Question

A particle of mass $1Kg$ & charge $\frac{1}{3}\mu C$ is projected towards a nonconducting fixed spherical shell having the same charge uniformly distributed on its surface. Find the minimum initial velocity of projection required if the particle just grazes the shell.

Answer 1

Let the initial velocity of particle at infinity be u and the final velocity when particle is grazing the shell be v. Since, there is no external torque, so angular momentum will be conserved.

Hence, applying conservation of angular momentum about the centre of spherical shell, we have

$mu\left(0.5\right)=mv\left(1\right)⟹v=\frac{u}{2}$

Since there is no loss in energy, energy of the system will be conserved, hence

applying conservation of energy equation, we have

Initial kinetic energy = work done in bringing the particle from infinity + final kinetic energy

$\therefore \frac{1}{2}m{u}^2=\left(\frac{kq}{R}\right)q+\frac{1}{2}m{v}^2$

On substituting values,

$\frac{1}{2}\left(1\right)\left(u^2\right)=\frac{9\times {10}^9\times \frac{1}{3}\times {10}^{-6}\times \frac{1}{3}\times {10}^{-6}}{1\times {10}^{-6}}+\frac{1}{2}\left(1\right){\left(\frac{u}{2}\right)}^2$

On solving for u, we get

$u^2=\frac{8}{3}⟹u=2\sqrt{\frac{2}{3}}m/s$