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.

 

[nlightner]

I would start by clarifying the problem with the person who assigned it. The mention of "no hint to the relative polarity of the charges" is concerning, as the polarity of the charges is crucial in determining the distribution and capacitance of the shells. Without this information, it is impossible to accurately solve the problem.

That being said, if we assume that the charges are of the same sign, then your proposed solution seems reasonable. The inner surface of the larger shell would have a charge of -q due to the repulsion of the inner charge q, while the outer surface would have a charge of q + Q due to the addition of the outer charge Q.

However, if we assume that the charges are of opposite signs, then the distribution and capacitance would be different. In this case, the inner surface of the larger shell would have a charge of -q + Q, while the outer surface would have a charge of q. The capacitance would also be different, as the potential difference between the shells would be the sum of the potentials due to each charge.

In conclusion, the problem needs to be clarified in order to accurately solve it. The polarity of the charges is crucial in determining the charge distribution and capacitance of the shells. Without this information, any proposed solution would be incomplete.