Magnetic Field of Circular Loop

The magnetic field of a circular current loop is a central result in magnetostatics with direct relevance to inductors, magnetic dipoles, MRI coils, and the modeling of atomic magnetic moments. Unlike the infinite wire, a circular loop does not possess full cylindrical symmetry, so the field varies with position along the axis and is most cleanly expressed on the axis of the loop where the integration simplifies substantially.

Core Concept Explanation

A circular loop of wire carrying a steady current I creates a magnetic field that is strongest at the center of the loop and decreases along the axis. The field on the axis is always directed along the axis itself. Off-axis, the field has both axial and radial components and must be expressed using elliptic integrals in general — this is why textbooks and GATE problems focus exclusively on the axial field and the center field.

The magnetic dipole moment of the loop is defined as m = I·A, where A is the area enclosed by the loop. For N turns, m = NIA. This quantity is fundamental because at distances far from the loop (z >> a), the field pattern becomes identical to that of a magnetic dipole — the same mathematical form as an electric dipole field, with field falling as 1/z³.

The center field is a special and important case: at z = 0, all current elements contribute equal and parallel dH components along the axis (by symmetry), and the result simplifies to H = I/(2a). This formula is frequently tested in GATE because it requires no integration to derive — just geometric symmetry.

Mathematical Expression

By integrating the Biot-Savart Law around the full circular loop, the axial field at height z above the center of a loop of radius a is:

H_z = I a² / [ 2 (a² + z²)^(3/2) ] [A/m]

Special cases that are most useful:

At the center (z = 0): H = I/(2a) — obtained by substituting z = 0 in the general formula.

Far from the loop (z >> a): H ≈ Ia²/(2z³) = m/(2πz³) — the magnetic dipole approximation, where m = IA.

For N turns wound as a coil: multiply by N, giving H = NI a² / [2(a² + z²)^(3/2)]. The magnetic moment becomes m = NIA.

Practical Understanding

Circular loop geometry is the building block of real inductors, transformer coils, and electromagnets. A Helmholtz coil pair — two circular loops separated by a distance equal to their radius — produces a nearly uniform magnetic field in the region between them. This is used in calibration labs and MRI gradient coils.

In atomic physics, electrons orbiting the nucleus are modeled as tiny current loops with magnetic moment m = evr/2. The concept of magnetic moment from m = NIA is therefore a bridge between macroscopic circuit theory and quantum mechanical spin concepts encountered in semiconductor band theory.

Given:
A circular loop of radius a = 4 cm carries current I = 2 A.
Find H at the center of the loop, and at z = 3 cm on the axis.

Why this formula applies:
For z = 0: H = I/(2a)  [center field, Biot-Savart simplified by symmetry]
For z ≠ 0: H = Ia²/[2(a²+z²)^(3/2)]  [axial field]

Formula:
H_center = I / (2a)
H_axis   = I·a² / [2·(a² + z²)^(3/2)]

Substitution (center):
H_center = 2 / (2 × 0.04) = 2 / 0.08 = 25 A/m

Substitution (z = 0.03 m):
a² = 0.0016, z² = 0.0009, a²+z² = 0.0025
(a²+z²)^(3/2) = (0.0025)^1.5 = 1.25 × 10⁻⁴
H = (2 × 0.0016) / (2 × 1.25×10⁻⁴)
H = 0.0032 / 0.00025

Final Answer:
H_center = 25 A/m  |  H at z = 3 cm ≈ 12.8 A/m  (axial direction)

Exam Tip: For the center of a single circular loop, H = I/(2a). For N turns, H = NI/(2a). The axial field formula H = Ia²/[2(a²+z²)^(3/2)] has a maximum at z = 0 and falls as 1/z³ far away. Do not confuse the loop radius a with the observation distance z.

Mechanism — Field Pattern and Dipole Behavior

• All dH contributions from loop elements at the center are parallel (by symmetry), giving maximum field H = I/(2a) at z = 0.
• Along the axis, the field has only a z-component; radial components cancel by symmetry.
• Field decreases monotonically as z increases, following (a²+z²)^(-3/2) dependence.
• At large z, the loop behaves as a magnetic dipole with moment m = IA, and H ~ 1/z³ — same spatial dependence as an electric dipole.
• For N-turn coils, every result scales by N: H = NI/(2a) at center, m = NIA.

Quick Revision

• Center field of circular loop: H = I/(2a) — direction along axis by right-hand rule.
• Axial field: H = Ia²/[2(a²+z²)^(3/2)].
• Magnetic dipole moment: m = NIA (units: A·m²).
• Far-field (z >> a): H ≈ m/(2πz³) — pure magnetic dipole.
• Helmholtz coil uses two loops at separation = radius to produce uniform field in between.
• Exam trap: At z = 0 the formula Ia²/[2(a²+z²)^(3/2)] reduces to I/(2a) — verify this by substitution, not memory.
• Off-axis field requires elliptic integrals and is not expected in GATE — only axial field is examinable.