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

The correct answer is/are option(s):
(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 2

Understanding Bohr's Theory
1. Bohr's Theory for Radius (r):
From Bohr's theory, the radius r of an electron's orbit in a magnetic field is given by:
r = mv / eB

2. Quantization of Angular Momentum:
According to Bohr's quantization rule, the angular momentum mvr is quantized:
mvr = nh or mv = nh / r

3. Expressing Radius (r):
Substitute mv = nh / r into the radius equation:
r = nh / eB
Rearranging for r:
r = √(nh / eB)

4. Finding Velocity (v):
Substitute r = nh / eB back into the velocity equation:
v = nh / mr = nh / (m √(nh / eB))
Simplifying this expression:
v = √(nheB / m²)

Summary

The radius r is given by:
r = mv / eB
The quantized angular momentum mvr leads to:
r = nh / eB
The velocity v is:
v = √(nheB / m²)