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 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 \frac{23 ρR}{16 \textrm{ } \left(kε\right)

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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 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 $\frac{23 ρR}{16 {kε}_{0}}$ . The value of k is

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$E_{1} : 2 π (2 R) ℓ = \frac{ρ π R^{2} l}{ε_{0}} ∴ E_{1} = \frac{ρR}{4 ε_{0}}$

$E_{2} : 4 π {(2 R)}^{2} = \frac{ρ \frac{4}{3} π {(\frac{R}{2})}^{3}}{ε_{0}}$

${∴ E}_{2} = \frac{ρR}{24 \times 4 ε_{0}}$

$E_{1} - E_{2} = \frac{ρR}{4 ε_{0}} (1 - \frac{1}{24}) = \frac{23 ρR}{4 \times 24 ε_{0}}$

$∴$ k = 6

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