A small particle of mass $m$ moves in such a way that the potential energy $U=\frac{1}{2}m{b}^2{r}^2$, where $b$ is a constant and $r$ is the distance of the particle from the origin (Nucleus). Assuming Bohr model of quantization of angular momentum and circular orbits, show that the radius of the $n^{\mathrm{th}}$ allowed orbit is proportional to $n$.

Suppose in certain conditions only those transitions are allowed to hydrogen atoms in which the principal quantum number $n$ change by $2$.

(i) Find the smallest wavelength emitted by hydrogen

(ii) List the wavelengths emitted by hydrogen in the visible range ($380 \mathrm{nm}$ to $780 \mathrm{nm}$) ($h=4.14\times {10}^{-15} \mathrm{eV} \mathrm{s} \text{\&} \mathrm{c}=3\times {10}^8 \mathrm{m} {\mathrm{s}}^{-1}$)

An electron in the ground state of hydrogen atoms is revolving in anti clock wise direction in a circular orbit of radius $R$.

(i) Obtain an expression for the orbital magnetic dipole moment of the electron.
(ii) The atom is placed in a uniform magnetic induction, such that the plane normal to the electron orbit makes an angle of $30°$ with the magnetic induction. Find the torque experienced by the orbiting electron.