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Is Electrostatic Force Conservative

The electric ...

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, Find the relation between $r_{n}$ and $n .$ where, $n$ being the principal quantum number.

r_{n} \propto n

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

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r_{n} \propto n^{2}

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

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Solution

The correct option is A $r_{n} \propto n$
Given, $V = V_{0} ln \frac{r}{r_{0}}$

$∵ E = - \frac{d V}{d r}$

$= - \frac{d}{d r} [V_{0} ln \frac{r}{r_{0}}] = - V_{0} (\frac{r_{0}}{r}) \times \frac{1}{r_{0}}$

$∴ E = - \frac{V_{0}}{r}$

$\Rightarrow | \begin{matrix} E \end{matrix} | = \frac{V_{0}}{r}$

Since, bohr's atomic model is applicable, thus necessary centripetal force required for the electron is,

$e E = \frac{m v^{2}}{r}$

$\frac{V_{0}}{r} e = \frac{m v^{2}}{r}$

$\Rightarrow v = \sqrt{\frac{e V_{0}}{m}} . . . . . . (1)$

From de-Broglie postulates,

$m v r_{n} = \frac{n h}{2 π}$

$\Rightarrow v = \frac{n h}{2 π m r_{n}} . . . . . . (2)$

From (1) and (2) we get,

$∴ r_{n} = \frac{n h}{2 π \sqrt{m e V_{0}}}$

$\Rightarrow r_{n} \propto n$

 Hence, $(A)$ is the correct answer.

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