Sources of Magnetic Field
Where Unit 206 treated B as given, this unit derives it: moving charges (currents) create magnetic fields, described by the Biot-Savart law in general and, for high-symmetry configurations, by Ampere's Law — the magnetic-field parallel to Unit 202's Gauss's Law, trading a flux integral for a circulation integral.
What you'll learn
- Apply the Biot-Savart law to find the magnetic field of a simple current configuration.
- Calculate the magnetic field of a long, straight current-carrying wire.
- Calculate the magnetic field at the center of a circular current loop.
- Apply Ampere's Law to find the field of a long straight wire, a solenoid, or a toroid.
- Calculate the force per unit length between two parallel current-carrying wires.
- Explain when Ampere's Law is a useful shortcut, and when it isn't.
1. The Biot-Savart Law and a Straight Wire
The Biot-Savart law, dB=(μ₀/4π)I dl×r̂/r², gives the magnetic field contribution from one small current-carrying element — the direct magnetic parallel to Unit 201's electric field integral, with I dl playing the role dq played there. Integrating over an infinite straight wire gives the compact result B=μ₀I/(2πr), circling around the wire according to the right-hand rule.
This is a genuinely different field geometry from anything in the electric case: rather than pointing toward or away from the source, B forms closed circles around the current, with no beginning or end — a preview of Gauss's Law for magnetism (Unit 210), which states there are no magnetic monopoles.
2. Ampere's Law
Ampere's Law, ∮B·dl=μ₀I_enc, is the magnetic analog of Gauss's Law: always true, but only a useful shortcut when a symmetric current configuration lets you choose an Amperian loop on which B is constant and consistently aligned with the path.
For a solenoid, a rectangular Amperian loop with one side inside the coil and one side far outside gives B=μ₀nI for the (uniform) interior field — vastly faster than trying to sum the Biot-Savart contribution of every individual winding.
3. Force Between Parallel Currents
Each of two parallel current-carrying wires sits in the magnetic field created by the other, so each feels a force via F=IL×B. Working through the geometry: currents in the same direction attract, and currents in opposite directions repel — genuinely the reverse of the "like charges repel" intuition from electrostatics, and worth deriving rather than guessing by analogy.
Key equations
- dB = (μ₀/4π) I dl×r̂/r² — The Biot-Savart law — the magnetic analog of Unit 201's ∫k dq/r² field integral, giving the field contribution from one small piece of current, to be integrated over the whole current-carrying path.
- B = μ₀I/(2πr) — Field of a long, straight current-carrying wire — circles around the wire, direction from the right-hand rule (thumb along current, fingers curl in B's direction).
- B = μ₀I/(2R) — Field at the center of a single circular current loop.
- ∮B·dl = μ₀I_enc — Ampere's Law — always true, but only useful for solving B directly when symmetry lets B be pulled outside the integral (straight wires, solenoids, toroids).
- B = μ₀nI (solenoid), F/L = μ₀I₁I₂/(2πd) (parallel wires) — Solenoid field (uniform inside, ideally zero outside, for a long tightly-wound coil) and the force per unit length between two parallel current-carrying wires — attractive if currents run the same direction, repulsive if opposite.