Within a spherical charge distribution of charge density $\rho(r)$,$N$ equipotential surfaces of potential $V_0, V_0 + \Delta V, V_0 + 2\Delta V, \dots, V_0 + N\Delta V$ (where $\Delta V > 0$) are drawn and have increasing radii $r_0, r_1, r_2, \dots, r_N$,respectively. If the difference in the radii of the surfaces is constant for all values of $V_0$ and $\Delta V$,then:

$A$ 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

An infinite line charge produces a field of $7.182 \times 10^8 \, N/C$ at a distance of $2 \, cm$. The linear charge density is

$A$ uniform rod $AB$ of mass $m$ and length $l$ is hinged at its midpoint $C$. The left half $(AC)$ of the rod has linear charge density $-\lambda$ and the right half $(CB)$ has $+\lambda$,where $\lambda$ is constant. $A$ large non-conducting sheet of uniform surface charge density $\sigma$ is also present near the rod. Initially,the rod is kept perpendicular to the sheet. The end $A$ of the rod is initially at a distance $d$. Now,the rod is rotated by a small angle $\theta$ in the plane of the paper and released. The time period of small angular oscillations is:

$A$ non-conducting solid sphere of radius $R$ has a uniform volume charge density $\rho$. The electric potential at the center of the sphere is related to the potential at the surface and outside the sphere due to this uniform charge distribution.
Statement-$1$: When a charge $q$ is moved from the surface to the center of the sphere,the change in its potential energy is $q\rho R^2 / 6\varepsilon_0$.
Statement-$2$: The electric field at a distance $r$ $(r < R)$ from the center of the sphere is $\rho r / 3\varepsilon_0$.

The electric field near a conducting surface having a uniform surface charge density $\sigma$ is given by: