Field of a Circular Loop
The loop that behaves like a magnet.
Lesson at a glanceShow the key points
- Centre: B = μ₀i/2a (n turns: × n).
- Axis: B = μ₀ia²/2(a² + d²)^(3/2) (35.8).
- Far: B = (μ₀/4π)(2μ/d³), μ = iπa² (35.9).
Bend a wire into a circle and every element contributes in the same direction at the centre. The result is simple, and far from the loop it becomes a magnetic dipole.
At the centre
For a loop of radius a, every element is perpendicular to the line to the centre (θ = 90°) and at distance a, and all their fields point the same way. So
Direction: curl your right-hand fingers along the current and your thumb points along B⃗. Equivalently, look along the axis: an anticlockwise current gives a field toward you, a clockwise one a field away from you.
A 50-turn circular coil of radius 4.0 cm carries 0.80 A. Find the field at its centre.
On the axis
For a point P on the axis at distance d from the centre, each element is at r = √(a² + d²) and perpendicular to r⃗, giving dB = μ₀i dl/4π(a² + d²). Diametrically opposite elements give fields whose sideways parts cancel, leaving only the components along the axis, dB cos α with cos α = a/√(a² + d²). Adding round the loop:
A single loop of radius 5.0 cm carries 3.0 A. Find the field on its axis 12 cm from the centre.
Far away: a magnetic dipole
When d ≫ a,
and iπa² is the loop’s magnetic moment μ, pointing along the axis with B⃗:
- 1Centre: B = μ₀i/2a (× n turns); right-hand rule for direction.
- 2Axis: B = μ₀ia²/2(a² + d²)^(3/2) (35.8).
- 3Far on the axis: B = (μ₀/4π)(2μ/d³), μ = iπa² (35.9), like an electric dipole.
Check yourself
Keep exploring
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