Q: 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 the applicable, write variation of r_{n} with n, n being the principal quantum number

(A) r_{n} ∝ n

(B) r_{n} ∝ 1/n

(C) r_{n} ∝ n^{2}

(D) r_{n} ∝ 1/n^{2}

Ans: (A)

Sol: $\large U = e V = e V_0 ln \frac{r}{r_0}$

$\large F = |-\frac{dU}{dr}| = \frac{e V_0}{r}$

This force will provide necessary centripetal force.

$\large \frac{m v^2}{r} = \frac{e V_0}{r} $

$\large v = \sqrt{\frac{e V_0}{m}}$ …(i)

According to Bohr’s theory;

$\large m v r = n\frac{h}{2\pi}$ …(ii)

Dividing (ii) by (i)

$\large m r = ( \frac{n h}{2\pi}) \sqrt{\frac{m}{e V_0}}$

r_{n} ∝ n