A particle of mass m moves in circular orbits with potential energy...

A particle of mass

m

moves in circular orbits with potential energy

U\left(r\right) = Fr

, 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)

Options

A $R ∝ {n}^{1 / 3}$ and $v ∝ {n}^{2 / 3}$
B $R ∝ {n}^{2 / 3}$ and $v ∝ {n}^{1 / 3}$
C $E = \dfrac{3}{2}{\left(\dfrac{{n}^{2}{h}^{2}{F}^{2}}{4{\pi}^{2}m}\right)}^{1 / 3}$
D $E = 2{\left(\dfrac{{n}^{2}{h}^{2}{F}^{2}}{4{\pi}^{2}m}\right)}^{1 / 3}$

B. $R ∝ {n}^{2 / 3}$ and $v ∝ {n}^{1 / 3}$

$U = F \cdot r$

${F}_{c} = - \dfrac{dU}{dr} = - F$

$\dfrac{m{v}^{2}}{r} = F$

$mur = \dfrac{nh}{2\pi}$

$⇒ \dfrac{\left(mu{r}^{2}\right)}{mu\left(4r\right)} = \dfrac{{n}^{2}{h}^{2}}{4{\pi}^{2} \cdot F}$

$⇒ m{r}^{3} = \dfrac{{n}^{2}{h}^{2}}{4{\pi}^{2}F}$

$⇒ r = {\left(\dfrac{{n}^{2}{h}^{2}}{4{\pi}^{2}mF}\right)}^{1 / 3}$

$⇒ r ∝ {n}^{2 / 3}$

$E = U + K = F \cdot r + 1 / 2m{v}^{2}$

$= F \cdot r + 1 / 2F \cdot r$

$E = 3 / 2F \cdot r$

Put value of $r$ ,

$E = \dfrac{3}{2}Fr{\left(\dfrac{{n}^{2}{h}^{2}}{4{\pi}^{2}mF}\right)}^{1 / 3}$