Question

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

Answer 1

Step 1: Given data

Particle of mass $=m$

Potential $U\left(r\right)=U_o{r}^4$

Step 2: Finding the value of $\alpha$

Formula used

$F=\frac{-dU}{dr}$

Where, $F=$Force,

$F=-4U_0{r}^3$
|F| = 4U₀r³……… $\left(i\right)$

Now also

$F=\frac{m{v}^2}{r}$ …………. $\left(ii\right)$

$\Rightarrow 4U_0{r}^3=\frac{m{v}^2}{r}$

$m{v}^2=4U_0{r}^4$

$v\propto r^2$

Also we that,

$mvr=\frac{nh}{2\pi }$

$r^3\propto n$

$r\propto n{}^{\frac{1}{3}}$

Hence, The value of $\alpha$ in $n{}^{\frac{1}{\alpha }}$ is $3$.