An infinitely long solid cylinder of radius R has a uniform volume charge density ρ. It has a spherical cavity of radius R/2 with its center 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 23ρR/16kε₀. The value of k is:

The given system of cylinder with cavity can be expressed as superposition of Infinite cylinder with charge density +ρ and a sphere with charge density −ρ.
Field due to infinite cylinder is given by
Ecyl = λ/2πε₀d
Here, λ is charge per unit length
λ = ρ × A = πR²ρ
and d = 2R
Thus, Ecyl = πR²ρ/2π(2R)ε₀ = ρR/4ε₀
Field due to sphere is given by
Esph = 1/4πε₀ Q/d²
Here, Q is the total charge in the sphere.
Q = −ρ × V = −(4/3)π(R/2)³ρ
and d = 2R
Thus, Esph = −(4/3)π(R/2)³ρ / 4πε₀(2R)² = −ρR/96ε₀
Thus, the net electric field is
E = ρR/ε₀(1/4 − 1/96) = 23ρR/96ε₀
Thus 16k = 96 ⇒ k = 6