A point charge `q = 3.0 m,uC` is located at the centre of a spherical layer of uniform isotropic dielectric with permittivity `epsilon = 3.0`. The inside radius of the layer is equal to `a = 250 mm`, the outside radius is `b = 500 mm`. Find the electrostatic energy inside the dielectric layer.

Explanation

(a) By exprerssion `int (1)/(2) epsilon_(0) epsilon E^(2) dV = int (1)/(2) epsilon_(0) E_(2) dV = int (1)/(2) epsilon E^(2) 4pi r^(2) dr`, for a spherical layer. To find the electrostativ energy inside the dielectric layer, we have to intergrate the upper expression in the limit `[a,b]`
`U = (1)/(2) epsilon_(0) epsilon int_(a)^(b) ((q)/(4pi epsilon_(0) epsilon r^(2)))^(2) 4pi r^(2) dr = (q^(2))/(8pi epsilon_(0) epsilon) [(1)/(a) - (1)/(b)] = 27 mJ`