Question

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

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

Answer 1

Answer: (B)

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