5.3 Magnetic Fields and Ampere's Law

Ampere's Law basics

Ampere's Law relates the circulation of B around a closed loop to the current enclosed: ∮B·dl = μ₀I_enc. It's most useful when symmetry (cylindrical, planar, or solenoidal) lets you pull B outside the integral, turning it into simple algebra like Gauss's Law does for E-fields.

Choosing an Amperian loop

Pick a closed path (Amperian loop) matching the symmetry of the current distribution — usually a circle centered on a wire, or a rectangle for a solenoid/sheet. B must be constant in magnitude and parallel (or perpendicular) to dl along segments for the integral to simplify. Only current passing through the loop counts as I_enc.

Key field results

Long straight wire: B = μ₀I/(2πr). Inside an ideal solenoid: B = μ₀nI (n = turns/length), and B ≈ 0 outside. Toroid: B = μ₀NI/(2πr) inside the coils, zero outside. These come directly from applying Ampere's Law to loops matching each geometry's symmetry.

Key terms

Ampere's Law

∮B·dl = μ₀I_enc — relates the circulation of B around a closed loop to the enclosed current.

Amperian loop

A closed path chosen to match the symmetry of a current distribution, simplifying Ampere's Law into algebra.

B field of a long straight wire

B = μ₀I/(2πr), circling the wire, direction given by the right-hand rule.

B field inside an ideal solenoid

B = μ₀nI, uniform and axial, where n = turns per unit length; ≈0 outside.

Sign convention for I_enc

Determined by the right-hand rule: curl fingers along the loop direction, thumb points in the direction of positive current.

Try an AP-style question

What does Ampère's law state (for steady currents)?

1. ∮ B · dA = 0
2. ε = −dΦ/dt
3. ∮ B · dl = μ₀ I_enc
4. ∮ E · dA = Q/ε₀