Magnetism and electricity appear to be separate phenomena — a bar magnet attracts iron, while a charged balloon attracts paper. But they are two aspects of a single fundamental force: electromagnetism. Moving electric charges create magnetic fields. Magnetic fields exert forces on moving charges. This deep connection, fully unified by James Clerk Maxwell in 1865, underlies every electric motor, every generator, every transformer, and every electromagnetic wave — including light.
The magnetic field B (magnetic flux density) at a point describes the magnetic force on moving charges there. It is a vector quantity measured in teslas (T). A magnetic field exerts a force on: (1) moving charged particles: F = qvB sinθ; (2) current-carrying conductors: F = BIL sinθ. The direction is given by the right-hand rule (or Fleming's left-hand rule for motors).
The Magnetic Force on a Moving Charge
A charged particle moving through a magnetic field experiences the Lorentz force:
where q is the charge (C), v is the particle's speed (m/s), B is the magnetic flux density (T), and θ is the angle between the velocity vector and the magnetic field vector. Key features:
• The force is maximum (F = qvB) when v ⊥ B (θ = 90°).
• The force is zero when v is parallel to B (θ = 0°).
• The direction of the force is always perpendicular to both v and B — given by the right-hand rule (or left-hand for negative charges).
• Because F ⊥ v always, the magnetic force does no work — it changes direction but not speed, and therefore not kinetic energy.
The Right-Hand Rule
To find the direction of the magnetic force on a positive charge:
1. Point the fingers of your right hand in the direction of v (velocity).
2. Curl them toward B (magnetic field).
3. Your thumb points in the direction of F (force on positive charge).
For negative charges, the force is in the opposite direction (or use the left hand). In Fleming's left-hand rule for motors: first finger = field direction; second finger = current direction; thumb = force (thrust) direction.
Circular Motion of Charged Particles in Magnetic Fields
When a charged particle moves perpendicular to a uniform magnetic field, the constant perpendicular force creates circular motion. Setting magnetic force equal to centripetal force:
The radius of circular motion r is proportional to momentum mv and inversely proportional to charge q and field strength B. This is the basis of:
Mass spectrometers: ions of different masses have different radii in the same B field — separating them by mass to identify chemical composition.
Particle accelerators: the LHC uses 8.3 T superconducting magnets to bend protons of momentum 6.5 TeV/c into a circular path of radius 2,804 m.
Cyclotrons: alternating electric fields accelerate particles; magnetic fields bend them into circular paths. Each half-circle, the particles are accelerated again by the electric field — reaching high energies in a compact design.
The Force on a Current-Carrying Conductor
A current-carrying wire in a magnetic field also experiences a force. Current is moving charge, so the same physics applies. For a straight conductor of length L carrying current I in a field B:
where θ is the angle between the current direction and the field. Maximum force when I ⊥ B (θ = 90°).
Worked Example: Force on a wire
A 0.5 m wire carries 3 A perpendicular to a 0.2 T field. Force on the wire:
Magnetic Flux Density and the Tesla
| Source | Approximate B (T) |
|---|---|
| Earth's magnetic field | ~5 × 10⁻⁵ T |
| Small bar magnet | ~0.01–0.1 T |
| Strong permanent magnet (NdFeB) | ~1–1.5 T |
| MRI scanner | 1.5–3 T |
| LHC bending magnets | 8.3 T |
The Motor Effect and Electric Motors
The force on a current-carrying conductor in a magnetic field (F = BIL) is the motor effect — the principle behind all electric motors. In a simple DC motor:
1. A rectangular coil carries current in a magnetic field.
2. The two sides of the coil perpendicular to B experience forces in opposite directions (by Fleming's left-hand rule), creating a torque that rotates the coil.
3. A split-ring commutator reverses the current direction every half turn, maintaining the torque in the same rotational direction.
The torque on a coil of N turns, area A, carrying current I in field B (when the coil plane is parallel to B): τ = NBAI. This is the basis of every DC motor — from tiny vibration motors in phones to the massive motors in electric vehicles.
Magnetic Flux
Magnetic flux Φ is the "amount" of magnetic field passing through a surface area A:
where θ is the angle between B and the normal to the surface. Unit: weber (Wb = T·m²). When θ = 0° (field perpendicular to surface), Φ = BA (maximum). When θ = 90° (field parallel to surface), Φ = 0.
Magnetic flux is central to Faraday's law of electromagnetic induction: a changing magnetic flux through a circuit induces an electromotive force (EMF). This is how generators, transformers, and induction charging work — but that is the subject of electromagnetic induction.
Frequently Asked Questions
What is a magnetic field?
A magnetic field B (measured in teslas) is a region where magnetic forces act on moving charges and magnetic materials. It is a vector quantity. Magnetic fields are created by moving charges (electric currents) and by magnetic materials (permanent magnets). Field lines run from north to south poles outside a magnet.
What is the formula for magnetic force on a moving charge?
F = qvB sinθ, where q is charge (C), v is speed (m/s), B is magnetic flux density (T), and θ is the angle between velocity and field. The force is maximum (F = qvB) when velocity is perpendicular to B. The force is always perpendicular to both v and B — so it does no work and changes only the direction of motion.
What is Fleming's left-hand rule?
Fleming's left-hand rule gives the direction of force on a current-carrying conductor in a magnetic field: hold the left hand with the first finger pointing in the field direction (B), the second finger pointing in the current direction (I), and the thumb points in the direction of the force (thrust). Used for electric motors.
Why does a magnetic force do no work?
The magnetic force F = qvB sinθ is always perpendicular to the velocity v. Work = F·d requires force and displacement to be parallel (or have a parallel component). A purely perpendicular force does zero work — it changes the direction of motion but not the speed, so kinetic energy is unchanged. This is why a magnetic field alone cannot accelerate a particle.
How does an electric motor work?
An electric motor uses the force on a current-carrying conductor in a magnetic field (F = BIL). A current-carrying coil in a magnetic field experiences a torque that rotates it. A commutator reverses current direction every half turn so torque always acts in the same rotational direction. Electrical energy is converted to rotational kinetic energy.
What is magnetic flux?
Magnetic flux Φ = BA cosθ is the "amount" of magnetic field passing through a surface, measured in webers (Wb). It depends on field strength B, area A, and the angle between B and the surface normal. Changing magnetic flux through a circuit induces an EMF — the basis of Faraday's law and all generators and transformers.
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Written by
Dr. Sarah KimThermodynamics researcher with a PhD from MIT, specializing in statistical mechanics and energy transfer. Passionate about connecting molecular physics to everyday phenomena.
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