Magnetic Field of Circular Loop
When electric current flows around a circular loop of wire, it produces a magnetic field whose shape is far more interesting than that of a straight wire. Close to the wire itself, the field lines curl in tight circles — just as they do around any current-carrying conductor. But away from the wire, all those individual contributions add together in a surprisingly elegant way: along the central axis of the loop, the fields from opposite sides of the ring reinforce each other, producing a strong, well-defined field that points straight along the axis. This axial field is of enormous practical importance. Stack many such loops together and you have a solenoid; bend the solenoid into a torus and you have a toroid. The circular loop is therefore the fundamental building block of most electromagnetic devices, from MRI scanners to electric motors. The mathematics that describes the field on the axis — derived using the Biot–Savart law — yields a clean formula that every physics student should know and understand, not merely memorise. The field pattern of a circular loop also resembles that of a bar magnet when viewed from a distance. The two faces of the loop act like a north pole and a south pole, making the loop a magnetic dipole. This concept links microscopic current loops inside atoms to the macroscopic magnetism we observe in permanent magnets.
- Applying the Biot–Savart Law to a Loop
- Geometry of the Axial Calculation
- Special Cases and Physical Insight
- Field Pattern Away from the Axis
Applying the Biot–Savart Law to a Loop
The Biot–Savart law states that a tiny current element Idl contributes a magnetic field dB = (μ₀/4π)(Idl × r̂)/r² at a point in space. For a circular loop of radius R carrying current I, every element of the loop is at the same distance from a point P on the axis. Because of the circular symmetry, the components of dB perpendicular to the axis cancel in pairs from opposite sides of the loop. Only the axial components survive and add constructively, which is why the net field points purely along the axis.
Geometry of the Axial Calculation
Place the loop in the xy-plane, centred at the origin. A point P on the axis is at distance x from the centre. The distance from any point on the loop to P is r = √(R² + x²). Each current element Idl is tangential to the loop, so it is perpendicular to the radial vector from that element to P. The angle between dB and the axis equals arctan(R/x). After integrating around the full circumference (total arc length 2πR), the axial components sum to give the final formula.
Special Cases and Physical Insight
At the centre of the loop (x = 0), the formula simplifies to B = μ₀I/(2R). This makes intuitive sense: a smaller loop concentrates the field more strongly at its centre, so B is larger for smaller R. Far from the loop (x ≫ R), the field falls off as 1/x³ — exactly the behaviour of a magnetic dipole. The magnetic dipole moment is m = IA, where A = πR² is the area of the loop. Multiple loops (N turns) multiply B by N, since each turn contributes independently.
Field Pattern Away from the Axis
Off the axis, the field lines emerge from one face of the loop, spread outward, loop around the outside, and re-enter through the other face — identical in shape to the field of a bar magnet. The face from which field lines emerge is the north pole: you can identify it by the right-hand rule. Curl the fingers of your right hand in the direction of current flow; your thumb points toward the north pole face. This dipole nature means the loop creates no magnetic monopoles, consistent with Maxwell's equations (∇·B = 0 everywhere).
The Key Equation
- B — Magnetic field strength (Tesla (T)): The magnitude of the magnetic field at point P on the axis, directed along the axis.
- \mu_0 — Permeability of free space (T·m·A⁻¹): A fundamental constant equal to 4π × 10⁻⁷ T·m·A⁻¹, relating magnetic fields to the currents that produce them.
- I — Current (Ampere (A)): The steady electric current flowing around the loop.
- R — Loop radius (Metre (m)): The radius of the circular current loop.
- x — Axial distance (Metre (m)): The distance from the centre of the loop to point P, measured along the central axis.
Worked Example
Problem. A circular loop of radius R = 5.0 cm carries a current I = 3.0 A. Calculate (a) the magnetic field at the centre of the loop, and (b) the magnetic field on the axis at a distance x = 12.0 cm from the centre.
Answer. At the centre: approximately 37.7 μT; on the axis at 12 cm: approximately 2.1 μT.
Origins · 1 minJean-Baptiste Biot and how this was discovered
In 1820, just weeks after Ørsted's discovery that currents deflect compass needles, Jean-Baptiste Biot and Félix Savart performed careful experiments measuring the magnetic force on a magnet near a long straight wire. They formulated the Biot–Savart law — the fundamental relationship between a current element and the magnetic field it produces. This law is the essential tool used to derive the axial field formula for a circular loop and underpins the entire mathematical theory of magnetostatics. Biot was also a pioneering aeronaut: in 1804 he ascended to over 4,000 metres in a hot-air balloon to study Earth's atmosphere, collecting data on air composition and temperature at altitude — a remarkably adventurous scientific expedition for its era.
Related Concepts
- Biot–Savart Law — The axial field formula for a circular loop is derived directly by applying the Biot–Savart law to each current element around the ring and integrating.
- Magnetic Field of a Straight Wire — Understanding the field of a straight wire using Biot–Savart builds the same mathematical intuition needed to tackle the circular loop geometry.
- Magnetic Field of a Solenoid — A solenoid is a series of closely-wound circular loops; its uniform interior field is found by summing the axial contributions of many loops.
- Magnetic Dipole Moment — A current loop is the simplest magnetic dipole; its dipole moment m = IA determines the far-field behaviour and links atomic current loops to macroscopic magnetism.