Three infinitely long charged non-conducting sheets are placed as shown in the figure. The electric field at point $P$ is ($\sigma$ - charge density,$\epsilon_0$ - permittivity of free space).

The line $AA^{\prime}$ lies on a charged infinite conducting plane which is perpendicular to the plane of the paper. The plane has a surface charge density $\sigma$. $B$ is a ball of mass $m$ with a like charge of magnitude $q$. $B$ is connected by a string to a point on the line $AA^{\prime}$. The tangent of the angle $\theta$ formed between the line $AA^{\prime}$ and the string is:

Three infinite plane sheets carrying uniform charge densities $-\sigma, 2 \sigma, 4 \sigma$ are placed parallel to the $XZ$ plane at $Y=a, 3a, 4a$ respectively. The electric field at the point $(0, 2a, 0)$ is

The nuclear charge $(Ze)$ is non-uniformly distributed within a nucleus of radius $R$. The charge density $\rho(r)$ (charge per unit volume) is dependent only on the radial distance $r$ from the center of the nucleus as shown in the figure. The electric field is only along the radial direction.
$1.$ The electric field at $r=R$ is
$(A)$ independent of $a$
$(B)$ directly proportional to $a$
$(C)$ directly proportional to $a^2$
$(D)$ inversely proportional to $a$
$2.$ For $a=0$,the value of $d$ (maximum value of $\rho$ as shown in the figure) is
$(A)$ $\frac{3Ze}{4\pi R^3}$ $(B)$ $\frac{3Ze}{\pi R^3}$ $(C)$ $\frac{4Ze}{3\pi R^3}$ $(D)$ $\frac{Ze}{3\pi R^3}$
$3.$ The electric field within the nucleus is generally observed to be linearly dependent on $r$. This implies
$(A)$ $a=0$ $(B)$ $a=\frac{R}{2}$ $(C)$ $a=R$ $(D)$ $a=\frac{2R}{3}$
Give the answer for questions $1, 2,$ and $3.$

$A$ hollow sphere of charge does not produce an electric field at any

An infinite nonconducting sheet of charge has a surface charge density of $10^{-7} \ C/m^2$. The separation between two equipotential surfaces near the sheet whose potential differ by $5 \ V$ is: