Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ $\varepsilon_0$ is the permittivity of free space.]

Two charges each equal to $\eta q({\eta ^{ – 1}} < \sqrt 3 )$ are placed at the corners of an equilateral triangle of side $a$. The electric field at the third corner is ${E_3}$ where $({E_0} = q/4\pi {\varepsilon _0}{a^2})$

Charges $Q _{1}$ and $Q _{2}$ arc at points $A$ and $B$ of a right angle triangle $OAB$ (see figure). The resultant electric field at point $O$ is perpendicular to the hypotenuse, then $Q _{1} / Q _{2}$ is proportional to

If the net electric field at point $\mathrm{P}$ along $\mathrm{Y}$ axis is zero, then the ratio of $\left|\frac{q_2}{q_3}\right|$ is $\frac{8}{5 \sqrt{x}}$, where $\mathrm{x}=$. . . . . .

A liquid drop having $6$ excess electrons is kept stationary under a uniform electric field of $25.5\, k\,Vm^{-1}$ . The density of liquid is $1.26\times10^3\, kg\, m^{-3}$ . The radius of the drop is (neglect buoyancy)