An infinitely 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 cylinder, as shown in the figure. The magnitude of the electric field at the point `P`, which is at a distance `2 R` form the axis of the cylinder, is given by the expression `( 23 r R)/( 16 k e_0)` . The value of `k` is .
.

Options

A. `2`
B. `4`
C. `6`
D. `8`

Answer

3

Explanation

At `P`, due to cylinder
`E_(1) = (lambda)/(2 pi in_(0) . 2 R) = (rho pi R^(2))/(2 pi in_(0) . 2 R) = (rho R)/(4 in_(0))`
Due to sphere
`E_(1) = (1)/(4 pi in_(0)) .(rho . (4)/(3) (R//2)^(3))/((2R)^(2))= (rho R)/( 4 in_(0))`
`E_(1) - E_(2) = (rho R)/(in_(0)) [ (1)/(4) - (1)/(96)] = (23 rho R)/(96 in_(0)) = (23 rho R)/(16k in_(0))`
`k = 6`