An infinitely long solid cylinder of the radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of the radius $\frac{R}{2}$ 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 $\frac{23\rho R}{16k{\epsilon }_0}$. The value of $k$ is