Question

A particle with mass \(m\), charge \(q\), and velocity \(v\) enters a magnetic field of magnitude \(B\) and with direction perpendicular to the initial velocity of the particle. What is the work done by the magnetic field on the particle? How does this affect the particle's motion?

Short answer

Answer: The work done by the magnetic field on the charged particle is 0, as the force and the displacement are perpendicular to each other. The magnetic force does not change the particle's kinetic energy, keeping its speed constant. However, the magnetic force changes the particle's direction, causing it to move in a circular path in a plane perpendicular to the magnetic field.

Step-by-step solution

Determine the Lorentz force on the particle

The force exerted by a magnetic field on a charged particle with velocity \(v\) is given by the Lorentz force formula, which is F = q(v × B) Since the velocity of the particle is perpendicular to the magnetic field, the formula can be simplified to F = qvB sin(90°) But, sin(90°) = 1, so F = qvB

Calculate the work done by the magnetic field

The work done by a force is given by the equation W = F × d × cos(θ) where W is the work done, F is the force applied, d is the distance over which the force is applied, and θ is the angle between the force and the displacement. In our case, the force and the displacement are perpendicular to each other (with an angle of 90° between them). Therefore, the work done by the magnetic field on the particle can be expressed as W = qvB × d × cos(90°) Since cos(90°) = 0, the work done by the magnetic force is: W = 0

Analyze the effect of the magnetic field on the particle's motion

Since the work done by the magnetic field on the particle is zero, this means the magnetic force does not change the particle's kinetic energy. As a result, the particle's speed remains constant. However, the magnetic force acts perpendicular to the velocity of the particle, causing it to change direction. This implies that the magnetic field causes the charged particle to move in a circular path in a plane perpendicular to the magnetic field. The centripetal force required for this circular motion is provided by the magnetic force, and the radius of the circular path depends on the mass, charge, velocity of the particle, and the magnetic field strength.

Key concepts

Lorentz Force

The Lorentz force is a fundamental concept explaining how charged particles interact with magnetic fields. When a charged particle, like an electron or proton, moves through a magnetic field, it experiences a force. The direction of this force is perpendicular to both the velocity vector of the particle and the magnetic field vector, influencing the particle's path without altering its speed.

This perpendicular relationship between force, magnetic field, and velocity is crucial as it means that even though the Lorentz force is constantly acting on the particle, it doesn't do any work on it. In our exercise, for a particle with charge 'q' and velocity 'v' entering a magnetic field 'B' at a right angle, we express the Lorentz force as \( F = qvB \). Since this force is always perpendicular to the displacement of the particle, it only alters the particle's direction while maintaining its kinetic energy constant.

Work-Energy Principle

The work-energy principle links the work done on a particle to changes in its kinetic energy. Work is calculated as the product of the force applied to an object, the displacement it causes, and the cosine of the angle between the force and displacement vectors. The formula is \( W = F \cdot d \cdot \cos(\theta) \).

However, when the force is perpendicular to displacement, like with the Lorentz force in a magnetic field, cosine of 90 degrees (\( \cos(90^\circ) \)) is zero, leading to no work done (\( W = 0 \)). This means that while the particle's path is redirected, its speed doesn't change because the magnetic field isn't adding or removing energy from the particle, and its kinetic energy remains unchanged.

Centripetal Force in Magnetic Fields

In the context of magnetic fields, centripetal force is the force that keeps a charged particle moving in a circular path. The Lorentz force, acting perpendicular to the particle's velocity, provides the necessary centripetal force to keep the particle in circular motion. It's calculated using the equation \( F_c = \frac{mv^2}{r} \), where \( F_c \) is the centripetal force, \( m \) is the mass of the particle, \( v \) is the velocity, and \( r \) is the radius of the circular path.

Since the centripetal force is offered by the magnetic force (\( F = qvB \) from the Lorentz force), we can use these equations to determine the radius of the particle's path. Such an understanding is essential for applications like cyclotrons and mass spectrometry, where charged particles are manipulated using magnetic fields.

Motion of Charged Particles in Magnetic Fields

Understanding the motion of charged particles in magnetic fields is crucial in fields ranging from astrophysics to medical imaging. As illustrated in the exercise, a charged particle entering a magnetic field perpendicular to it experiences a force (Lorentz force) that compels the particle to move in a circular path. The Lorentz force serves as the centripetal force, ensuring the particle's velocity vector is constantly changing direction but not magnitude.

This results in what we call 'circular motion.' The radius of this motion can be derived from the relationship between the mass, charge, velocity of the particle, and the strength of the magnetic field. If any of these parameters change, the radius and frequency of the particle's circular motion will also change. As such, scientists can manipulate these variables to control the behavior of charged particles in a magnetic field for various applications.