Calculating Capacitance

The equation Q = C ΔV makes it look like the capacitance depends on potential difference, which is not true. Interpret the equation above as saying that the charge stored on a capacitor is proportional to the potential difference across the capacitor. You should be able to determine that the capacitance is actually determined by the geometry of the capacitor.

Take, for example, a cylindrical capacitor of length L. A solid cylinder of radius a and charge +Q lies inside a cylindrical shell of radius b and charge -Q. What is the capacitance of this device?

We know that C = Q/ΔV.

We can use the relationship Vb - Va = -∫Eds
and integrate along a radial line from a to b. The field is parallel to this line, and has a value:

E = 2kλ/r

Vb - Va = -∫ E dr = -2kλ ∫ (1/r) dr

the limits on the integral are a to b.

This gives Vb - Va = -2kλ [ ln(b) - ln(a) ] = -[2kQ/L] ln(b/a)

The ΔV in the capacitance equation is the absolute value of Vb - Va, so:

C = Q/ΔV = L / [2k ln(b/a)]

The capacitance is only determined by the geometry of the capacitor (and, again, what material is placed between the capacitor plates, but we'll get to that).