Question

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

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

Hint:

Quantise angular momentum m v r = n h / 2 pi and use q v B = m v^2 / r.

Answer 1

The Correct Option is D

Step 1: Use the cyclotron frequency:
A charge in a field moves with angular velocity $\omega=\dfrac{qB}{m}$, independent of radius.

Step 2: Quantise the angular momentum:
$L=I\omega$ with $I=mr^2$ gives $L=mr^2\omega=\dfrac{nh}{2\pi}$. Kinetic energy is $E=\dfrac12m r^2\omega^2=\dfrac12L\omega$.

Step 3: Substitute:
$E=\dfrac12\cdot\dfrac{nh}{2\pi}\cdot\dfrac{qB}{m}=\dfrac{nhqB}{4\pi m}$. Option D.

Final Answer:
Using L = n h/2 pi and E = L omega / 2 with omega = qB/m gives n h q B/(4 pi m). \[ \boxed{\text{(D) }\dfrac{nhqB}{4\pi m}} \]