A solid uniformly charged fixed non-conducting sphere of total charge $Q$ and radius $R$ contains a tunnel of negligible diameter. If a point charge $–q$ of mass $m$ is released at rest from the point $P$ as shown in the figure then find out its velocity at the following points.

$\left(\mathrm{i}\right)$ At the surface of the sphere

$\left(\mathrm{ii}\right)$ At the centre of the sphere

Two identical charges, $5 \mathrm{\mu C}$ each is fixed at a distance $8 \mathrm{cm}$ and a charged particle of mass $9\times {10}^{-6} \mathrm{kg}$ and charge $–10 \mathrm{\mu C}$ is placed at a distance $5 \mathrm{cm}$ from each of them and is released. Find the speed of the particle when it is nearest to the two charges.

A particle of mass $m$, charge $q>0$ and initial kinetic energy $K$ is projected from infinity towards a heavy nucleus of the charge $Q$ assumed to have a fixed position.

$\left(\mathrm{a}\right)$ If the aim is perfect, how close to the centre of the nucleus is the particle when it comes instantaneously to rest?

$\left(\mathrm{b}\right)$ With a particular imperfect aim, the particle’s closest approach to the nucleus is twice the distance determined in $\left(\mathrm{a}\right)$ Determine the speed of the particle at the closest distance of approach.

Three-point charges are arranged at the three vertices of a triangle as shown in the figure.

Given $q={10}^{–7} \mathrm{C}$, calculate the electrostatic potential energy of the system.

Eight equal point charges each of charge $q$ and mass $m$ is placed at eight corners of a cube of side $a$.

$\left(\mathrm{i}\right)$ Find out potential energy of charge system

$\left(\mathrm{ii}\right)$ Find out work done by an external agent against electrostatic forces and by electrostatic forces to increase all sides of the cube from $a$ to $2a$.

$\left(\mathrm{iii}\right)$ If all the charges are released at rest, then find out their speed when they are at the corners of the cube of the side $2a$.

$\left(\mathrm{iv}\right)$ If keeping all other charges fixed, the charge of the corner $A$ is released then find out its speed when it is at an infinite distance?

$\left(\mathrm{v}\right)$ If all charges are released simultaneously from rest then find out their speed when they are at a very large distance from each other.

Two concentric spherical shells of radius $R_1$ and $R_2\left(R_2>R_1\right)$ are having uniformly distributed charges $Q_1$ and $Q_2$, respectively. Find out the total energy of the system.