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 condition is applied, radii of possible orbitals 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 $r_n\propto \sqrt{n},E_n\propto n$

$F=\frac{dV}{dR}=kr=\frac{m{v}^2}{r}$
$mvr=\frac{nh}{2\pi }$
$r^2\infty n$
$r^2\infty \sqrt{n}$
$E=\frac{1}{2}k{r}^2+\frac{1}{2}m{v}^2\infty r^2$