Find the electrostatic energy stored in a cylindrical shell of length I, inner radius a and outer radius b, coaxial with a uniformly charged wire with linear charge density `lambda C//m`.

Explanation

For this we consider an elemental shell of radius x and width dx. The volume of this shell dV can be given as
`dV=2pixl.dx`
The electric field due to the wire at the shell is
`E=(2Klambda)/(x)`
The electrostatic field energy stored in the volume of this shell is
`dU=(1)/(2)in_(0)E^(2).dV`
or `dU=(1)/(2)in_(0)((2Klambda)/(x))^(2).2pix lambda.dx`
The total electrostatic energy stored in the above mentioned volume can be obtained by integrating the above expression within limits from a to as
`U=int dU=underset(b)overset(a)int(1)/(2)in_(0)((2Klambda)/(x))^(2)2pi x l.dx`
or `U=(lambda^(2)l)/(4pi in_(0))underset(a)overset(b)(1)/(x)dx`
or `U=(lambda^(2)l)/(4pi in_(0))underset(a)overset(b)int(1)/(x)dx`
or `U=(lambda^(2)l)/(4pi in_(0))ln((b)/(a))`