Question

Imagine an electron moving in the potential field of a proton given by $V = V _0 \ln \frac{ r }{ r _0}$ where $r_0$ is a positive constant and $r$ is the distance between the electron and proton. Assuming Bohr's model to be applicable, the variation of $r_n$ with $n$ (principal quantum number) is

• A. $r _{ n } \propto n$
• B. $r _{ n } \propto \frac{1}{ n }$
• C. $r_n \propto n^2$
• D. $r_n \propto \frac{1}{n^2}$

Answer 1

Answer: (A)

$ U=e V=e V_0 \ln \left(\frac{r}{r_0}\right) $
$| F |=\left|-\frac{ dU }{ dr }\right|=\frac{ eV _0}{ r }$
This force provide the necessary centripetal force.
Hence $\frac{ mv ^2}{ r } =\frac{ eV _0}{ r } $
$\text { or } v =\sqrt{\frac{ eV _0}{ m }} \ldots(1)$
More over, $mvr =\frac{ nh }{2 \pi} \ldots \text { (2) }$
Dividing Eq. (2) by Eq. (1), we have
$m r=\left(\frac{n h}{2 \pi}\right) \sqrt{\frac{m}{e V_0}} \text { or } r_n \propto n$