Question

An infinity long solid cylinder of radius R has a uniform volume charge density `rho`. It has a spherical cavity of radius `R//2` with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression `(23rhoR)/(16Kepsilon_0)`. The value of k is

Answer 1

We suppose that the cavity is filled up by a positive as well as negative volume charge of `rho`. So the electric field now produced at P is the superposition of two electric fields
(a) The electric field created due to the infinitely long solid cylinder is
`E_1=(rhoR)/(4epsilon_0)` directed towards the `+Y` direction
(b) The electric field created due to the spherical negative charge density
`E_2=(rhoR)/(96epsilon_0)` directed towards the `-Y` direction.
`:.` The net electric field is
`E=E_1-E_2=1/6[(23rhoR)/(16epsilon_0)]`