An infinitely long solid cylinder of radius R with uniform volume charge density ρ 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 a point P which is at a distance 2 R from the axis of the cylinder is given by 23 ρ R 6 K ε 0 . What is the value of K ?

Step-by-step solution

The electric field at the point P due to the entire cylinder is E 1 = λ 2 π ε 0 ( 2 R ) = ρ π R 2 4 π ε 0 R = ρ R 4 ε 0 The electric field at the P due to the spherical cavity is E 2 = 1 4 π ε 0 ρ · 4 3 π R 3 8 ( 2 R ) 2 = ρ R ( 24 × 4 ) ε 0 E n e t = E 1 - E 2 = ρ R 4 ε 0 - ρ R 96 ε 0 = ρ R 4 ε 0 1 - 1 24 = 23 96 ρ R ε 0 = 23 6 × 16 ρ R ε 0 ∴    K = 16