Question 4
Several \(\alpha\)-particles of different speeds enter a uniform magnetic field confined into a cylindrical region. If all the \(\alpha\)-particles enter the field radially, what can you say about time intervals spent by them in the magnetic field?
Solution
When a charged particle enters a magnetic field, it follows a circular path. The radius of curvature is \( R = \frac{mv}{qB} \).
Time spent in the field:
The time \(t\) spent in the magnetic field is related to the angle of deviation \(\theta\) (the angle subtended by the arc at the center of curvature):
$$ t = \frac{\theta}{\omega} = \frac{m\theta}{qB} $$Here, \(\frac{m}{qB}\) is a constant for all \(\alpha\)-particles.
Relationship with Speed:
- Faster particles (High \(v\)): They have a larger radius of curvature \(R\). Their path is “flatter” (closer to a straight line), resulting in a smaller deviation angle \(\theta\). Therefore, \(t\) is smaller.
- Slower particles (Low \(v\)): They have a smaller radius of curvature \(R\). They curve significantly more inside the field, resulting in a larger \(\theta\). Therefore, \(t\) is larger.
Thus, faster particles spend less time, and slower particles spend more time.
Correct Answers: (a) and (c)