Question

A particle of mass \(m\) and charge \(q\) moves in a uniform, constant magnetic field \(\mathbf{B}\). Show that if \(\mathbf{v}\) is perpendicular to \(\mathbf{B},\) the particle moves in a circle of radius $$r=|\mathbf{p} / q B|$$ [This result agrees with the nonrelativistic result \((2.81),\) except that \(\mathbf{p}\) is now the relativistic momentum \(\mathbf{p}=\gamma m \mathbf{v} .]\)

Short answer

The particle moves in a circle with radius \( r = \frac{|\mathbf{p}|}{qB} \).

Step-by-step solution

Understand Lorentz Force

The motion of a charged particle in a magnetic field is affected by the Lorentz force, given by \( \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \). This is the force acting on a charge \( q \) moving with velocity \( \mathbf{v} \) in a magnetic field \( \mathbf{B} \).

Apply Newton's Second Law

From Newton's second law, \( \mathbf{F} = \frac{d\mathbf{p}}{dt} \), where \( \mathbf{p} \) is the momentum. For a charged particle in a magnetic field, \( \frac{d\mathbf{p}}{dt} = q(\mathbf{v} \times \mathbf{B}) \).

Relate Perpendicular Motion to Circular Motion

When \( \mathbf{v} \) is perpendicular to \( \mathbf{B} \), the force is always perpendicular to the velocity, hence the particle undergoes circular motion. In circular motion, the centripetal force needed is provided by the magnetic force.

Define Relativistic Momentum

Relativistic momentum is defined as \( \mathbf{p} = \gamma m \mathbf{v} \), where \( \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \). This expression accounts for relativistic effects at high speeds.

Equate Centripetal and Magnetic Forces

For circular motion, centripetal force \( F_c \) is \( \frac{\gamma m v^2}{r} \). Equating this to the magnetic force \( qvB \), since \( F_m = qvB \) for \( v \perp B \), gives: \( \frac{\gamma m v^2}{r} = qvB \).

Solve for Radius of the Circle

Rearranging \( \frac{\gamma m v^2}{r} = qvB \) gives \( r = \frac{\gamma m v}{qB} \). Substitute \( \mathbf{p} = \gamma m \mathbf{v} \) to get \( r = \frac{|\mathbf{p}|}{qB} \).

Key concepts

Relativistic Momentum

Relativistic momentum is a critical concept when dealing with particles moving at speeds close to the speed of light. Classical mechanics describes momentum simply as the product of mass and velocity:

• Classical momentum formula: \[ \mathbf{p}_{\text{classical}} = m \mathbf{v} \]

However, as speeds approach the speed of light, relativistic effects become significant. This is where Einstein's theory of relativity comes into play. In special relativity, the momentum of an object is not merely straightforward:

• Relativistic momentum formula: \[ \mathbf{p} = \gamma m \mathbf{v} \]

Here, \( \gamma \) (the Lorentz factor) is defined as:

• \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

This factor accounts for the increase in mass that an object experiences as it moves faster. It’s this factor that differentiates relativistic momentum from classical momentum. The implication is that as a particle's speed increases, much more force is needed to maintain or increase its velocity. Understanding this helps explain why nothing can speed-up infinitely to reach or exceed the speed of light.

Centripetal Force

Centripetal force is pivotal in any scenario involving circular motion. When an object moves in a circle, it constantly changes direction, meaning it is accelerating even if its speed remains constant.

• Acceleration and velocity in circular motion:

This acceleration is due to a force acting at right angles to the velocity of the object, directed towards the center of the circle. This force is called the centripetal force.

• Centripetal force formula: \[ F_c = \frac{m v^2}{r} \]

For a particle moving perpendicularly to a magnetic field, the magnetic force serves as the centripetal force:

• Magnetic force as centripetal force: \[ qvB = \frac{\gamma m v^2}{r} \]

By equating the magnetic force to the centripetal force, you confirm the circle of motion is sustained by the magnetic force. This relationship is necessary to find the radius of the circle the particle describes.

Magnetic Field

A magnetic field exerts a force on moving charged particles, forcing them to move in specific paths or patterns. To understand how this happens, we use the Lorentz force formula:

• Force due to a magnetic field: \[ \mathbf{F} = q(\mathbf{v} \times \mathbf{B}) \]

This formula shows that the force is strongest when velocity \( \mathbf{v} \) is perpendicular to the magnetic field \( \mathbf{B} \). This right-angle interaction explains why a charged particle moves in a circle. The force direction is perpendicular to both \( \mathbf{v} \) and \( \mathbf{B} \), maintaining a circular path.

• Key properties of magnetic fields:
• They do not change the speed of particles, only the direction.
• Forces are orthogonal to the direction of momentum.

When a particle enters a magnetic field, it curves into circular motion if the velocity vector is perpendicular to the magnetic field lines. That's the reason why the particle loops instead of flying off, anchored by the magnetic pull.