How To Find the Capacitance of a Conductor Configuration

Capacitance is the amount of charge something can hold for a given applied potential difference between separated parts of the conductor: $C=Q/\Delta V$. So the overall strategy here is to find the potential difference $\Delta V$ corresponding to a particular $Q$ on an object, then take the ratio.

1. First, assign charge $+Q$ and $-Q$ to the ``plates'' of the capacitor (``plates'' meaning the separated charged surfaces... of course they need not really be flat plates). Don't try to find $Q$: you assume a particular $Q$ and find the $\Delta V$ corresponding to that $Q$; then when you calculate the capacitance $Q/\Delta V$, the $Q$ will cancel out.

2. Find the electric field $\vec{E}$ in the region between the plates. Often it's appropriate to use Gauss' Law. See the first howto.

3. Using the electric field you have calculated, find the potential difference $\Delta V_{AB}= -\int_A^B{\vec{E}\cdot d\vec{l}}$ between the plates. See the first howto.

4. Now calculate $C=Q/\Delta V$. The arbitrary $Q$ you assigned in step 1 should cancel out.

5. Capacitance is positive, so check your signs if you end up with a negative number.