NTA Abhyas 2020
- Exam
- NTA Abhyas
- Year
- 2020
- Subject
- Physics
- Chapter
- Atomic Physics
NTA Abhyas 2020 · Physics · Atomic Physics
Imagine an electron moving in the potential field of a proton given...
Imagine an electron moving in the potential field of a proton given by \text{V} = {V}_{0} \text\ln \dfrac{\text{r}}{{r}_{0}} where is a positive constant and is the distance between the electron and proton. Assuming Bohr's model to be applicable, the variation of with & (principal quantum number) is
Options
- A
- B{r}_{n} ∝ \dfrac{1}{\text{n}}
- C{r}_{n} ∝ n^{2}
- D{r}_{n} ∝ \dfrac{1}{n^{2}}
Correct answer
A.
Step-by-step solution
\text{U} = \text{eV} = {eV}_{0} \text\ln \left(\dfrac{\text{r}}{{r}_{0}}\right)<br /><br />\left|\text{F}\right| = \left|- \dfrac{\text{dU}}{\text{dr}}\right| = \dfrac{{eV}_{0}}{\text{r}}<br /><br />This force provide the necessary centripetal force.<br /><br />Hence <br /><br /> \dfrac{{mv}^{2}}{\text{r}} = \dfrac{{eV}_{0}}{\text{r}}<br /><br />or \text{v} = \sqrt{\dfrac{{eV}_{0}}{\text{m}}}...(1)<br /><br />Moreover,<br /><br />\text{mvr} = \dfrac{\text{nh}}{2 \pi}...(2)<br /><br />Dividing Eq. (2) by Eq. (1), we have<br /><br />\text{mr} = \left(\dfrac{\text{nh}}{2 \pi}\right) \sqrt{\dfrac{\text{m}}{{eV}_{0}}} or{r}_{n} ∝