Question

A uniform magnetic field b exists in a region. An electron of charge q and mass m moving with velocity v enters the region in a direction perpendicular to the magnetic field. Considering Bohr angular momentum quantization, which of the following statement(s) is/are true?

• A. The radius of n^m orbit \(r_n\)∞\(\sqrt n\)
• B. The minimum velocity of the electron is \(\sqrt{\frac{qBℏ}{m}}\)
• C. The energy of the nth level E_n∞n
• D. Transition frequency ω between two successive levels is independent of n.

Answer 1

The Correct Option is A, B, C, D

Radius of the circular path of moving electron:
\[ r_n = \frac{mv_n}{Bq} \tag{i} \] By Bohr's quantisation:
\[ mv_n r_n = \frac{n\hbar}{2\pi} = n\hbar \tag{ii} \] From (i) and (ii):
\[ Bq r_n \cdot r_n = n\hbar \Rightarrow r_n^2 = \frac{n\hbar}{Bq} \Rightarrow r_n \propto \sqrt{n} \] Also, using the quantisation relation again:
\[ \frac{mv_n^2}{Bq} = n\hbar \Rightarrow v_n^2 = \frac{Bq n \hbar}{m^2} \] So, velocity:
\[ v_n = \frac{\sqrt{Bq \hbar n}}{m} \] For minimum velocity (n = 1):
\[ v_{\text{min}} = \sqrt{\frac{Bq \hbar}{m}} \] Also, as energy levels are quantized: 
\[ h\nu \propto n \Rightarrow \nu \propto n \]