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

$A$ cubical volume is bounded by the surfaces $x = 0, x = a, y = 0, y = a, z = 0, z = a$. The electric field in the region is given by $\overrightarrow{E} = E_0 x \hat{i}$,where $E_0 = 4 \times 10^4 \text{ N C}^{-1} \text{m}^{-1}$. If $a = 2 \text{ cm}$,the charge contained in the cubical volume is $Q \times 10^{-14} \text{ C}$. The value of $Q$ is $...........$ (Take $\varepsilon_0 = 9 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{m}^{-2}$)

$A$ solid ball of radius $R$ has a charge density $\rho$ given by $\rho = \rho_0 \left( 1 - \frac{r}{R} \right)$ for $0 \leq r \leq R$. The electric field outside the ball is

$A$ spherical shell with an inner radius $a$ and an outer radius $b$ is made of conducting material. $A$ point charge $+Q$ is placed at the centre of the spherical shell and a total charge $-q$ is placed on the shell. Charge $-q$ is distributed on the surfaces as:

The total charge enclosed in an incremental volume of $2 \times 10^{-9} \, m^{3}$ located at the origin is ...... $nC$,if the electric flux density of its field is given by $\vec{D} = e^{-x} \sin y \hat{i} - e^{-x} \cos y \hat{j} + 2z \hat{k} \, C/m^{2}$.

If a small sphere of mass $m$ and charge $q$ is hung from a silk thread at an angle $\theta$ with the surface of a vertical charged conducting plate,then for equilibrium of the sphere,the surface charge density of the plate is