Modern Physics 1 Question 21
18. The electric potential between a proton and an electron is given by $V=V _0 \ln \frac{r}{r _0}$, where $r _0$ is a constant. Assuming Bohr’s model to be applicable, write variation of $r _n$ with $n, n$ being the principal quantum number.
(2003, 2M)
(a) $r _n \propto n$
(b) $r _n \propto \frac{1}{n}$
(c) $r _n \propto n^{2}$
(d) $r _n \propto \frac{1}{n^{2}}$
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Solution:
$$ \begin{gathered} U=e V=e V _0 \ln \frac{r}{r _0} \\ |F|=\left|-\frac{d U}{d r}\right|=\frac{e V _0}{r} \end{gathered} $$
This force will provide the necessary centripetal force. Hence, $\quad \frac{m v^{2}}{r}=\frac{e V _0}{r}$
$$ \begin{aligned} & \text { or } \quad v=\sqrt{\frac{e V _0}{m}} \\ & \text { Moreover, } \quad m v r=\frac{n h}{2 \pi} \end{aligned} $$
Dividing Eq. (ii) by (i), we have
$$ \begin{aligned} m r & =\frac{n h}{2 \pi} \sqrt{\frac{m}{e V _0}} \\ \text { or } \quad r _n & \propto n \end{aligned} $$
