Question

The work done by the magnetic field on a charged particle in motion in a cyclotron is zero. How, then, can a cyclotron be used as a particle accelerator, and what essential feature of the particle's motion makes it possible? A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One \(\bullet\) and two \(\bullet\) indicate increasing level of problem difficulty.

Short answer

Answer: A cyclotron accelerates charged particles by using a varying electric field applied between two D-shaped electrodes, known as Dees, which increases the kinetic energy and velocity of the particles as they pass through the gap. The magnetic field maintains the circular motion of the particles without doing any work, while the essential feature of the constant period of their motion ensures proper synchronization with the accelerating electric field.

Step-by-step solution

1. Understand the Cyclotron Principle

A cyclotron is a type of particle accelerator that uses a constant magnetic field and a varying electric field to increase the kinetic energy of charged particles, such as protons, electrons, or ions. It consists of two D-shaped hollow metal electrodes-called "Dees"-arranged facing each other, with a gap between them where the electric field oscillates. An external magnetic field is applied perpendicular to the plane of the Dees.

2. Motion of Charged Particles in a Magnetic Field

When a charged particle moves through a magnetic field, it experiences a Lorentz force given by the equation \(F_B = q(v \times B)\), where \(q\) is the charge of the particle, \(v\) is the velocity of the particle (vector), and \(B\) is the magnetic field (vector). The force is always perpendicular to both the velocity and the magnetic field, which means it will not do any work on the charged particle because the angle between the direction of motion and the force is 90 degrees (\(W = Fd \cos(\theta)\), and \(\cos(90°) = 0\)).

3. Circular Motion of the Particle in Cyclotron

In a cyclotron, the charged particle moves in a circular path due to the Lorenz force. This is because the force is always perpendicular to the velocity of the particle, causing it to move in a circle in the horizontal plane. The period of the circular motion is given by the equation \(T = \frac{2\pi m}{qB}\), where \(m\) is the mass of the charged particle.

4. Particle Acceleration Using Electric Field

The acceleration of the charged particle in a cyclotron occurs due to the varying electric field applied between the Dees. As the particle passes through the gap between the Dees, the electric field accelerates it by increasing its kinetic energy and thus its velocity. The electric field oscillates in such a way that it always accelerates the particles in the gap, regardless of their current position within the Dees.

5. Essential Feature of Particle's Motion for Acceleration

The essential feature of the particle's motion in a cyclotron is that the period of its circular motion remains constant (\(T = \frac{2\pi m}{qB}\)), even as its radius and velocity increase due to the acceleration provided by the electric field. This constant period ensures that the particle always encounters the accelerating electric field at the right time and in the proper phase, enabling continuous acceleration and the increase of its kinetic energy, despite the zero work done by the magnetic field. In conclusion, a cyclotron can be used as a particle accelerator because the charged particle acquires energy from the electric field in the gaps between the Dees, while the magnetic field maintains its circular motion. The essential feature that makes this possible is the constant period of the particle's motion, ensuring the proper synchronization with the accelerating electric field.

Key concepts

Lorentz Force

The Lorentz force is a fundamental concept when examining the behavior of charged particles in electromagnetic fields. This force describes the combined effect of electric and magnetic fields on a moving charge. When a charged particle enters a magnetic field, it experiences this force which can be expressed by the equation
\( F_B = q(v \times B) \),
where \( q \) represents the charge of the particle, \( v \) is the velocity of the particle, and \( B \) is the strength of the magnetic field. In the context of a cyclotron, the Lorentz force is crucial as it does not do any work on the particle; instead, it steers the particle along a curved path. By maintaining the particle's motion perpendicular to the field lines, the Lorentz force ensures that energy can be efficiently added to the particle in the electric field gaps between the Dees.

Circular Motion in Magnetic Field

The circular motion of charged particles in a magnetic field is a captivating phenomenon rooted in the interplay between the Lorentz force and the particle's momentum. This force always acts perpendicular to the particle's velocity, thereby causing the particle to follow a circular path without changing its speed. The radius of this circular path depends on the particle’s velocity and the magnetic field strength.

The period of the particle's motion can be determined using the formula
\( T = \frac{2\pi m}{qB} \),
where \( m \) is the particle's mass, \( q \) its charge, and \( B \) the magnetic field. Uniform circular motion arises in a cyclotron because the Lorentz force only affects the particle's direction, not its speed, providing a stable and predictable trajectory that is key for synchronized acceleration.

Electric Field Acceleration

Electric field acceleration is the process that actually imparts energy to charged particles in a cyclotron. As particles cross the gap between the Dees, they are subject to an electric field that accelerates them, increasing their kinetic energy. The electric field alternates to ensure it always pushes the particle forward regardless of its position, which is essential for the particle to gain energy in each half cycle of its motion.

The cyclotron capitalizes on the timely application of this accelerating electric field, coinciding with the particle's arrival at the gap. This precise timing is governed by the particle's constant period of cycling, which is orchestrated by the magnetic field—showing the elegant coordination between steady circular motion and intermittent acceleration in a cyclotron.

Kinetic Energy of Charged Particles

The kinetic energy of charged particles is of central interest in the operation of a cyclotron. It's the energy associated with the motion of the particles and an indicator of the efficiency of acceleration. Each time the charged particle travels through the gap between the Dees, it is subject to an electric force in the direction of its velocity. This results in an increase in speed, and consequently, an increase in kinetic energy which is given by the equation
\( KE = \frac{1}{2}mv^2 \),
where \( m \) is the mass of the particle and \( v \) is its velocity. With every acceleration cycle, as the particle's velocity increases within the magnetic field, so too does its kinetic energy, leading to higher energy particles being output by the cyclotron for applications in physics experiments or medical treatments.