Magnetic field due to a current element
In 1820 Hans Christian Oersted noticed that a compass needle near a current-carrying wire turned. This was the first evidence that an electric current produces a magnetic field.
The Biot-Savart law gives the magnetic field dB produced by a small current element I dl at a point at distance r: dB = (μ₀/4π) × I dl sin θ / r² where θ is the angle between dl and the line joining the element to the point, and μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space. The direction of dB is perpendicular to both dl and r (given by the right-hand rule or the cross product dl × r). dB is zero along the line of the current (θ = 0).
Field on the axis of a circular loop of radius R carrying current I, at distance x from the centre: B = μ₀ I R² / [2(x² + R²)^(3/2)] At the centre (x = 0): B = μ₀I/(2R). For N turns: B = μ₀NI/(2R).
The direction is given by the right-hand thumb rule: curl the fingers along the current and the thumb points along the field at the centre. Seen from one face, an anticlockwise current makes that face a north pole.
Worked example: a loop of radius 0.1 m carries 5 A. B = 4π × 10⁻⁷ × 5 / (2 × 0.1) = π × 10⁻⁵ ≈ 3.14 × 10⁻⁵ T.
Key points
- Oersted (1820): current produces a magnetic field.
- dB ∝ I dl sin θ / r²; zero along the axis of the element.
- Loop centre: B = μ₀NI/(2R).
- SI unit of B: tesla (T); 1 G = 10⁻⁴ T.
Exam tips
- Substitution into μ₀I/2R is very common; μ₀/4π = 10⁻⁷ simplifies arithmetic.
- Questions often ask how B at the centre changes when R is doubled or I is halved.
Common mistakes
- Using μ₀I/(2πR) (the straight-wire formula) for the loop centre.
- Forgetting the factor N for a coil of many turns.
- Thinking Biot-Savart gives a field along the direction of the current element.