Let $P\left( r \right) = \frac{Q}{{\pi {R^4}}}r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q$. For a point $P$ inside the sphere at distance $r_1$ from the centre of the sphere, the magnitude of electric field is

An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho $. It has a spherical cavity of radius $R/2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2R$ from the axis of the cylinder, is given by the expression $\frac{{23\rho R}}{{16K{\varepsilon _0}}}$ .The value of $K$ is

At a point $20\, cm$ from the centre of a uniformly charged dielectric sphere of radius $10\, cm$, the electric field is $100\, V/m$. The electric field at $3\, cm$ from the centre of the sphere will be.......$V/m$

A positive charge $q$ is placed in a spherical cavity made in a positively charged sphere. The centres of sphere and cavity are displaced by a small distance $\vec l $ . Force on charge $q$ is :

Consider a sphere of radius $\mathrm{R}$ which carries a uniform charge density $\rho .$ If a sphere of radius $\frac{\mathrm{R}}{2}$ is carved out of it, as shown, the ratio $\frac{\left|\overrightarrow{\mathrm{E}}_{\mathrm{A}}\right|}{\left|\overrightarrow{\mathrm{E}}_{\mathrm{B}}\right|}$ of magnitude of electric field $\overrightarrow{\mathrm{E}}_{\mathrm{A}}$ and $\overrightarrow{\mathrm{E}}_{\mathrm{B}}$ respectively, at points $\mathrm{A}$ and $\mathrm{B}$ due to the remaining portion is

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