Physics › ⚡ Electricity & Magnetism

Magnetic Fields & Forces

A moving charge feels an invisible sideways shove — and that single fact runs every electric motor on Earth.

High schoolAP Physics 2 level
💡
The big idea: Magnetism is a force that only acts on charge in motion, and it always pushes perpendicular to both the charge's velocity and the magnetic field. That one geometric fact explains why compass needles align, why a current-carrying wire feels a push in a field, and why looping that wire into a coil makes it spin — the motor principle.
🎯 By the end, you'll be able to
  • Describe how magnets produce fields and how field lines represent them
  • Calculate the magnetic force on a moving charge using F = qvB sin θ
  • Calculate the force on a current-carrying wire in a magnetic field using F = BIL sin θ
  • Explain the motor principle: how forces on a current loop produce a spinning torque
📎 You should already know
  • Electric Fields & Coulomb's Law
  • Vectors and the right-hand rule
  • Basic circuits: current and Ohm's law

The Invisible Push

Hold two magnets close together and you feel it before you see anything: a push, a pull, a force reaching across empty space. That force isn't magic — it's the same electromagnetic force that holds atoms together, just organized around moving electric charge. Every magnet, from a fridge magnet to Earth's core, ultimately traces back to charges in motion, whether that's electrons orbiting and spinning within atoms or current flowing through a wire.

🔑 The core rule: magnetism only pushes on moving charge

A stationary charge feels nothing from a magnetic field. Only a charge in motion feels a magnetic force — and that force always points perpendicular to both the charge's velocity and the field itself. This is the strangest and most important fact in this lesson: a magnetic field never speeds a charge up or slows it down. It only steers it, bending its path.

\[ F = qvB\sin\theta \]
Force on a single moving charge, where θ is the angle between the velocity v and the field B.
\[ F = BIL\sin\theta \]
Force on a straight current-carrying wire; I is the current, L is the length of wire inside the field, θ is the angle between the wire and B.
🎮 Interactive: Explore a Magnetic Field LIVE
Drag a moving charge or current-carrying wire through the field and watch how the force direction flips as you change the velocity, current, or field orientation.

Finding the direction: your right hand knows

Point the fingers of your right hand along the velocity \(v\) (or the current \(I\)), then curl them toward \(B\); your thumb points along the resulting force for a positive charge or conventional current. Flip the direction for a negative charge, like an electron. This right-hand rule is the fastest way to check direction without memorizing a formula for every case.

\[ \tau = NIAB\sin\theta \]
The motor principle: torque on a coil of N turns and area A carrying current I in field B — this is what spins the coil in every electric motor.
📝 Worked example: A proton moves at 3.0×10⁵ m/s perpendicular to a 0.80 T magnetic field. Find the magnitude of the magnetic force on it.
  1. Identify the givens: \(q = 1.6\times10^{-19}\,C\), \(v = 3.0\times10^5\,m/s\), \(B = 0.80\,T\), and \(\theta = 90^\circ\) so \(\sin\theta = 1\).
  2. Apply \(F = qvB\sin\theta = (1.6\times10^{-19})(3.0\times10^5)(0.80)\).
  3. Multiply step by step: \((3.0\times10^5)(0.80) = 2.4\times10^5\), then \((1.6\times10^{-19})(2.4\times10^5) = 3.84\times10^{-14}\,N\).
✓ F ≈ 3.8×10⁻¹⁴ N, directed perpendicular to both v and B (direction found with the right-hand rule).
📝 Worked example: A rectangular coil with 50 turns and area 0.020 m² carries a current of 2.0 A. It sits in a 0.30 T magnetic field, oriented so its plane is parallel to B (the position of maximum torque). Find the torque on the coil.
  1. When the coil's plane is parallel to B, the angle between the loop's normal and B is 90°, so \(\sin\theta = 1\) — this is where torque is largest.
  2. Apply \(\tau = NIAB\sin\theta = (50)(2.0)(0.020)(0.30)(1)\).
  3. Multiply step by step: \(50\times2.0 = 100\); \(100\times0.020 = 2.0\); \(2.0\times0.30 = 0.60\).
✓ τ ≈ 0.60 N·m — this is the twisting force that starts the coil spinning, the basis of the motor principle.
⚠️ Common trap: magnetic force isn't like gravity or electric force

It's tempting to picture a magnetic force pulling a charge straight toward a pole, the way gravity pulls you down. It doesn't. Magnetic force is always perpendicular to the charge's velocity, which is why a charged particle in a uniform field moves in a circle (or a helix) instead of heading straight toward the source. Also remember: \(F = qvB\sin\theta\) needs the velocity component perpendicular to B — a charge moving exactly parallel to the field feels zero magnetic force.

Check your understanding

1. Outside a bar magnet, magnetic field lines point from the magnet's ___ pole to its ___ pole.
Field lines emerge from the north pole, loop around outside the magnet, and enter the south pole — then continue from south to north inside the magnet to form closed loops.
2. An electron moves at 1.0×10⁶ m/s perpendicular to a 0.40 T magnetic field. What is the magnitude of the magnetic force on it?
F = qvB sin θ = (1.6×10⁻¹⁹)(1.0×10⁶)(0.40) with sin 90° = 1, giving 6.4×10⁻¹⁴ N.
3. Why doesn't the magnetic force ever change the speed of a moving charge, only its direction?
Work requires a force component along the direction of motion. Since magnetic force is always perpendicular to velocity, it can redirect a charge but never do work on it or change its speed.
4. Which change would increase the torque produced by a current loop spinning inside a motor?
Torque follows τ = NIAB sin θ, so increasing the number of turns N directly increases torque, while decreasing current, shrinking the area, or weakening the field would all decrease it.
✅ Key takeaways
  • Magnetic force acts only on moving charges and is always perpendicular to velocity, so it changes direction, never speed.
  • For a single charge, F = qvB sin θ; for a current-carrying wire, F = BIL sin θ.
  • The right-hand rule gives the force direction from the velocity (or current) and the field.
  • The motor principle: forces on the sides of a current loop create a torque, τ = NIAB sin θ, that spins the coil — the basis of every electric motor.