As shown in the figure,a ball $B$ is suspended by a string $S$ from a large charged plate $P$ such that it makes an angle $\theta$ with the plate. The surface charge density of the plate is proportional to which of the following?

The potential due to an electrostatic charge distribution is $V(r) = \frac{q e^{-\alpha r}}{4 \pi \varepsilon_{0} r}$,where $\alpha$ is positive. The net charge within a sphere centred at the origin and of radius $1/\alpha$ is

Let there be a spherically symmetric charge distribution with charge density varying as $\rho (r) = \rho _0 \left( \frac{5}{4} - \frac{r}{R} \right)$ for $r \le R$,and $\rho (r) = 0$ for $r > R$,where $r$ is the distance from the origin. The electric field at a distance $r (r < R)$ from the origin is given by:

An electric field converges at the origin whose magnitude is given by the expression $E = 100\,r\,N/C$,where $r$ is the distance measured from the origin.

An infinite non-conducting sheet has a surface charge density $\sigma = 0.10 \, \mu C/m^2$ on one side. How far apart are equipotential surfaces whose potentials differ by $50 \, V$?

An electric dipole is placed on the $x$-axis in proximity to a line charge with a linear charge density of $3.0 \times 10^{-6} \, C/m$. The line charge is placed on the $z$-axis. The positive and negative charges of the dipole are at distances of $10 \, mm$ and $12 \, mm$ from the origin,respectively. If a total force of $4 \, N$ is exerted on the dipole,find the magnitude of the positive or negative charge of the dipole.