Question

The electric potential between a proton and an electron is given by $\text{V} = \text{V}_{0} \text{ ln } \frac{\text{r}}{\text{r}_{0}}$ where $r_{0}$ is a constant. Assuming Bohr's model to be applicable, write a variation of $r_{n}$ with $n$ , $n$ being the principal quantum number.

• A. $r_{\text{n}} \propto n$
• B. $r_{\text{n}} \propto \frac{1}{n}$
• C. $r_{\text{n}} \propto n^{2}$
• D. $r_{\text{n}} \propto \frac{1}{n^{2}}$

Answer 1

Answer: (A)

$U=\text{e}\textit{V}=\text{e}\left(\textit{V}\right)_{0}\text{ ln}\left(\frac{\textit{r}}{\left(\textit{r}\right)_{0}}\right)$
$\left|F\right|=\left|- \frac{\text{d} \textit{U}}{\text{d} \textit{r}}\right|=\frac{\text{e} \textit{V}_{0}}{r}$
This force provides the necessary centripetal force.
Hence
$\frac{m v^{2}}{r}=\frac{\text{e} \textit{V}_{0}}{r}$
or $v=\sqrt{\frac{\text{e} \textit{V}_{0}}{m}}$ ...(1)
Moreover,
$\textit{mvr}=\frac{\textit{nh}}{2 \pi }$ ...(2)
Dividing Eq. (2) by Eq. (1), we have
$\textit{mr}=\left(\frac{\textit{nh}}{2 \pi }\right)\sqrt{\frac{\textit{m}}{\text{e} \left(\textit{V}\right)_{0}}}$ or $\text{r}_{\text{n}} \propto \text{n}$