A cyclotron with dee radius 53.0 cm is operated at an oscillator frequency of 12.0 MHz to accel

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A cyclotron with dee radius 53.0 cm is operated at an...

A cyclotron with dee radius $53.0 \mathrm{~cm}$ is operated at an oscillator frequency of $12.0 \mathrm{MHz}$ to accelerate protons. (a) What magnitude $B$ of magnetic field is required to achieve resonance? (b) At that field magnitude, what is the kinetic energy of a proton emerging from the cyclotron? Suppose, instead, that $B=1.57 \mathrm{~T}$. (c) What oscillator frequency is required to achieve resonance now? (d) At that frequency, what is the kinetic energy of an emerging proton?

Key Concepts

Magnetic Lorentz Force and Centripetal Motion

The motion of a charged particle in a magnetic field is governed by the Lorentz force, which acts as the centripetal force that keeps the particle moving in a circular path. The balance between the magnetic force (qvB) and the required centripetal force (mv²/r) defines the particle’s orbit and is essential for understanding particle acceleration in a cyclotron.

Kinetic Energy in Accelerators

The kinetic energy of particles in a cyclotron is a measure of the energy gained as the particles are accelerated by the electric field and can be related to their velocity and orbital radius. In the non-relativistic regime, the kinetic energy is given by K = ½ mv², and by substituting the expression for v from the circular motion relations, one can relate this energy to the magnetic field strength and the radius of the particle’s orbit.

Cyclotron Frequency

The cyclotron frequency is the frequency at which a charged particle orbits within a uniform magnetic field and is given by the relation f = qB/(2?m) in the non-relativistic limit. It is a fundamental parameter in cyclotron design because it ensures that the timing of the accelerating field remains in phase with the particle’s circular motion.

Cyclotron Resonance

Cyclotron resonance is the condition where the oscillating electric field used to accelerate charged particles has a frequency that exactly matches the natural circular motion (cyclotron frequency) of those particles in a magnetic field. Meeting this condition allows the particles to consistently gain energy as they pass through the accelerating gap at the optimal phase.

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