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
$e^2 / 4 \pi \varepsilon_0 m v^2$
$5 e^2 / 4 \pi \varepsilon_0 m v^2$
$2 e^2 / 4 \pi \varepsilon_0 m v^2$
$4 e^2 / 4 \pi \varepsilon_0 m v^2$
A metallic sphere has a charge of $10\,\mu C$. A unit negative charge is brought from $A$ to $B$ both $100\,cm$ away from the sphere but $A$ being east of it while $B$ being on west. The net work done is........$joule$
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?
A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $xy-$ plane centered at origin. A point charge $q$ is moving towards the ring along the $z-$ axis and has speed $v$ at $z = 4a$. The minimum value of $v$ such that it crosses the origin is
On rotating a point charge having a charge $q$ around a charge $Q$ in a circle of radius $r$. The work done will be
Calculate potential energy of a point charge $-q$ placed along the axis due to a charge $+ Q$ uniformly distributed along a ring of radius $R$. Sketch $P.E.$ as a function of axial distance $z$ from the centre of the ring. Looking at graph, can you see what would happen if $-q$ is displaced slightly from the centre of the ring (along the axis) ?