r is -Q²/r.

Fact 3:

The work needed to expand the outer shell is equal to the gain in potential energy.

Facts 1 and 2 can be derived using calculus.

Fact 3 is the definition of potential energy.

Question (1):

Using facts 2 and 3 the work is -Q²/r3 - (-Q²/r2) = Q²/r2 - Q²/r3

Question (2):

If the voltage (i.e., difference in electrical potential) between the two spheres

remains the same, then so does the potential energy, and hence the work is 0.