Question 406: Atoms & Nuclei

Question

A particle of mass m moves in a circular orbit in a central potential field U(r) = U0r4. If Bohr's quantization conditions are applied, radii of possible orbitals rn vary with $${n^{{1 \over \alpha }}}$$, where $$\alpha$$ is ____________.

Accepted Answer

$$\overrightarrow F  =  - {{d\overrightarrow u } \over {dr}}$$
$$ =  - {d \over {dr}}({U_0}{r^4})$$
$$\overrightarrow F  =  - 4{U_0}{r^3}$$
$$ \because $$ $${{m{v^2}} \over r} = 4{U_0}{r^3}$$
$$m{v^2} = 4{U_0}{r^4}$$
Then $$v \propto {r^2}$$
$$ \because $$ $$mvr = {{nh} \over {2\pi }}$$
Then $${r^3}\propto\,n$$
$$r\,\propto \,{(n)^{{1 \over 3}}}$$
So the value of $$\alpha  = 3$$

Atoms & Nuclei

A particle of mass m moves in a circular orbit in a central potential field U(r) = U0r4. If Bohr's quantization conditions are applied, radii of possible orbitals rn vary with ${n^{{1 \over \alpha }}}, where \alpha$ is ____________.

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