Nuclear Physics And Radioactivity Question 46
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 [IIT-JEE (Screening) 2003]
Options:
A) $ r_{n}\propto n $
B) $ r_{n}\propto 1/n $
C) $ r_{n}\propto n^{2} $
D) $ r_{n}\propto 1/n^{2} $
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Answer:
Correct Answer: A
Solution:
Potential energy $ U=eV=eV_{0}\ln \frac{r}{r_{0}} $ Force $ F=-| \frac{dU}{dr} |=\frac{eV_{0}}{r} $ . The force will provide the necessary centripetal force. Hence $ \frac{mv^{2}}{r}=\frac{eV_{0}}{r} $
Þ $ v=\sqrt{\frac{eV_{0}}{m}} $ ?..(i) and $ mvr=\frac{nh}{2\pi } $ ?..(ii) From equation (i) and(ii) $ mr=( \frac{nh}{2\pi } )\sqrt{\frac{m}{eV_{0}}} $ or r µ n
BETA
