A simple pendulum of mass $m$ and charge $+q$ is suspended vertically by a massless thread of length $\ell$. At the point of suspension, a point charge $+q$ is also fixed. If the pendulum is displaced slightly from equilibrium position, its time period will be

The diagram shows the arrangement of three small uniformly charged spheres $A\text{,} B$ and $C$. The arrows indicate the direction of the electrostatic forces acting between the spheres(for example, the left arrow on sphere $A$ indicates the electrostatic force on sphere $A$ due to sphere $B$). At least two of the spheres are positively charged. Which sphere, if any, could be negatively charged?

Two semi-circular rings lying in the same plane, of uniform linear charge density $\mathrm{\lambda }$ have radius $r$ and $2r$. They are joined using two straight uniformly charged wires of linear charge density $\mathrm{\lambda }$ and length $r$ as shown in the figure. The magnitude of the electric field at the common center of semi-circular rings is

The figure above shows a closed Gaussian surface in the shape of a cube of edge length $3.0 \mathrm{m}$. There exists an electric field given by, $\overrightarrow{\mathrm{E}}=\left[\left(2.0x+4.0\right)\hat{\mathrm{i}}+8.0\hat{\mathrm{j}}+3.0\hat{\mathrm{k}}\right] \mathrm{N} {\mathrm{C}}^{-1}$, where $x$ is in meters, in the region in which it lies. The net charge in coulombs enclosed by the cube is equal to

A hollow conducting sphere is placed in an electric field produced by a point charge placed at $P$ as shown in the figure. Let $V_{\mathrm{A}}\text{,} V_{\mathrm{B}} \text{and} V_{\mathrm{C}}$ be the potentials at points $A\text{,} B$ and $C$ respectively. Then