Magnetic Force on Moving Charges

Introduction

Magnetism is one of the fundamental forces of nature, and it arises from the motion of electric charges. The magnetic force on moving charges is a cornerstone of electromagnetism, bridging the concepts of electric current, magnetic fields, and particle motion. Understanding this phenomenon is crucial in fields ranging from particle physics and electrical engineering to medical imaging and space science.

This article explores the magnetic force on moving charges, detailing its definition, mathematical formulation, properties, visualization, applications, and experimental verification.

1. Fundamental Concepts

Before discussing magnetic forces, it is important to recall key concepts from physics and electromagnetism.

1.1 Electric Charge

A property of matter responsible for electric interactions.
Types: Positive (+) and Negative (−).
Unit: Coulomb (C).

1.2 Electric Current

Flow of electric charge through a conductor.
Unit: Ampere (A).
Current creates a magnetic field, a fundamental observation by Ørsted.

1.3 Magnetic Field

Region where a magnetic force acts on moving charges or magnetic materials.
Denoted by B, unit Tesla (T).
Can be produced by magnets or current-carrying conductors.

2. Definition of Magnetic Force on a Moving Charge

A moving charge in a magnetic field experiences a force perpendicular to both its velocity and the field.

2.1 Vector Formulation

F⃗=q (v⃗×B⃗)\vec{F} = q \, (\vec{v} \times \vec{B})F=q(v×B)

Where:

F⃗\vec{F}F = magnetic force
qqq = charge of the particle
v⃗\vec{v}v = velocity vector of the particle
B⃗\vec{B}B = magnetic field vector
×\times× = vector cross product

2.2 Magnitude of Force

F=qvBsin⁡θF = q v B \sin \thetaF=qvBsinθ

Where θ\thetaθ is the angle between velocity and magnetic field.

Key Cases:

θ=0°\theta = 0°θ=0° or 180°180°180°: F=0F = 0F=0 (parallel motion).
θ=90°\theta = 90°θ=90°: Fmax=qvBF_\text{max} = q v BFmax=qvB (perpendicular motion).

3. Direction of Magnetic Force

3.1 Right-Hand Rule for Positive Charges

Thumb points along velocity.
Fingers point along magnetic field.
Palm pushes in the direction of force.

3.2 Negative Charges

Force direction is opposite to the right-hand rule.

3.3 Perpendicular Nature

Magnetic force is always perpendicular to velocity → does no work on the particle.
Energy of the particle remains constant, but direction changes.

4. Motion of Charged Particles in Magnetic Fields

4.1 Circular Motion

When velocity is perpendicular to uniform magnetic field:

F=qvB=mv2r ⟹ r=mvqBF = qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB}F=qvB=rmv2⟹r=qBmv

Particle moves in circular path of radius rrr.

4.2 Helical Motion

When velocity has parallel and perpendicular components:

v=v∥+v⊥v = v_\parallel + v_\perpv=v∥+v⊥

Circular motion in plane perpendicular to B, translation along B → helix.

4.3 Cyclotron Frequency

Angular frequency of circular motion:

ω=qBm\omega = \frac{qB}{m}ω=mqB

Used in particle accelerators: Cyclotrons.

5. Magnetic Force in a Current-Carrying Conductor

5.1 Lorentz Force on Charge Carriers

Current III in wire: charges qqq move with drift velocity vdv_dvd.
Force on wire segment length lll:

F⃗=I(l⃗×B⃗)\vec{F} = I (\vec{l} \times \vec{B})F=I(l×B)

5.2 Fleming’s Left-Hand Rule

Thumb: Force on conductor (motion)
First finger: Magnetic field (B)
Second finger: Current (I)

5.3 Applications

Electric motors
Galvanometers
Relays and actuators

6. Work Done by Magnetic Force

Magnetic force is always perpendicular to velocity, thus:

W=F⃗⋅d⃗=0W = \vec{F} \cdot \vec{d} = 0W=F⋅d=0

Particle’s speed remains constant, only direction changes → circular or helical motion.

7. Energy Considerations

7.1 Kinetic Energy

Constant in purely magnetic fields:

K.E.=12mv2K.E. = \frac{1}{2} m v^2K.E.=21mv2

7.2 Magnetic Potential Energy

Magnetic field does not contribute to potential energy of free charges, unlike electric fields.

8. Deflection of Charges in Mass Spectrometer

8.1 Principle

Mass spectrometer uses magnetic and electric fields to separate charged particles.

8.2 Magnetic Force Role

Magnetic field deflects ions in circular arcs.
Radius depends on mass-to-charge ratio:

r = \frac{mv}{qB} \