Two nonconducting solid spheres of radii $R$ and $2R$ having uniform volume charge densities ${\rho }_1$ and ${\rho }_2$, respectively, touch each other. The net electric field at a distance $2R$ from the centre of the smaller sphere, along the line joining the centre of the spheres, is zero. The ratio $\frac{{\rho }_1}{{\rho }_2}$ can be:

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin $O$. In this distribution, a spherical cavity of radius $R_2$, centred at $P$ with distance $OP =a =R_1 –\mathit{ }{R}_2$ (see figure) is made. If the electric field inside the cavity at position $\stackrel{\mathit{\rightharpoonup }}{\mathit{r}}$ is $\stackrel{\rightharpoonup }{E }\left(\stackrel{\rightharpoonup }{r}\right)$, then the correct statement(s) is(are) 

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\mathrm{\lambda }$ are kept parallel to each other. In their resulting electric field, point charges $q$ and $-q$ are kept in equilibrium between them. The point charges are confined to move in the $x$-direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is (are):

A cubical region of side a has its centre at the origin. It encloses three fixed point charges, $-q$ at $\left(0, -\frac{a}{4}, 0\right), +3q$ at $\left(0, 0, 0\right), -q$ at $\left(0, +a/4, 0\right)$. Choose the correct option(s).

An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y-z$ plane at $z =\frac{\sqrt{3}}{2}a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $ABCD$ lying in the $x-y$ plane with the centre at the origin is $\frac{\lambda l}{n{\epsilon }_0}$ (${\epsilon }_0$ is permittivity of free space ), then the value of $n$ is ____.