Q145 / 180 · JIPMER 2017 · Physics

The potential of an atom is given by $V=V_0 \log _e\left(r / r_0\right)$ where $r_0$ is a constant and $r$ is the radius of the orbit. Assuming Bohr's model to be applicable, which variation of $r_n$ with $n$ is possible ( $n$ being principal quantum number)?

  • Correct answer
    A
    $r_n \propto n$
  • B
    $r_n \propto 1 / n$
  • C
    $r_n \propto n^2$
  • D
    $r_n \propto 1 / n^2$

Solution

Given that, $V=V_0 \log _e\left(r / r_0\right)$ Field $E=-\frac{d V}{d r}=-V_0\left(\frac{r}{r_0}\right)$ $\begin{aligned} & e E=\frac{m V^2}{r} \text { or } \frac{e V_0 I_0}{r}=\frac{m V^2}{r} \\ & V=\left(\frac{e V_0 I_0}{m}\right)^{1 / 2}\end{aligned}$ $m V=\left(m_e V_0 r_0\right)^{1 / 2}=$ constant $\therefore m v r=\frac{n h}{2 \pi}$, Bohr's quantum condition or $r \propto n$

Hence, the correct answer is option: 'A'.

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