Question

Show that the radius of the orbit in hydrogen atoms varies as $n^2$, where n is the principal quantum number of the atom.

Answer 1

The centripetal acceleration of the electron in a Bohr's atom is due to the Coulomb attraction of electron towards nucleus.

Thus, $\frac{m{v}^2}{r}=\frac{1}{4\pi {ϵ}_0}\frac{Z{e}^2}{r^2}$

Also Bohr's postulate states, $mvr=\frac{nh}{2\pi }$

Substituting $v$ from second equation into the first gives,
$r\propto n^2$