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Question

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 23ρR16kϵ0. The value of k is:
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A
6
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B
7
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C
8
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D
9
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Solution

The correct option is A 6
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=λ2πdϵ0
Here, λ is charge per unit length λ=ρ×A=πR2ρ
and d=2R
Thus, Ecyl=πR2ρ2π(2R)ϵ0=ρR4ϵ0

Field due to sphere is given by Esph=14πϵ0Qd2
Here, Q is the total charge in the sphere. Q=ρ×V=43π(R2)3ρ
and d=2R
Thus, Esph=4π(R2)3ρ3×4πϵ0(2R)2=ρR96ϵ0

Thus, the net electric field is E=ρRϵ0(14196)=23ρR96ϵ0

Thus 16k=96k=6

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