${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are two hollow concentric spheres with charge $q$ and $2q.$ Space between ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ is filled with a dielectric of dielectric constant, $K=5$. The ratio of flux through $S_2$ and flux through $S_1$ is $K\text{'}.$ Then find the value of $15K\text{'}.$

This questions has Statement $1$ and Statement $2.$ Of the four choices given after the statements, choose the one that best describes the two statements.

An insulating solid sphere of radius $R$ has a uniformly positive charge density $\rho .$ As a result of this uniform charge distribution there is a finite value of the electric potential at the centre of the sphere, at the surface of the sphere and also at a point outside the sphere. The electric potential at infinity is zero.

Statement $1$: When a charge is taken from the centre to the surface of the sphere its potential energy changes by $\frac{q\rho }{3{\epsilon }_0}$
Statement $2$: The electric field at a distance $r(r<R)$ from the centre of the sphere is $\frac{\rho r}{3{\epsilon }_0}$

The region between two concentric spheres of radii $a$ and $b$, respectively (see figure) has a volume charge density $\rho = \frac{A}{r},$ where $A$ is a constant and $r$ is the distance from the centre. At the centre of the spheres is a point charge $Q$. The value of $A$ such that the electric field in the region between the spheres is constant is

A disc of radius $\frac{a}{4}$ having a uniformly distributed charge of $6 \mathrm{C}$ is placed in the $xy$ plane with its centre at $\left(\frac{-a}{2}, 0, 0\right)$. A rod of length carrying a uniformly distributed charge $8 \mathrm{C}$ is placed on the $x$- axis  from $x = \frac{a}{4}$ to $x\mathit{ }=\frac{5a}{4}$. Two point charges $-7 \mathrm{C}$ and $3 \mathrm{C}$ are placed at $\left(\frac{a}{4}, \frac{-a}{4}, 0\right)$ and $\left(-\frac{3a}{4}, \frac{3a}{4}, 0\right)$ respectively. Consider a cubical surface formed by six surfaces $x= \pm \frac{a}{2}, y = \pm \frac{a}{2}, z= \pm \frac{a}{2}.$The electric flux through this cubical surface is