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

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$r_{n} \propto n$

$r_{n} \propto 1 / n$

$r_{n} \propto n^{2}$

$r_{n} \propto 1 / n^{2}$

answer is A.

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Detailed Solution

potential energy $U = eV = {eV}_{o} l n \frac{r}{r_{o}}$

$∴ Force F = - |\frac{dU}{dr}| = \frac{{eV}_{o}}{r} .$

$∴$ The force will provide the necessary centripetal

$\frac{m v^{2}}{r} = \frac{e v_{0}}{r} v = \sqrt{\frac{e v_{0}}{m}} \dots \dots \dots \dots \dots \dots (1) a n d m v r = \frac{n h}{2 π} \dots \dots \dots \dots \dots \dots (2) S o l v i n g r \infty n$

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