Prove that the time for one revolution of a charged particle...

Prove that the time for one revolution of a charged particle moving perpendicular to a uniform magnetic field is independent of its speed. (This is the principle on which the cyclotron operates.) In doing so, write an expression that gives the period $T$ (the time for one revolution) in terms of the mass of the particle, the charge of the particle, and the magnetic field strength.

Key Concepts

Lorentz Force

This concept explains that a charged particle moving in a magnetic field experiences a force given by the cross product of its velocity and the magnetic field. The resulting magnetic force is always perpendicular to the particle's velocity, ensuring that while the direction of motion changes, the speed remains constant. This principle is central to understanding why the magnetic field causes circular motion rather than speeding up or slowing down the particle.

Centripetal Force and Circular Motion

In a circular motion scenario, a constant inward force, known as the centripetal force, is required to keep an object moving along a circular path. For a charged particle in a uniform magnetic field, the Lorentz force takes on the role of the centripetal force. By equating the magnetic force (qvB) to the necessary centripetal force (mv²/r), one can deduce that the radius of the circular path is proportional to the particle’s mass and speed, and inversely proportional to the product of its charge and the magnetic field strength.

Cyclotron Motion and Period Independence from Speed

The derived relationship from equating the centripetal and Lorentz forces leads to the expression for the period of one complete revolution, T = 2?m/(qB). This expression shows that the period T is independent of the particle’s speed, which is a critical characteristic of cyclotron operation. The constant period irrespective of energy is what allows cyclotrons to accelerate particles efficiently, as particles can be synchronized with a constant frequency alternating electric field.

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