An electrostatic field in a region is radially outward with magnitude $E = \alpha r$,where $\alpha$ is a constant and $r$ is the radial distance. The charge contained in a sphere of radius $R$ in this region (centered at the origin) is:

The graphical variation of the electric field $E$ due to a uniformly charged insulating solid sphere of radius $R$ with respect to the distance $r$ from the centre $O$ is represented by:

$A$ charged ball $B$ hangs from a silk thread $S$,which makes an angle $\theta$ with a large charged conducting sheet $P$,as shown in the figure. The surface charge density $\sigma$ of the sheet is proportional to

Three infinitely long charged thin sheets are placed as shown in the figure. The magnitude of the electric field at point $P$ is $\frac{x \sigma}{\epsilon_0}$. The value of $x$ is . . . . . . . (All quantities are measured in $SI$ units).

$A$ small bob of mass $100 \ mg$ and charge $+10 \ \mu C$ is connected to an insulating string of length $1 \ m$. It is brought near to an infinitely long nonconducting sheet of charge density $\sigma$ as shown in the figure. If the string subtends an angle of $45^{\circ}$ with the sheet at equilibrium,the charge density of the sheet will be (Given,$\varepsilon_0 = 8.85 \times 10^{-12} \ F/m$ and acceleration due to gravity,$g = 10 \ m/s^2$): (in $nC/m^2$)

An infinite sheet carrying a uniform surface charge density $\sigma$ lies on the $xy$-plane. The work done to carry a charge $q$ from the point $A = a(\hat{i} + 2\hat{j} + 3\hat{k})$ to the point $B = a(\hat{i} - 2\hat{j} + 6\hat{k})$ (where $a$ is a constant with the dimension of length and $\varepsilon_{0}$ is the permittivity of free space) is