In free space, a particle $A$ of charge $1\,\mu C$ is held fixed at a point $P.$ Another particle $B$ of the same charge and mass $4\,\mu g$ is kept at a distance of $1\,mm$ from $P$. If $B$ is released, then its velocity at a distance of $9\,mm$ from $P$ is [ Take $\frac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}\,N{m^2}{C^{ - 2}}$ ]

Two positive point charges of $12\,\mu C$ and $8\,\mu C$ are $10\,cm$ apart. The work done in bringing them $4\, cm$ closer is

Explain electric potential energy. Show that the sum of kinetic energy and electric potential energy remains constant.

Derive the formula for the electric potential energy of system of two charges.

Consider a spherical shell of radius $R$ with a total charge $+ Q$ uniformly spread on its surface (centre of the shell lies at the origin $x=0$ ). Two point charges $+q$ and $-q$ are brought, one after the other from far away and placed at $x=-a / 2$ and $x=+a / 2( < R)$, respectively. Magnitude of the work done in this process is 

Prove that electrostatic forces are conservative in nature and define electrostatic potential energy.