Biot-Savart law

Currents make magnetic fields. The Biot–Savart law gives the field dB made by a tiny piece dl of a wire carrying current I, at a point a distance r away:

dB = (µ₀/4π) × I dl sin θ / r², with µ₀/4π = 10⁻⁷ T m/A.

• Like Coulomb's law, it falls as 1/r².
• It is largest at right angles to the element (θ = 90°) and zero along the line of the element (θ = 0).
• Its direction is given by dl × r: perpendicular to both.

To find the field of a whole wire, add up (integrate) the pieces.

At the centre of a circular loop of radius R, every piece is at the same distance and at right angles, so all the contributions add:

B = µ₀I / (2R) for one turn, and B = µ₀NI / (2R) for a coil of N turns.

Example: one loop of radius 5 cm with 5 A: B = 4π × 10⁻⁷ × 5 / 0.1 ≈ 6.3 × 10⁻⁵ T.

On the axis at distance x from the centre: B = µ₀IR² / [2(x² + R²)^(3/2)]. Put x = 0 and it becomes µ₀I/(2R) again.

A part of a loop gives that fraction of the field: an arc that is a semicircle gives half. Straight wire pieces pointing straight at the centre give nothing there (θ = 0).

Direction: curl the fingers of your right hand along the current; the thumb points along B at the centre. Looking at the loop, a clockwise current makes a field going away from you.