Two long thin charged rods with charge density $\lambda$ each are placed parallel to each other at a distance $d$ apart. The force per unit length exerted on one rod by the other will be $\left(\right.$ where $\left.k=\frac{1}{4 \pi \varepsilon_0}\right)$

Three infinitely long charge sheets are placed as shown in figure. The electric field at point $P$ 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

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

A sphere of radius $R$ has a uniform distribution of electric charge in its volume. At a distance $x$ from its centre, for $x < R$, the electric field is directly proportional to

Let $\rho (r)\, = \frac{Q}{{\pi {R^4}}}\,r$ be the volume 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