The electric potential between a proton and an electron is 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 $r_{n}$ with $n,\ n$ being the principal quantum number?

Question

Dr. Abubacker · Accepted Answer

(a) $\ \ \ \ \ \ \ U\  = eV\  = eV_{0}\ln\left( \frac{r}{r_{0}} ight)$
$	herefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |F| = \ \left| - \frac{dU}{dr} ight| = \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 Eq. (i), we have
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr = \left( \frac{nh}{2\pi} ight)\sqrt{\frac{\ m}{{eV}_{0}}}$
Or $r_{n} \propto n$

Q. The electric potential between a proton and an electron is 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 $r_{n}$ with $n,\ n$ being the principal quantum number?

Explanation

Reviewed by Dr. Abubacker

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