An electron enters in high potential region ${V_2}$ from lower potential region ${V_1}$ then its velocity

A proton of mass $m$ and charge $e$ is projected from a very large distance towards an $\alpha$-particle with velocity $v$. Initially $\alpha$-particle is at rest, but it is free to move. If gravity is neglected, then the minimum separation along the straight line of their motion will be

In a hydrogen atom, the electron and proton are bound at a distance of about $0.53\; \mathring A:$

$(a)$ Estimate the potential energy of the system in $eV$, taking the zero of the potential energy at infinite separation of the electron from proton.

$(b)$ What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in $(a)?$

$(c)$ What are the answers to $(a)$ and $(b)$ above if the zero of potential energy is taken at $1.06\;\mathring A$ separation?

Two positrons $(e^+)$ and two protons $(p)$ are kept on four corners of a square of side $a$ as shown in figure. The mass of proton is much larger than the mass of positron. Let $q$ denotes the charge on the proton as well as the positron then the kinetic energies of one of the positrons and one of the protons respectively after a very long time will be-

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

Three point charges $q, q$ and $-2 q$ are placed at the corners of an equilateral triangle of side '$L$'. Calculate work done by extemal force in moving all the charges far apart without acceleration