JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Modern Physics

A particle of mass

m

is moving in a circular orbit under the influence of the central force

F (r) = - k r

, corresponding to the potential energy

V (r) = k r^{2} /2

, where

k

is a positive force constant and

r

is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by

L = n ℏ

, where

ℏ = h / (2 π), h

is the Planck's constant, and

n

a positive integer. If

v

and

E

are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

Answer: (A), (B), (C)

Solution

The central force provides centripetal force:

k r = \frac{m v ^{2}}{r}

, giving

k r^{2} = m v^{2}

. From quantization:

n ℏ = m v r

. Dividing these equations:

\frac{k r ^{4}}{n ^{2} ℏ ^{2}} = \frac{1}{m}

, yielding

r^{2} = n ℏ \frac{1}{mk}

(A is correct). Substituting back:

v^{2} = n ℏ \frac{k}{m ^{3}}

(B is correct). Also,

\frac{L}{m r ^{2}} = \frac{v}{r} = \frac{k}{m}

(C is correct). For total energy:

E = \frac{1}{2} m v^{2} + \frac{1}{2} k r^{2} = n ℏ \frac{k}{m}

, not

\frac{n ℏ}{2} \frac{k}{m}

(D is incorrect).

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · One or more correct options