homework and exercises - Capacitor with different charges on each plate

I am confused as to whether/how capacitance changes when each plate has a different charge. For example, consider a coaxial cable and put $20Q$ on the outer cable, and $-Q$ on the inner. Or how about concentric spheres, grounding either the inner or the outer?

Answer

Systems of plates are not typically considered capacitors unless they are globally neutral. Nevertheless, capacitance is a geometric property that is to do with the system more than the actual voltages and charges you apply to it, so that your question still makes sense: the capacitance is the same as it would be with symmetric charges.

More specifically, the (mutual) capacitance of two conducting surfaces is defined as the charge that must be applied to one surface so that the potential in the other one will rise by one unit; by energy considerations it must be symmetric.

If the charges on both surfaces are antisymmetric (i.e. $+Q$ and $-Q$) then there will be a potential difference between the plates of $V=Q/C$. If they are asymmetric, a similar statement holds: if plate 1 has charge $Q_1$ and plate 2 has charge $Q_2$, then there will still be a potential difference between them of $V=(Q_1-Q_2)/2C$.

The problem with this, though, is that you're no longer seeing the whole picture, and you'll also have to deal with the self capacitance of the plates, which wasn't a problem before: if you put 100 C of charge on one plate and 99 C on the other, there will still be some potential difference between the plates, but they are also at a very high potential with respect to anything else you might consider, and you will have a host of other problems. This is why the situation is hardly ever considered.

In the general case, then, you have not one but two independent voltages to consider; that is, the mean and the difference, or the two voltages of the plates separately. To deal with this appropriately, you need to use a general capacitance matrix as Suresh describes in his answer.

Finally, you also need to worry about what other charged systems you must consider, and where they are. You can't just conjure a large charge on one plate without taking it from somewhere, and depending on where you're grounding there may also be some significant energy of interaction there.