Question

An electron and a proton are separated by a distance $r$ so that the potential energy between them is $u=k\log r$, where $k$ is a constant. In such an atom, radius of $nth$ Bohr's orbit is :

• A. $\frac{2nh}{\pi \sqrt{mk}}$
• B. $\frac{nh}{2\pi \sqrt{2mk}}$
• C. $\frac{nh}{2\pi \sqrt{mk}}$
• D. $\frac{nh}{4\pi \sqrt{mk}}$

Answer 1

The correct option is C $\frac{nh}{2\pi \sqrt{mk}}$

$\begin{array}{cc}u& =k\log r\\ F& =\frac{-du}{dr}=\frac{k}{r}\end{array}$

$\begin{array}{c}\text{cas we know centripital force}F=\frac{m{v}^2}{r}\\ \frac{k}{r}=\frac{m{v}^2}{r}\\ v=\sqrt{\frac{k}{m}}\end{array}$

$\text{now by angular momentum conservation}$

$\begin{array}{c}mvr=\frac{nh}{2\pi }\\ mr\sqrt{\frac{k}{m}}=\frac{nh}{2\pi }\\ r=\frac{nh}{2\pi \sqrt{km}}\end{array}$