Question

The figure to the right shows a cylindrical capacitor with inner radius b and outer radius a. Between the cylinders (shaded region) is a dielectric of constant k. If the inner cylinder contains charge +Q and out charge -Q determine an expression for: 

1. The electric field in the region between the cylinders. 

2. The potential difference between in the region between the cylinders. 

3. The capacitance of the capacitor. 

4. The energy density of the capacitor

Answer 1

Here,
a) electric field between the region

E = (1/(4pi * epsilon_0 * k)) * Q/r^2

as the electrcic field due to outer charge is zero

b)

the potential difference at the plates is

V = 2* 1/(4pi * epsilon_0 * k) * Q * ln(b/a)

the potential difference is 2k * Q * ln(b/a)

C)

as Q = C * V

C = V/Q

C = (4pi * epsilon_0 * k)/(2 * ln(b/a))

d)

energy density = 0.5 * epsilon* k * E^2

energy density = 0.5 * epsilon* k * ((1/(4pi * epsilon_0 * k)) * Q/r^2)^2

the energy density is 0.5 * epsilon* k * ((1/(4pi * epsilon_0 * k)) * Q/r^2)^2