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 \(2 R\) from the axis of the cylinder, is given by the expression \(\frac{23 \rho R}{16 K \varepsilon_{0}}\). The value of \(k\) is

The magnitude of the electric field at the point \(P\) which is at a distance \(2 R\) from the axis of the cylinder
\(\mathrm{E}=\mathrm{E}_{\text {total }}-\mathrm{E}_{\text {cavity }} \)
\( =\frac{\gamma}{2 \pi \varepsilon_{0}(2 R)}-\frac{1}{4 \pi \varepsilon_{0}} \frac{Q}{(2 R)^{2}} \)
\( \mathrm{Q}_{\text {sphere }}=\frac{4}{3} \pi\left(\frac{R}{2}\right)^{3} \rho=\frac{\pi R^{3} \rho}{6} \)
\( \lambda_{\text {cylinder }}=\pi R^{2} \rho \)
\( \therefore E=\frac{\pi R^{2} \rho}{4 \pi \varepsilon_{0} R}-\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\pi R^{3} \rho / 6}{4 R^{2}} \)
\( =\frac{23 \rho R}{96 \varepsilon_{0}}=\frac{23 \rho R}{16 \times 6 \times \varepsilon_{0}} \)
\( \therefore k=6\)