Question

If potential energy between a proton and an electron is given by $|U|=k e^{2} / 2 R^{3}$, where $e$ is the charge of electron and $R$ is the radius of atom, then radius of Bohr's orbit is given by $(h=$ Planck's constant, $k=$ constant $)$

• A. $\frac{k e^{2} m}{h^{2}}$
• B. $\frac{6 \pi^{2}}{n^{2}} \frac{k e^{2} m}{h^{2}}$
• C. $\frac{2 \pi}{n} \frac{k e^{2} m}{h^{2}}$
• D. $\frac{4 \pi^{2} k e^{2} m}{n^{2} h^{2}}$

Answer 1

Answer: (B)

$U=-\frac{k e^{2}}{2 R^{3}}, $
$F=-\frac{d U}{d R}=-\frac{3 k e^{2}}{2 R^{4}}$
But, $F=\frac{m v^{2}}{R} $
$\Rightarrow \frac{m v^{2}}{R}=\frac{3 k e^{2}}{2 R^{4}}$
Also, $m v R=\frac{n h}{2 \pi}$
Solve to get: $R=\frac{6 \pi^{2} k e^{2} m}{n^{2} h^{2}}$