Question

The electron of a hydrogen atom revolves round the proton in a circular $n^{\text {th }}$ orbit of radius $r_{n}=\frac{\varepsilon_{0} n^{2} h^{2}}{\left(\pi m e^{2}\right)}$ with a speed $v_{n}=\frac{e^{2}}{2 \varepsilon_{0} n h}$. The current due to the circulating charge is proportional to

• A. $e^{2}$
• B. $e^{3}$
• C. $e^{5}$
• D. $e^{6}$

Answer 1

Answer: (C)

Time period of electron,
$T=\frac{2 \pi r}{v}=\frac{\frac{2 \pi \times \varepsilon_{0} n^{2} h^{2}}{\pi m e^{2}}}{\frac{e^{2}}{2 \varepsilon_{0} n h}}$
$T=\frac{4 \varepsilon_{0}^{2} n^{2} h^{3}}{m e^{4}}$
Current, $i=\frac{e}{T}=\frac{e}{\frac{4 \varepsilon_{0}^{2} n^{2} h^{2}}{m e^{4}}}$
$i=\frac{m e^{5}}{4 \varepsilon_{0}^{2} n^{2} h^{5}}=$
i.e., $i \propto e^{5}$