PHYS 2112 Ch 12 – Biot-Savart and Ampère's Laws

Biot-Savart law

dB = (μ₀/4π) · I dl × r̂ / r²

Field on the axis of a circular loop

B = μ₀IR² / [2(R² + z²)^(3/2)] (reduces to μ₀I/2R at center)

Ampère's law

∮ B · dl = μ₀ I_enclosed

When is Ampère's law useful?

High-symmetry current distributions (long straight wire, solenoid, toroid)

Direction of dB in the Biot-Savart law

Perpendicular to both dl and r̂ (given by the cross product dl × r̂)

What the Biot-Savart law is used for

Finding B from any current distribution by adding up (integrating) the contributions dB from each small current element

Biot-Savart result: infinitely long straight wire

B = μ₀I / (2πR) at distance R from the wire

Biot-Savart result: center of a circular loop

B = μ₀I / (2R)

Amperian loop

The imaginary closed path you choose for Ampère's law, picked so B is constant in magnitude and parallel (or perpendicular) to the path

Which current counts in Ampère's law?

Only I_enclosed — net current passing through the surface bounded by the Amperian loop (sign by right-hand rule)

Ampère's law applied to a long straight wire

Circle of radius r: B(2πr) = μ₀I → B = μ₀I / (2πr)

Biot-Savart vs Ampère's law

Biot-Savart works for any geometry (but needs an integral); Ampère's law is quick but only useful with high symmetry