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A source at the center of a cylindrical tube of radius R emits positively charged particles, each of charge q and mass m, with equal speed v, uniformly spread over the upper semicircle in the plane of the diagram. A detector of arc length l = pi*R/6 is fixed on the inner surface of the tube. A uniform magnetic field B (directed out of the page) exists throughout the tube. For what values of B will the detector record the maximum possible number of particles per unit time?
- mv / (qR)
- 3mv / (2qR)
- 2mv / (qR)
- 5mv / (2qR)
Correct answer: mv / (qR)
Solution
Particles are emitted from the center with the same speed v in all directions in the upper half. In field B, each follows a circle of radius r = mv/(qB). The landing position on the tube wall depends on the emission angle and r. For B = mv/(qR), r = R. A particle emitted at angle theta from the positive x-axis (0 to pi) follows a circle of radius R. It lands at angle (pi - theta) from the starting direction. Analysis shows that all particles from the upper semicircle focus symmetrically, and for r = R, the detector of arc pi*R/6 (spanning 30 deg) at the bottom intercepts the maximum number of particles.
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