Question

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

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

Answer 1

Answer: (A)

$r_{n} \propto n$

Answer 2

Potential energy $U = eV = eV_{0}\ln\frac{r}{r_{0}}$

∴ Force $F = - \left| \frac{dU}{dr} \right| = \frac{eV_{0}}{r}$.

The force will provide the necessary centripetal force.

Hence $\frac{mv^{2}}{r} = \frac{eV_{0}}{r}$ ⇒ $v = \sqrt{\frac{eV_{0}}{m}}$…..(i) and

$mvr = \frac{nh}{2\pi}$ …..(ii)

Dividing equation (ii) by (i) we have $mr = \left( \frac{nh}{2\pi} \right)\sqrt{\frac{m}{eV_{0}}}$

or r ∝ n