Magnetic Dipole Moment Calculator (μ = N·I·A, torque and energy in a field)

Understanding the magnetic dipole moment calculator — concept and principle

This calculator finds the magnetic dipole moment of a current loop or coil, μ = N I A, and the torque and potential energy it has in a uniform magnetic field. It can also give the loop's own field far away on its axis. It is used in electromagnetism courses and for estimating forces in motors, meters and magnetic sensors.

Turns × current × area gives the moment; the field twists it hardest when it lies at 90°.

How it is calculated

μ = N × I × A

Torque τ = μ B sin θ; maximum at 90° = μ B

Potential energy U = −μ B cos θ; energy to turn from 0° to 180° = 2 μ B

Far field on the axis: B = μ₀ μ ÷ (2π z³), with μ₀ = 4π × 10⁻⁷ T·m/A

Area from a circle (diameter), rectangle or a given area; dimensions in cm, B in T, z in cm.

The principle

A loop of current behaves like a small bar magnet. Its strength and direction are summed up in the dipole moment, which points along the loop's axis by the right-hand rule. In a uniform field, the forces on opposite sides of the loop cancel, but they form a couple that twists the loop to line up its moment with the field. Turning it away from alignment takes work, stored as potential energy. Far from the loop its own field looks like that of a point dipole and falls off as 1/z³.

Worked example

Default inputs: circular loop, N = 50, I = 0.5 A, diameter 4 cm, B = 0.2 T, θ = 30°, z = 20 cm.

A = π × (0.02 m)² = 12.57 cm².
μ = 50 × 0.5 × 0.0012566 = 0.03142 A·m².
τ = 0.03142 × 0.2 × sin 30° = 0.003142 N·m; maximum 0.006283 N·m.
U = −0.03142 × 0.2 × cos 30° = −0.005441 J; 0° to 180° needs 0.01257 J.
Field at 20 cm = 4π × 10⁻⁷ × 0.03142 ÷ (2π × 0.2³) = 0.7854 µT.

Each point is the calculator re-run with only “Size 1 (cm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Size 1 (cm)	Magnetic dipole moment μ (A·m²)
1	0.001963
2	0.007854
3	0.01767
4	0.03142
5	0.04909
6	0.07069
8	0.1257

How to read the result

The torque is largest when the loop's axis is perpendicular to the field and zero when aligned. This is the operating principle of electric motors and moving-coil meters. The far field falls quickly with distance: doubling z divides it by eight.

Common mistakes

Using the diameter as the radius in the area.
Measuring θ from the loop plane instead of from its axis.
Using the far-field formula close to the loop, where it is inaccurate.

Keep in mind: The formulas assume a uniform external field and a flat loop. Real coils of finite length need more detailed calculation.

Frequently asked questions

What units is the magnetic moment in?

A·m², which equals J/T.

When is the torque zero?

When the moment is parallel or antiparallel to the field.

How far away does the dipole formula work?

Roughly beyond three times the loop size; closer, the calculator shows a dash.

Does a stronger field change the moment?

No. The moment depends on N, I and A; the field changes torque and energy.

Sources: OpenStax, University Physics Volume 2 (2016), section 11.5 "Force and Torque on a Current Loop"

Checked 9 Nov 2026. OpenStax, University Physics Volume 2 (2016), §11.5 “Force and Torque on a Current Loop”, Eqs. 11.21–11.23 (μ = NIA, τ = μ × B, U = −μ·B). On-axis dipole field B = μ₀μ/(2πz³): standard result.

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