Question

The electron of a hydrogen atom revolves round the proton in a circular $n^{\text{th }}$ orbit of radius $r_n=\frac{{\epsilon }_0{n}^2{h}^2}{\left(\pi m{e}^2\right)}$ with a speed $v_n=\frac{e^2}{2{\epsilon }_{0}nh}$. 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,
$T=\frac{2\pi r}{v}=\frac{\frac{2\pi \times {\epsilon }_0{n}^2{h}^2}{\pi m{e}^2}}{\frac{e^2}{2{\epsilon }_{0}nh}}$
$T=\frac{4{\epsilon }_0^2{n}^2{h}^3}{m{e}^4}$
Current, $i=\frac{e}{T}=\frac{e}{\frac{4{\epsilon }_0^2{n}^2{h}^2}{m{e}^4}}$
$i=\frac{m{e}^5}{4{\epsilon }_0^2{n}^2{h}^5}=$
i.e., $i\propto e^5$