A sphere of the radius $R$ carries charge such that its volume charge density is proportional to the square of the distance from the centre. What is the ratio of the magnitude of the electric field at a distance $2R$ from the centre to the magnitude of the electric field at a distance of $\frac{R}{2}$ from the centre?

An uncharged conducting large plate is placed as shown. Now an electric field $E$ towards the right is applied. Find the induced charge density on the right surface of the plate.

Figure shows a uniformly charged hemisphere of radius $R$. It has a volume charge density $\rho$. If the electric field at a point $2R$ above its centre is $E$, then what is the electric field at the point $2R$ below its centre?

Four very large metal plates are given charges as shown in the figure. The middle two are then connected through a wire. Find the charge that will flow through the wire.

A conic surface is placed in a uniform electric field $E$ as shown in the figure such that the field is perpendicular to the surface on the side $AB$. The base of the cone is of the radius $R$, and the height of the cone is $h$. The angle of the cone is $\mathrm{\theta }$. Find the magnitude of the flux that enters the cone’s curved surface from the left side. Do not count the outgoing flux $\left(\mathrm{\theta }<45°\right)$.

A non-conducting sphere of the radius $R$ is filled with uniform volume charge density $-\rho$. The centre of this sphere is displaced from the origin by $\overrightarrow{d}$. The electric field $\overrightarrow{E}$ at any point $P$ having a position vector inside the sphere is,