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Ionisation Energy

A particle of...

Question

A particle of mass m moves along circular orbit in a centro-symmetrical potential field $U (r) = \frac{k r^{2}}{2}$ where k is some constant. Using Bohr quantization condition, radius of $n^{t h}$ orbit is given as $r = \frac{C n^{a} h^{b}}{m^{c}}$ , where C is some constant, n is $n^{t h}$ orbit, m is mass and h is Planck's constant, then $(a + b) \times c$ is
(Write upto two digits after the decimal point.)

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Solution

$F = - \frac{d v}{d r} = - K r$
$\Rightarrow K r = \frac{m v^{2}}{r}$ ...(1)
and $m v r = \frac{n h}{2 π}$ ...(2)
From (1) and (2)
$r = \sqrt{\frac{n h}{2 π \sqrt{m K}}} \Rightarrow r = \frac{C n^{\frac{1}{2}} h^{\frac{1}{2}}}{m^{\frac{1}{4}}}$
So, a = $\frac{1}{2}$ , b = $\frac{1}{2}$ , and c = $\frac{1}{4}$

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