Modern Physics Ques 29
- 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) $r _n \propto \frac{1}{n}$
(c) $r _n \propto n^{2}$
(d) $r _n \propto \frac{1}{n^{2}}$
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Answer:
Correct Answer: 29.(a)
Solution:
Formula:
Radius And Speed Of Electron In Hydrogen Like Atoms:
$ \begin{gathered} U=e V=e V _0 \ln \frac{r}{r _0} \\ |F|=\left|-\frac{d U}{d r}\right|=(\frac{e V _0}{r}) \end{gathered} $
This force will provide the necessary centripetal force.
Hence, $\quad \frac{m v^{2}}{r}=\frac{e V _0}{r}$
$ \begin{aligned} & \text { or } \quad v=\sqrt{\frac{e V _0}{m}} \quad …….(i)\\ & \text { Moreover, } \quad m v r=\frac{n h}{2 \pi} \quad …….(ii) \end{aligned} $
Dividing Eq. (ii) by (i), we have
$ \begin{aligned} m r & =(\frac{n h}{2 \pi}) \sqrt{\frac{m}{e V _0}} \\ \text { or } \quad r _n & \propto n \end{aligned} $
BETA
