Un fisico está diseñando un ciclotrón para acelerar protones...

Name: Problem 53, Chapter 11: Física Universitaria Volumen 2
Uploaded: 2020-07-09T17:24:31-08:00
Duration: 1 min 24 s
Channel: Joseph Petrullo
Description: Un fisico está diseñando un ciclotrón para acelerar protones a una décima parte de la velocidad de la luz. El campo magnético tendrá una intensidad de 1,5 T. Determine (a) el periodo de rotación de los protones que circulan y (b) el radio máximo de la órbita de los protones.

Un fisico está diseñando un ciclotrón para acelerar protones a una décima parte de la velocidad de la luz. El campo magnético tendrá una intensidad de 1,5 T. Determine (a) el periodo de rotación de los protones que circulan y (b) el radio máximo de la órbita de los protones.

Key Concepts

Charged Particle Motion in Magnetic Fields

This concept explains that when a charged particle enters a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field. This force, known as the Lorentz force, causes the particle to move in a circular or spiral path. Understanding this motion is fundamental in designing devices like cyclotrons where the controlled circular motion is used to accelerate charged particles.

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), indicating that the force is perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for the centripetal acceleration that causes a charged particle to follow a curved trajectory, making it essential in the operation of accelerators such as cyclotrons.

Uniform Circular Motion

Uniform circular motion refers to the motion of a particle traveling with constant speed along a circular path. For a charged particle moving in a uniform magnetic field, the Lorentz force provides the necessary centripetal force to keep the particle in a circular orbit. The balance between the magnetic force and the centripetal force defines key parameters such as the period and the radius of the orbit.

Cyclotron Frequency and Period

The cyclotron frequency (or gyrofrequency) is the rate at which a charged particle orbits in a uniform magnetic field, and is independent of its speed for non-relativistic particles. It is determined by the charge-to-mass ratio and the magnetic field strength, and is given by the formula f = qB/(2?m). The period of rotation, being the reciprocal of the frequency, plays a critical role in timing the accelerating electric fields used in cyclotrons.

Radius of Charged Particle Orbit

The radius of the orbit of a charged particle moving in a magnetic field is defined by the balance between the centripetal force required to maintain circular motion and the magnetic force acting on the particle. It can be calculated using the equation r = mv/(qB), where m is the mass, v is the speed of the particle, q is its charge, and B is the magnetic field strength. This relation is crucial in designing magnetic confinement systems and particle accelerators.

Non-relativistic Approximation

In many practical accelerator applications, especially when particle speeds are a small fraction of the speed of light, relativistic effects can be neglected. This approximation allows the use of classical formulas for cyclotron frequency, period, and orbit radius. It simplifies the calculations by assuming that the mass of the particle remains constant regardless of its speed.

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