An infinite line of charge with uniform line charge density of $\lambda = 1 \ C \ m^{-1}$ is placed along the $y$-axis. $A$ point charge $q = 1 \ C$ is placed on the $x$-axis at a distance of $d = 3 \ m$ from the origin. At what distance $r$ from the origin on the $x$-axis,between the origin and the point charge,is the total electric field zero (in $m$)?

$A$ solid ball of radius $R$ has a charge density $\rho$ given by $\rho = \rho_0 \left( 1 - \frac{r}{R} \right)$ for $0 \leq r \leq R$. The electric field outside the ball is

There is a solid sphere of radius $R$ having uniformly distributed charge throughout it. What is the relation between electric field $E$ and distance $r$ from the centre (where $r < R$)?

$A$ hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is $(\sigma / 2 \varepsilon_{0}) \hat{n}$,where $\hat{n}$ is the unit vector in the outward normal direction,and $\sigma$ is the surface charge density near the hole.

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:

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