$A$ non-conducting solid sphere of radius $R$ has a uniform volume charge density $\rho$. The electric potential at the center of the sphere is related to the potential at the surface and outside the sphere due to this uniform charge distribution.
Statement-$1$: When a charge $q$ is moved from the surface to the center of the sphere,the change in its potential energy is $q\rho R^2 / 6\varepsilon_0$.
Statement-$2$: The electric field at a distance $r$ $(r < R)$ from the center of the sphere is $\rho r / 3\varepsilon_0$.

What will be the total electric flux through the faces of the cube of side length '$a$' if a charge '$Q$' is placed at '$B$', the midpoint of an edge of the cube (see figure)?

The volume charge density of a sphere of radius $6 \, m$ is $2 \, \mu C \, m^{-3}$. The number of lines of force per unit surface area coming out from the surface of the sphere is $.... \times 10^{10} \, N C^{-1}$. [Given: Permittivity of vacuum $\epsilon_{0} = 8.85 \times 10^{-12} \, C^{2} N^{-1} m^{-2}$]

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 $3.$

Two infinite plane parallel sheets separated by a distance $d$ have equal and opposite uniform charge densities $\sigma$ and $-\sigma$. The electric field at a point between the sheets is

The electric field intensity $(E)$ at a distance of $3 \ m$ from a uniform long straight wire of linear charge density $0.2 \ \mu Cm^{-1}$ is