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

Show 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 magnitude.

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Key Concepts

Cyclotron Frequency and Period

The cyclotron frequency is determined by the balance between the magnetic and centripetal forces, leading to an expression where the period of revolution, T, is given by a formula that depends only on the mass, charge, and magnetic field. Specifically, T = 2?m/(qB), a result which shows that the period is independent of the particle's speed. This property forms the fundamental operating principle of cyclotrons.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It acts perpendicular to both the particle's velocity and the magnetic field direction, ensuring that while it changes the direction of the velocity, it does not alter its magnitude. This perpendicular nature is crucial as it means the work done by the magnetic force is zero.

Centripetal Force and Circular Motion

In uniform circular motion, a centripetal force acts on the particle, directing it towards the center of its circular path. When a charged particle moves in a magnetic field, the Lorentz force serves as this centripetal force. Equating the magnetic force to the centripetal force allows one to relate the particle's radius of motion to its mass, speed, charge, and the magnetic field magnitude.

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