Obtain the expression of electric field by charged spherical shell on a point outside it.

The electric field at a distance $r$ from the centre in the space between two concentric metallic spherical shells of radii $r_1$ and $r_2$ carrying charge $Q_1$ and $Q_2$ is $(r_1 < r < r_2)$

An infinite plane sheet of charge having uniform surface charge density $+\sigma_5 \mathrm{C} / \mathrm{m}^2$ is placed on $\mathrm{x}-\mathrm{y}$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e \mathrm{C} / \mathrm{m}$ is placed at $z=4 \mathrm{~m}$ plane and parallel to $y$-axis. If the magnitude values $\left|\sigma_s\right|=2\left|\lambda_{\mathrm{e}}\right|$ then at point $(0,0,2)$, the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi \sqrt{\mathrm{n}}: 1$. The value of $n$ 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

Let $E_1(r), E_2(r)$ and $E_3(r)$ be the respective electric fields at a distance $r$ from a point charge $Q$, an infinitely long wire with constant linear charge density $\lambda$, and an infinite plane with uniform surface charge density $\sigma$. if $E_1\left(r_0\right)=E_2\left(r_0\right)=E_3\left(r_0\right)$ at a given distance $r_0$, then