Calculate magnetic dipole moment, axial and equatorial fields, torque, and potential energy for current loops and coils.
The magnetic dipole moment is a fundamental quantity in electromagnetism that characterizes the strength and orientation of a magnetic source. For a current-carrying loop, the magnetic moment is simply the product of the number of turns, the current, and the enclosed area: m = NIA. This elegant relationship connects a macroscopic measurement (the moment) to the microscopic parameters of the current distribution.
At distances much larger than the size of the loop, any current distribution looks like a magnetic dipole, and its field falls off as the inverse cube of distance. This dipole approximation is remarkably powerful — it describes everything from the field of a tiny bar magnet to the geomagnetic field of the Earth, which is well-approximated by a dipole with moment 7.94 × 10²² A·m².
This calculator computes the magnetic moment for single loops and multi-turn coils, then calculates the resulting dipole field at any observation distance. It also determines the torque and potential energy when the dipole is placed in an external magnetic field — essential for understanding motors, compass needles, MRI physics, and atomic-scale magnetic behavior.
The magnetic dipole moment is central to understanding electromagnetic devices from compass needles to electric motors to MRI machines. Calculating it by hand requires careful unit management (SI vs CGS) and remembering the correct field formulas for axial vs equatorial positions.
This calculator handles the complete dipole calculation chain: from current and geometry to moment, from moment to field at any distance, and from moment to torque and energy in an external field. The visual field comparison and reference table of common moments provide physical intuition for the numbers.
Magnetic Dipole Moment: • Moment: m = N × I × A (A·m²) • Axial field: B_axis = (µ₀/4π) × 2m/r³ • Equatorial field: B_eq = (µ₀/4π) × m/r³ • Torque: τ = m × B × sin(θ) • Potential energy: U = −m · B = −mB cos(θ) Where µ₀ = 4π × 10⁻⁷ T·m/A, r = distance from dipole center
Result: m = 2.0 A·m², B_axis = 6.40 µT at 0.5 m
A 100-turn coil carrying 2A with 0.01 m² area has moment m = 100 × 2 × 0.01 = 2.0 A·m². The axial field at 0.5 m is B = (µ₀/4π) × 2 × 2.0 / 0.5³ = 6.40 µT. The equatorial field at the same distance is exactly half: 3.20 µT.
The magnetic dipole is the simplest non-trivial magnetic source. Just as the electric dipole (two equal and opposite charges) produces an electric field that falls off as 1/r³, the magnetic dipole (a current loop) produces a magnetic field with the same spatial dependence. This is not a coincidence — both are the lowest-order multipole terms in their respective multipole expansions.
For a current loop of area A carrying current I, the magnetic moment vector is m = IA n̂, where n̂ is the unit normal to the loop (determined by the right-hand rule). For multiple turns, m = NIA. The field of this dipole in spherical coordinates is B_r = (µ₀/4π)(2m cos θ)/r³ and B_θ = (µ₀/4π)(m sin θ)/r³, which gives the familiar 2:1 ratio between axial and equatorial fields.
Electric motors rely on the torque τ = m × B experienced by current-carrying coils in a magnetic field. The moment is maximized by using many turns of wire (large N), high current (large I), and large coil area (large A). Practical motors optimize the trade-off between these parameters, wire resistance, and mechanical constraints.
In atomic physics, orbital and spin magnetic moments determine atomic behavior in magnetic fields. The Stern-Gerlach experiment demonstrated quantized angular momentum by deflecting atoms with specific magnetic moments. Magnetic resonance (NMR and MRI) exploits the precession of nuclear magnetic moments in strong fields.
The dipole approximation is the far-field description of any localized current distribution. A solenoid, a bar magnet, a magnetized sphere, and a current loop all produce identical dipole fields at large distances — they differ only in their near-field structure. This universality makes the dipole moment the most important single number characterizing a magnetic source.
The magnetic dipole moment is a vector quantity that measures the strength and direction of a magnetic source. For a current loop, it points perpendicular to the loop plane (right-hand rule) with magnitude m = NIA. It determines the torque the loop experiences in an external field and the field the loop produces.
The dipole approximation is accurate when the observation distance is much larger than the size of the current loop (r >> √A). At close range, higher-order multipole terms become significant. As a rule of thumb, the approximation is good to within 10% when r > 3× the loop radius.
This 2:1 ratio is a fundamental property of the dipole field geometry. Along the axis, both "poles" of the dipole contribute fields in the same direction. In the equatorial plane, the fields from the two poles partially cancel, resulting in half the axial value.
Magnetization (M) is the magnetic moment per unit volume: M = m/V. In a permanent magnet, the total moment is the integral of M over the volume. For a uniformly magnetized material, m = M × V.
The Bohr magneton (µ_B = 9.274 × 10⁻²⁴ A·m²) is the fundamental unit of magnetic moment for electrons. It arises from the electron's orbital motion and spin. Atomic magnetic moments are typically expressed in multiples of µ_B.
MRI relies on the magnetic moments of hydrogen nuclei (protons). In a strong external field, these moments precess at the Larmor frequency. RF pulses tip the moments, and the precessing magnetization generates the MRI signal as it relaxes back to equilibrium.