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)

$U=\text{e}\text{V}=\text{e}{\left(\text{V}\right)}_0\text{ ln}\left(\frac{\text{r}}{{\left(\text{r}\right)}_0}\right)$
$\left|F\right|=\left|-\frac{\text{d}\text{U}}{\text{d}\text{r}}\right|=\frac{\text{e}{\text{V}}_0}{r}$
This force provides the necessary centripetal force.
Hence
$\frac{m{v}^2}{r}=\frac{\text{e}{\text{V}}_0}{r}$
or $v=\sqrt{\frac{\text{e}{\text{V}}_0}{m}}$ ...(1)
Moreover,
$\text{mvr}=\frac{\text{nh}}{2\pi }$ ...(2)
Dividing Eq. (2) by Eq. (1), we have
$\text{mr}=\left(\frac{\text{nh}}{2\pi }\right)\sqrt{\frac{\text{m}}{\text{e}{\left(\text{V}\right)}_0}}$ or ${\text{r}}_{\text{n}}\propto \text{n}$