Question

The electric potential between a proton and an electron is r given by $V = V_0 in\frac{r}{r_0}$, where $r_0$ is a constant. Assuming Bohr's model to be applicable, write variation of rn with n, n being the principal quantum number.

• 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)

$U=eV=eV_0 in\bigg(\frac{r}{r_0}\bigg)$
$|F|=\bigg|-\frac{dU}{dr}\bigg|=\frac{eV_0}{r}$
This force will provide the necessary centripetal force.
Hence, $\frac{mv^2}{r}=\frac{eV_0}{r}$
or $v=\sqrt{\frac{eV_0}{m}}$...(i)
Moreover, $mvr=\frac{nh}{2\pi}$...(ii)
Dividing Eq. (ii) by (i), we have
or $r_n \propto n$