A particle of mass moves in circular orbits with potential energy where is a positive constant and is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by and its speed an

Question

A particle of mass $m$ moves in circular orbits with potential energy $V (r) = F r,$ where $F$ is a positive constant and $r$ is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by $R$ and its speed and energy are denoted by $v$ and $E$ , respectively, then for the $n^{th}$ orbit (here $h$ is the Planck's constant) [2020]

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(2, 3)

Given: Potential energy of the particle of mass $V (r) = F r,$ moving in a circular orbit, $V (r) = F r$

$\Rightarrow F = \frac{V (r)}{r}$

Also, centripetal force,

$F = \frac{m v^{2}}{R}$ $\dots (1)$

According to Bohr's second postulate,

$m v R = \frac{n h}{2 π}$ $\Rightarrow v = \frac{n h}{2 π m R}$

Putting this value of $v$ in Eq. (1),

$F = \frac{m}{R} \times \frac{n^{2} h^{2}}{2 π^{2}} \times \frac{1}{m^{2} R^{2}}$ $\Rightarrow R = {(\frac{n^{2} h^{2}}{4 π^{2} m F})}^{1 / 3}$ $∴$ $R \propto n^{2 / 3}$

Now putting this value of $R$ in $v = \frac{n h}{2 π m R}$

$v = \frac{n h}{2 π m} {(\frac{4 π^{2} m F}{n^{2} h^{2}})}^{1 / 3}$ $∴$ $v \propto n^{1 / 3}$

Hence, option (2) is correct.

Total energy, $E = \frac{1}{2} m v^{2} + V (r) = \frac{1}{2} m v^{2} + F R$

$\Rightarrow E = \frac{1}{2} m (\frac{n^{2 / 3} h^{2 / 3} F^{2 / 3}}{2^{2 / 3} π^{2 / 3} m^{4 / 3}}) + F {(\frac{n^{2} h^{2}}{4 π^{2} m F})}^{1 / 3}$

$\Rightarrow E = {(\frac{n^{2} h^{2} F^{2}}{4 π^{2} m})}^{1 / 3} [\frac{1}{2} + 1]$ or, $E = \frac{3}{2} {(\frac{n^{2} h^{2} F^{2}}{4 π^{2} m})}^{1 / 3}$

Hence, option (3) is correct.

Solution

1 $R \propto n^{1 / 3}$ and $v \propto n^{2 / 3}$

2 $R \propto n^{2 / 3}$ and $v \propto n^{1 / 3}$

3 $E = \frac{3}{2} {(\frac{n^{2} h^{2} F^{2}}{4 π^{2} m})}^{1 / 3}$

4 $E = 2 {(\frac{n^{2} h^{2} F^{2}}{4 π^{2} m})}^{1 / 3}$

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Solution

1 R∝n1/3 and v∝n2/3

2 R∝n2/3 and v∝n1/3

3 E=32(n2h2F24π2m)1/3

4 E=2(n2h2F24π2m)1/3

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1 $R \propto n^{1 / 3}$ and $v \propto n^{2 / 3}$

2 $R \propto n^{2 / 3}$ and $v \propto n^{1 / 3}$

3 $E = \frac{3}{2} {(\frac{n^{2} h^{2} F^{2}}{4 π^{2} m})}^{1 / 3}$

4 $E = 2 {(\frac{n^{2} h^{2} F^{2}}{4 π^{2} m})}^{1 / 3}$