Consider a cylindrical capacitor (see figure 1) of length L and composed of two concentric cylinders: a solid inner conducing wire of radius r1, and an outer conducting shell with inner radius r2.

The capacitance for this arrangement (you may consider the inner conductor to have a charge per unit length of +λ and the outer conductor to have a charge per unit length -λ) is derived in the text and is equal to

C = 2Πε₀L/ln( r₂/ r₁),            (1)

where ε₀ is the permittivity of free space, and L is the length of the capacitor.

Now consider the following question, which are variations of this original situation.

1. Single dielectric with constant k

Now we imagine taking the previous capacitor and completely filling the space with a dielectric as shown in Figure 2.

(a) The applied electric field from the existing positively charged inner conductor polarizes the dielectric and induces surface charge densities on the inner and outer surfaces of the dielectric. Assuming that we have a linear dielectric so that

E(r) = E₀(r) - E_induced(r) = E0(r)/k,

where E_induced is the electric field from the induced surface charge, derive an expression for the magnitude of the potential difference (ΔV) between the inner conductor and the outer shell.

b) Using the fact that C = Q/ΔV , write down an expression for the capacitance, C' .