Question

The electron of hydrogen atom is considered to be revolving round a proton in circular orbit of radius $h^2/m{e}^2$ with velocity $e^2/h,$ where h=$\hslash /2\pi$. The current $i$ is

• A. $\frac{4{\pi }^{2}m{e}^5}{{\hslash }^2}$
• B. $\frac{4{\pi }^{2}m{e}^5}{{\hslash }^3}$
• C. $\frac{4{\pi }^2{m}^2{e}^2}{{\hslash }^3}$
• D. $\frac{4{\pi }^2{m}^2{e}^5}{{\hslash }^3}$

Answer 1

Answer: (B)

Current, $i=\frac{e}{t}=\frac{e}{2\pi r/v}=\frac{ev}{2\pi r}$
Here, $v=e^2/h$ and $r=h^2/m{e}^2$
$\therefore i=\frac{e\left(e^2/h\right)}{2\pi \left(h^2/m{e}^2\right)}=\frac{e^3\times m{e}^2}{2\pi h^3}$
$=\frac{m{e}^5}{2\pi h^3}$
$\because h=\hslash /2\pi$ (given)
$\therefore i=\frac{m{e}^5}{2\pi {\left(\frac{\hslash }{2\pi }\right)}^3}=\frac{4{\pi }^{2}m{e}^5}{{\hslash }^3}$