Question

Suppose, the electron in a hydrogen atom makes transition from n = 3 to n = 2 in 10⁻⁸ s. The order of the torque acting on the electron in this period, using the relation between torque and angular momentum as discussed in the chapter on rotational mechanics is

(a) 10⁻³⁴ N m
(b) 10⁻²⁴ N m
(c) 10⁻⁴² N m
(d) 10⁻⁸ N m

Answer 1

(c) 10⁻⁴² N-m

The angular momentum of the electron for the nth state is given by
$L_{\mathrm{n}}=\frac{n\mathrm{h}}{2\mathrm{\pi }}$
Angular momentum of the electron for n = 3, $L_{\mathrm{i}}=\frac{3h}{2\mathrm{\pi }}$

Angular momentum of the electron for n = 2, $L_{\mathrm{f}}=\frac{2h}{2\mathrm{\pi }}$

The torque is the time rate of change of the angular momentum.
$$\begin{gathered} \mathrm{Torque},\tau =\frac{L_{\mathrm{f}}-L_{\mathrm{i}}}{t} \\ =\frac{(2\mathrm{h}/2\mathrm{\pi })-(3\mathrm{h}/2\mathrm{\pi })}{{10}^{-8}} \\ =\frac{-(\mathrm{h}/2\mathrm{\pi })}{{10}^{-8}} \\ =\frac{-{10}^{-34}}{{10}^{-8}}\left[\because \frac{\mathrm{h}}{2\mathrm{\pi }}\approx {10}^{-34}\mathrm{J}-\mathrm{s}\right] \\ =-{10}^{-42}\mathrm{N}-\mathrm{m} \end{gathered}$$
The magnitude of the torque is 10^$-$42 N-m.