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Energy Levels

The electric ...

Question

The electric potential between a proton and an electron is given by $V = V_{0} l n \frac{r}{r_{0}}$ , 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

r_{n} \infty n

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r_{n} α \frac{1}{n}

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r_{n} α n^{2}

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r_{n} α \frac{1}{n^{2}}

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Solution

The correct option is A $r_{n} \infty n$
Potential energy $U = e V = e V_{0} l n \frac{r}{r_{0}}$
$∴$ Force $F = - ∣ ∣ \frac{d U}{d r} ∣ ∣ = \frac{e V_{0}}{r}$ . The force will provide the necessary centripetal force. Hence
$\frac{m v^{2}}{r} = \frac{e V_{0}}{r} \Rightarrow v = \sqrt{\frac{e V_{0}}{m}}$ .....(i) and $m v r = \frac{n h}{2 π}$ ....... (ii)
Dividing equation (ii) by (i) we have $m r = (\frac{n h}{2 π}) \sqrt{\frac{e V_{0}}{m}}$ or $r \infty n$

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