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JEE Main & Advanced Physics Nuclear Physics And Radioactivity Question Bank Critical Thinking

  • question_answer
    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]

    A)            \[{{r}_{n}}\propto n\]      

    B)            \[{{r}_{n}}\propto 1/n\]

    C)            \[{{r}_{n}}\propto {{n}^{2}}\]                                           

    D)            \[{{r}_{n}}\propto 1/{{n}^{2}}\]

    Correct Answer: A

    Solution :

               Potential energy \[U=eV=e{{V}_{0}}\ln \frac{r}{{{r}_{0}}}\] Force \[F=-\left| \frac{dU}{dr} \right|=\frac{e{{V}_{0}}}{r}\].  The force will provide the necessary centripetal force. Hence \[\frac{m{{v}^{2}}}{r}=\frac{e{{V}_{0}}}{r}\] Þ \[v=\sqrt{\frac{e{{V}_{0}}}{m}}\]   ?..(i)                    and \[mvr=\frac{nh}{2\pi }\]                                                                    ?..(ii)            From equation (i) and(ii)  \[mr=\left( \frac{nh}{2\pi } \right)\sqrt{\frac{m}{e{{V}_{0}}}}\] or r µ n


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