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Several alpha particles of different speeds enter a uniform magnetic field confined to a cylindrical region, all entering radially. Each alpha particle has mass m and charge q. Which statement(s) is/are correct?
- (A) Faster is the particle, lesser is the time spent.
- (B) Slower is the particle, lesser is the time spent.
- (C) Slower the particle, greater is the time spent.
- (D) Time spent by particle which have speed v is t = 2m/qB * tan⁻¹(BqR/mv).
Correct answer: (D) Time spent by particle which have speed v is t = 2m/qB * tan⁻¹(BqR/mv).
Solution
A particle with radius r = mv/(qB) entering radially traces a circular arc inside the cylinder. The geometry gives the half-angle subtended as sin(alpha) = R/r or using tan, leading to t = (2m/(qB)) * arctan(BqR/(mv)). Slower particles have smaller r, spend more time inside (larger arc angle), so option C and D are correct, but the specific formula in D captures the full answer.
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