A physicist is designing a cyclotron to accelerate protons...

Name: Problem 53, Chapter 11: University Physics
Uploaded: 2020-07-09T17:24:31-08:00
Duration: 1 min 24 s
Channel: Joseph Petrullo
Description: A physicist is designing a cyclotron to accelerate protons to one-tenth the speed of light. The magnetic field will have a strength of 1.5 T. Determine (a) the rotational period of the circulating protons and (b) the maximum radius of the protons' orbit.

A physicist is designing a cyclotron to accelerate protons to one-tenth the speed of light. The magnetic field will have a strength of 1.5 T. Determine (a) the rotational period of the circulating protons and (b) the maximum radius of the protons' orbit.

Key Concepts

Cyclotron Operation

A cyclotron is a type of particle accelerator that employs a constant magnetic field and an alternating electric field to increase the energy of charged particles. The magnetic field causes the particles to follow a circular path while the electric field accelerates them each time they cross the gap between the dees. This synergy of magnetic and electric forces enables the particle to gain energy gradually, leading to higher speed and larger orbital radii over time.

Lorentz Force and Circular Motion

When a charged particle moves through a magnetic field, it experiences a force known as the Lorentz force, which acts perpendicular to both its velocity and the magnetic field. This force provides the necessary centripetal force for the particle to move in a circular path. The balance between the magnetic force and the required centripetal force determines the radius of the particle's orbit.

Cyclotron Frequency and Period

The cyclotron frequency is the rate at which a charged particle orbits in a magnetic field and is determined by the particle's charge-to-mass ratio and the strength of the magnetic field. This frequency is crucial, as it dictates the period of the particle's circular motion. Under non-relativistic conditions, this frequency remains constant even as the particle’s speed increases, which simplifies the synchronization with the accelerating electric field.

Relationship Between Particle Speed, Magnetic Field, and Orbit Radius

In a magnetic field, the radius of a charged particle's circular motion is directly related to its momentum and inversely related to both its charge and the magnetic field strength. As the particle accelerates, the increase in momentum results in a larger radius of curvature. This relationship is essential in determining the physical size limits of accelerators like cyclotrons.

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