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 $\frac{23\rho R}{16K\varepsilon_0}$. The value of $K$ is

An electric dipole is placed on the $x$-axis in proximity to a line charge with a linear charge density of $3.0 \times 10^{-6} \, C/m$. The line charge is placed on the $z$-axis. The positive and negative charges of the dipole are at distances of $10 \, mm$ and $12 \, mm$ from the origin,respectively. If a total force of $4 \, N$ is exerted on the dipole,find the magnitude of the positive or negative charge of the dipole.

$\sigma$ is the uniform surface charge density of a thin spherical shell of radius $R$. The electric field at any point on the surface of the spherical shell is:

$A$ long, straight wire is surrounded by a hollow, thin, long metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of $\lambda$, and the cylinder has a net charge per unit length of $2\lambda$. The radius of the cylinder is $R$. Which of the following statements is correct?

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(A)$ Electric field inside (distance $r < R$ from center) of a uniformly charged spherical shell with surface charge density $\sigma$ and radius $R$.	$(I)$ $\sigma / \varepsilon_0$
$(B)$ Electric field at distance $r$ from a uniformly charged infinite plane sheet with surface charge density $\sigma$.	$(II)$ $\sigma / 2 \varepsilon_0$
$(C)$ Electric field outside (distance $r > R$ from center) of a uniformly charged spherical shell with surface charge density $\sigma$ and radius $R$.	$(III)$ $0$
$(D)$ Electric field between $2$ oppositely charged infinite plane parallel sheets with uniform surface charge density $\sigma$.	$(IV)$ $\frac{\sigma R^2}{\varepsilon_0 r^2}$

Choose the correct answer from the options given below:

The expression for electric field intensity at a point outside a uniformly charged thin plane sheet is (where $d$ is the distance of the point from the plane sheet):