Question

A particle of mass $m$ moves in a circular orbit in a central potential field U(r) = $U_0{r}^4$. If Bohr's quantization conditions are applied, radii of possible orbital $r_n$ vary with $n^{\frac{1}{\alpha }}$, where $\alpha$ is

Answer 1

We know that, $F=-\frac{dU}{dr}$
$F=-\frac{d}{dr}\left(U_0{r}^4\right)$
$F=-4U_0{r}^3$

Magnitude of this force is equal to centripetal force.
$\therefore \frac{m{v}^2}{r}=4U_0{r}^3$
$m{v}^2=4U_0{r}^4$
So, $v\propto r^2$

Now, $\because mvr=\frac{nh}{2\pi }$
Thus, $r^3\propto n$
$\Rightarrow r\propto (n{)}^{\frac{1}{3}}=(n{)}^{\frac{1}{\alpha }}$

So the value of $\alpha =3$.