SATHEE: Modern Physics 1 Question 21

Modern Physics 1 Question 21

18. 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, 2M)

(a) $r_{n} \propto n$

(b) $r_{n} \propto \frac{1}{n}$

(d) $r_{n} \propto \frac{1}{n^{2}}$

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Solution:

$\begin{matrix} U = e V = e V_{0} \ln \frac{r}{r_{0}} \\ | F | = | - \frac{d U}{d r} | = \frac{e V_{0}}{r} \end{matrix}$

This force will provide the necessary centripetal force. Hence, $\frac{m v^{2}}{r} = \frac{e V_{0}}{r}$

$\begin{aligned} or v = \sqrt{\frac{e V_{0}}{m}} \\ Moreover, m v r = \frac{n h}{2 π} \end{aligned}$

Dividing Eq. (ii) by (i), we have

$\begin{aligned} m r & = \frac{n h}{2 π} \sqrt{\frac{m}{e V_{0}}} \\ or r_{n} & \propto n \end{aligned}$

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Modern Physics 1 Question 21

18. 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.

Solution:

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18. 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.