Using the Biot-Savart law, derive an expression for the magnetic field at a point on the axis of a circular current-carrying loop. Hence write the field at its centre.

1. 1 · State the law. dB = (μ₀/4π) I dl sin θ/r², perpendicular to the plane of dl and r.
2. 2 · Draw the figure. [Loop of radius R, centre O, axis through O, point P at distance x; element dl at the top; r to P; dB resolved into dBx and dB⊥; opposite element shown]
3. 3 · Field of one element. dl ⊥ r, so sin θ = 1 and dB = (μ₀/4π) I dl/(x² + R²).
4. 4 · Cancel and add.
   • dB⊥ parts cancel in diametrically opposite pairs.
   • dBx = dB cos α, with cos α = R/(x² + R²)^(1/2).
5. 5 · Integrate round the loop. ∮ dl = 2πR, so B = μ₀IR²/[2(x² + R²)^(3/2)], along the axis.
6. 6 · Special case. At the centre (x = 0): B = μ₀I/2R; for N turns, μ₀NI/2R.
7. 7 · Real life. Flat coils in induction cooktops and wireless chargers are strongest at the centre.
8. 8 · Close in one line. So, the axial field of a loop is μ₀IR²/[2(x² + R²)^(3/2)], pointing along the axis by the right-hand rule.

Scoring tip: Say clearly why dB⊥ cancels: opposite elements are symmetric. Examiners give a mark for this sentence.