$A$ hollow metal sphere of radius $R$ is uniformly charged. The electric field due to the sphere at a distance $r$ from the centre is:

$A$ large metal plate has a surface charge density of $8.85 \times 10^{-6} \ C \ m^{-2}$. An electron having initial kinetic energy of $8 \times 10^{-17} \ J$ is moving towards the center of the plate. If the electron stops just before reaching the plate,then the initial distance between the electron and the plate is [Take $\epsilon_{0} = 8.85 \times 10^{-12} \ C^{2} \ N^{-1} \ m^{-2}$]

An infinite line charge produces a field of $9 \times 10^4 \; N/C$ at a distance of $2 \; cm$. Calculate the linear charge density in $\mu C/m$.

The electric field due to a uniformly charged sphere of radius $R$ as a function of the distance $r$ from its centre is represented graphically by

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 $3.$

$A$ spherically symmetric charge distribution is considered with charge density varying as
$\rho(r)=\begin{cases} \rho_{0}\left(\frac{3}{4}-\frac{r}{R}\right) & \text{for } r \leq R \\ 0 & \text{for } r>R \end{cases}$
Where,$r (r < R)$ is the distance from the centre $O$ (as shown in figure). The electric field at point $P$ will be.