Capacitance of Concentric Shells (different charges)

[Hemmer]

Homework Statement

A thin conducting shell of radius a has charge q. Concentric to this is another shell of radius b (b > a) with a different charge Q. How is the charge distributed and what is the capacitance of the two shells. No hint to the relative polarity of the charges is given.

Homework Equations

$$C = \frac{Q}{V}$$

The Attempt at a Solution

The difficulty lies in the difference in the charges (I'm sure I'm missing something). Any reference to capacitance I see requires equal and opposite charges, leading me to think the the distribution must be something like:

Inside surface of small shell: no charge
Outside surface of small shell: +q

Inside surface of larger shell: -q (?)
Outside surface of larger shell: q + Q (?)


If I assume that it the charges are -q on inner and +q on outer then I think you could find it by:

$$E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$$

$$V = -\frac{q}{4\pi\epsilon_0} \int_a^b \frac{1}{r^2} = \frac{q}{4\pi\epsilon_0} \left(\frac{1}{a} - \frac{1}{b}\right)$$

The just find the capacitance simply by $$C = \frac{q}{V}$$

But there is no mention of Q which seems wrong. Any help or suggestions very much appreciated.

 

[rl.bhat]

When you connect the charged concentric shells, both must have the same potential.
The common potential is given by ( total charge/total capacity)
The capacitance of the spherical shell is proportional to its radius.