The electric potential between a proton and an electron is given by where is a constant. Assuming Bohr's model to be applicable, write variation of with , being the principal quantum number? [2003]

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? [2003]

Smart Achievers · Accepted Answer

(1)

Given potential energy $(U)$ between electron and proton

$U = e V_{0} \ln (\frac{r}{r_{0}})$ $[∵ | U | = e V]$

$∴$ $| F |$ $=$ $| - \frac{d U}{d r} |$ $= \frac{d}{d r}$ $[e V_{0} \ln (\frac{r}{r_{0}})] = \frac{e V_{0}}{r_{0}} \times \frac{1}{r}$

This force will provide the necessary centripetal force.

$∴$ $\frac{m v^{2}}{r} = \frac{e V_{0}}{r r_{0}} \Rightarrow m v^{2} = \frac{e V_{0}}{r_{0}}$ $\dots (i)$

As per Bohr's postulate, $m v r = \frac{n h}{2 π}$ $\dots (i i)$

From Eqs. (i) and (ii),

$\frac{m^{2} v^{2} r^{2}}{m v^{2}} = \frac{n^{2} h^{2} r_{0}}{4 π^{2} V_{0} e}$ $\Rightarrow r^{2} = \frac{n^{2} h^{2} r_{0}}{4 π^{2} V_{0} m e}$

$∴$ $r \propto n$

Solution

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 [2003]

1 $r_{n} \propto n$

2 $r_{n} \propto \frac{1}{n}$

3 $r_{n} \propto n^{2}$

4 $r_{n} \propto \frac{1}{n^{2}}$

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Solution

Q. The electric potential between a proton and an electron is given by V=V0ln(rr0), where r0 is a constant. Assuming Bohr's model to be applicable, write variation of rn with n, n being the principal quantum number [2003]

1 rn∝n

2 rn∝1n

3 rn∝n2