Question

A particle of mass $m$ moves in a circular orbit in a central potential field $U\left(r\right)=\frac{1}{2}k{r}^2.$ If Bohr's quantization conditions are applied, radii of possible orbitls and energy levels vary with quantum number $n$ as :

• A. $r_n\propto n^2,E_n\propto \frac{1}{n^2}$
• B. $r_n\propto \sqrt{n},E_n\propto \frac{1}{n}$
• C. $r_n\propto n,E_n\propto n$
• D. $r_n\propto \sqrt{n},E_n\propto n$

Answer 1

The correct option is D $r_n\propto \sqrt{n},E_n\propto n$

Let force of attraction towards the centre is $F$, then

$F=\frac{dU}{dr}=kr=\frac{m{v}^2}{r}=\text{centripetal force}$

$mvr=\frac{nh}{2\pi }\left[\text{Bohr's Quantization rule}\right]$

So, $\frac{m^2{v}^2}{m}=k{r}^2\Rightarrow {\left(\frac{nh}{2\pi r}\right)}^2\frac{1}{m}=k{r}^2$

$\Rightarrow r^2\propto n$

$\Rightarrow r\propto \sqrt{n}$

Also $E=K.E+P.E.$

$E=\frac{1}{2}k{r}^2+\frac{1}{2}m{v}^2=\frac{1}{2}k{r}^2+\frac{1}{2}\times \left(k{r}^2\right)$

$=k{r}^2\propto n[\text{as k is constant and}{r}^2\propto n]$