Magnetic Fields and Forces
This unit starts from a magnetic field as a given (Unit 207 covers where it comes from) and asks what it does to moving charges and currents. The single defining fact is that magnetic force is always perpendicular to velocity — F=qv×B — which means it can never speed a charge up or slow it down, only steer it, producing circular motion whose period is strikingly independent of speed.
What you'll learn
- Calculate the magnetic force on a moving charge using F=qv×B, including direction via the right-hand rule.
- Analyze the circular motion of a charged particle moving perpendicular to a uniform magnetic field.
- Explain why the cyclotron period of circular motion in a magnetic field doesn't depend on speed.
- Calculate the force on a current-carrying wire in a magnetic field.
- Calculate the torque on a current loop (magnetic dipole) in a uniform field.
- Explain why magnetic force does no work on a moving charge.
1. Force on a Moving Charge
A magnetic field exerts force only on *moving* charges: F=qv×B. Because this is a cross product, the force is always perpendicular to both the velocity and the field — meaning it can never point along the direction of motion, and therefore can never speed a charge up or slow it down, only redirect it.
Finding the direction requires the right-hand rule: point your fingers along v, curl them toward B, and your thumb points along v×B (the force direction for a positive charge; reverse it for a negative one).
2. Circular Motion in a Magnetic Field
When a charged particle moves perpendicular to a uniform field, the magnetic force is always perpendicular to the velocity — exactly the condition for uniform circular motion, with the magnetic force playing the role of the centripetal force. Setting qvB=mv²/r gives r=mv/(qB).
The period of this motion, T=2πr/v=2πm/(qB), remarkably contains no v at all: a faster particle traces a proportionally larger circle in exactly the same time as a slower one. This is the working principle behind a cyclotron particle accelerator.
3. Force on Currents and Torque on Loops
A current-carrying wire in a magnetic field feels a force F=IL×B — really just the collective F=qv×B force on all the individual moving charge carriers, added up. A closed current loop, meanwhile, feels zero *net* force in a uniform field (the forces on opposite sides cancel), but generally does feel a net torque, τ=μ×B, where μ=IA is the loop's magnetic dipole moment.
This torque acts to rotate the loop until μ aligns with B — the same behavior an electric dipole shows in a uniform electric field (Unit 201), just with a current loop's own magnetic moment standing in for the electric dipole's p.
Key equations
- F = qv×B — Force on a moving charge — direction from the right-hand rule (fingers along v, curl toward B, thumb gives F for positive q; reverse for negative q). Zero if v is parallel to B, maximum if perpendicular.
- r = mv/(qB) — A charged particle moving perpendicular to a uniform B field travels in a circle — the magnetic force provides exactly the centripetal force needed.
- T = 2πm/(qB) — The cyclotron period — notably independent of speed: a faster particle just moves in a proportionally bigger circle, taking the same time per revolution.
- F = IL×B — Force on a straight current-carrying wire segment in a field — the same v×B logic applied to the collective drift of many charge carriers.
- μ = IA, τ = μ×B — Torque on a current loop in a uniform field — tends to rotate the loop until μ aligns with B, exactly analogous to Unit 201's electric dipole torque.