Question

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 $\frac{23 \rho R}{16 n \varepsilon_{0}} .$ The value of $n$ is

Answer 1

Electric field due to cylinder $=\frac{2 K \lambda}{2 R}$
Electric field due to sphere $=\frac{K Q}{(2 R)^{2}}$
From principle of superposition,
$E =\frac{2 K(\lambda)}{2 R}-\frac{K(Q)}{(2 R)^{2}}=\frac{K\left(\rho \pi R^{2}\right)}{R}-\frac{K\left(\rho \times \frac{4 \pi}{3}\left(\frac{R^{3}}{8}\right)\right)}{4 R^{2}} $
$=K(\rho \pi R)-\frac{K(\rho)(\pi R)}{24}=\frac{23 \rho \pi R}{4 \pi \varepsilon_{0} \times 24}=\frac{23 \rho R}{16 \times 6 \varepsilon_{0}}$
$ \Rightarrow n=6$