12.6 Magnetic Force on a Moving Charge

Introduction

The magnetic force is the fundamental interaction between a magnetic field and a moving electric charge. This force is responsible for a wide range of phenomena, from the operation of electric motors to the aurora borealis. The force acts only on charges in motion and its direction is perpendicular to both the velocity of the charge and the magnetic field.

---

Definition and Formula

The magnetic force experienced by a particle with charge $q$ moving with velocity $\mathbf{v}$ through a magnetic field $\mathbf{B}$ is described by the Lorentz force equation:

$\mathbf{F}=q(\mathbf{v}\times \mathbf{B})$

Where:

• $\mathbf{F}$ is the magnetic force vector.
• $q$ is the electric charge of the particle (positive or negative).
• $\mathbf{v}$ is the velocity vector of the particle.
• $\mathbf{B}$ is the magnetic field vector.
• $\times$ denotes the vector cross product.

---

Characteristics of the Magnetic Force

Direction: The force $\mathbf{F}$ is always perpendicular to the plane formed by $\mathbf{v}$ and $\mathbf{B}$.

Magnitude: The magnitude of the force is calculated as:

$F=|q|vB\sin \theta$

where $\theta$ is the angle between the velocity and the magnetic field. The force is maximum when $\theta =90^{\circ }$ (perpendicular motion) and zero when $\theta =0^{\circ }$ or $180^{\circ }$ (parallel motion).

Work Done: Since the magnetic force is always perpendicular to the direction of motion, it does no work on the charged particle. It changes the particle's direction but not its kinetic energy or speed.

---

Right-Hand Rule

The direction of the magnetic force on a positive charge can be determined using the right-hand rule:

1. Point the fingers of your right hand in the direction of velocity $\mathbf{v}$.
2. Curl your fingers in the direction of the magnetic field $\mathbf{B}$.
3. Your thumb points in the direction of the magnetic force $\mathbf{F}$.

For a negative charge, the direction of the force is opposite to that indicated by the right-hand rule.

---

Motion of a Charged Particle in a Magnetic Field

The nature of the magnetic force leads to distinct trajectories for charged particles.

Circular Motion: If a charged particle's velocity is entirely perpendicular to a uniform magnetic field, the magnetic force provides the centripetal force, causing the particle to move in a uniform circular path.

• Radius of the circle: $r=\frac{mv}{qB}$
• Angular frequency (cyclotron frequency): $\omega =\frac{qB}{m}$

Helical Motion: If the particle's velocity has components both parallel and perpendicular to the magnetic field, its path becomes a helix. The perpendicular component creates circular motion while the parallel component remains constant, causing the particle to drift along the field lines.

The magnetic field concepts are further explored in Scalar And Vector Quantities→ and Rectangular Components Of A Vector→.

---

Applications

The force on moving charges is fundamental to many technologies:

• Cyclotron: A particle accelerator that uses a magnetic field to bend charged particles into a circular path, accelerating them to high energies.
• Auroras: Natural light displays created when charged particles from the sun are guided by Earth's magnetic field toward the poles, colliding with atmospheric gases.
• Magnetic Confinement: In fusion reactors, magnetic fields trap and control hot plasma, keeping it from touching reactor walls.

---

Possible Questions and Answers

Q: Why does the magnetic force not change the speed of a charged particle?

A: Because the force is always perpendicular to the particle's velocity. According to the work-energy theorem, only a force component parallel to the direction of motion can do work and change the kinetic energy.

Q: What happens if a charged particle is at rest in a magnetic field?

A: If the velocity is zero, the magnetic force ($F=qvB\sin \theta$) is also zero. A magnetic field exerts a force only on moving charges.

---

Significance: The magnetic force on a moving charge is a fundamental principle in electromagnetism that explains how magnetic fields interact with matter. It is essential for numerous applications in science, technology, and for understanding natural phenomena in the universe.