Question

Bohr model is applied to a particle of mass \(m\) and charge \(q\) is moving in a plane under the influence of a transverse magnetic field \(B\). The energy of the charged particle in \(n^{th}\) level will be (\(h\) = Planck's constant)

• A. \(\frac{2nhqB}{\pi m}\)
• B. \(\frac{nhqB}{2\pi m}\)
• C. \(\frac{nhqB}{4\pi m}\)
• D. \(\frac{nhqB}{\pi m}\)

Hint:

Equate the magnetic force to the centripetal force, apply the quantisation of angular momentum, and write the energy as purely kinetic.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The magnetic force on a charge moving perpendicular to \(B\) supplies the centripetal force: \(qvB=\dfrac{mv^2}{r}\), so \(mv=qBr\).

Step 2: Bohr condition
Quantised angular momentum gives \(mvr=\dfrac{nh}{2\pi}\).

Step 3: Find the radius
Substituting \(mv=qBr\): \(qBr^2=\dfrac{nh}{2\pi}\), so \(r^2=\dfrac{nh}{2\pi qB}\).

Step 4: Energy
The energy is kinetic: \(E=\dfrac{1}{2}mv^2=\dfrac{q^2B^2r^2}{2m}=\dfrac{q^2B^2}{2m}\cdot\dfrac{nh}{2\pi qB}=\dfrac{nhqB}{4\pi m}\).

Step 5: Compare the options
This is option (C).

Final Answer:
The energy in the nth level is n h q B divided by 4 pi m. \[ \boxed{\text{(C) }\dfrac{nhqB}{4\pi m}} \]