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

Question

Dr. Abubacker · Accepted Answer

(a) Potential energy $U = eV = eV_{0}\ln\frac{r}{r_{0}}$
$	herefore$ Force $F = - \left| \frac{dU}{dt} ight| = \frac{eV_{0}}{r}$
$	herefore$ The force will provide the necessary centripetal force. Hence $\frac{mv^{2}}{r} = \frac{eV_{0}}{r}$
$	herefore\  \Rightarrow v = \sqrt{\frac{eV_{0}}{m}}$ …(i)
and $mvr = \frac{nh}{2\pi}$ …(ii)
From equations (i) and (ii)
$mr = \left( \frac{nh}{2\pi} ight)\sqrt{\frac{m}{eV_{0}}}$ or $r \propto n$

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 be applicable, write variation of $r_{n}$ with $n,n$ being the principal quantum number

Explanation

Reviewed by Dr. Abubacker

Want more practice?