Write the Biot-Savart law for a small current element. Apply it to a circular coil of N turns and radius r carrying a current I to find the magnetic field at its centre.

1. 1 · State the law. ΔB = (μ₀/4π) I Δl sin θ/r², where θ is the angle between Δl and the line to the point.
2. 2 · Draw the coil. [A circle of radius r with centre O; a small piece Δl with a current arrow I; the radius r joining it to O; B at O at right angles to the plane]
3. 3 · Apply it to one piece. Each piece is at right angles to r, so θ = 90° and ΔB = (μ₀/4π) IΔl/r².
4. 4 · Add all the pieces. All ΔB point the same way; ΣΔl = 2πr, so B = (μ₀/4π)(I/r²)(2πr) = μ₀I/2r.
5. 5 · Write the result for N turns. B = μ₀NI/2r along the axis; its direction follows the end rule.
6. 6 · Close in one line. So, the field at the centre grows with the turns and the current, and falls as the radius grows.

Scoring tip: Say clearly that every piece gives a field in the same direction at the centre. That is why the pieces simply add.