Magnetic Moment Calculator
Enter the current and geometry of your loop or coil to get the magnetic dipole moment instantly. Switch modes to compute torque on a dipole in an external field, potential energy of alignment, magnetization of a material, or the quantum orbital and spin moments of an electron. Results appear as you type, with a full step-by-step derivation shown below.
Formula
Worked example
A 10-turn coil carries 2 A and encloses an area of 0.05 m². Its magnetic moment is mu = 10 x 2 x 0.05 = 1.0 A·m². Placed at 45 deg in a 0.2 T field, the torque is tau = 1.0 x 0.2 x sin(45) = 0.141 N·m and the potential energy is U = -1.0 x 0.2 x cos(45) = -0.141 J.
What is magnetic moment?
The magnetic (dipole) moment, symbol mu, is a vector quantity that describes the strength and orientation of a source of magnetism. For a flat current loop it equals the current times the enclosed area: mu = I * A. For a multi-turn coil or solenoid the turns multiply the effect, giving mu = N * I * A. The SI unit is the ampere-square metre (A·m²), sometimes written as joules per tesla (J/T) because the two are equivalent. Every magnet, from a compass needle to an MRI coil to an orbiting electron, can be characterised by its magnetic moment.
Torque and potential energy in an external field
When a magnetic dipole is placed in an external field B, two things happen. First, the field exerts a torque tau = mu * B * sin(theta), where theta is the angle between the moment vector and the field. This torque tries to align the dipole with the field, exactly as a compass needle aligns with Earth's magnetic field. Second, the dipole has a potential energy U = -mu * B * cos(theta). The energy is lowest (most negative) when the dipole is fully aligned (theta = 0) and highest when it is anti-aligned (theta = 180 deg). The difference between these two extremes, 2 * mu * B, sets the energy scale for phenomena like nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR).
Quantum orbital and spin magnetic moments
At the atomic scale, electrons carry two kinds of magnetic moment. The orbital moment arises from the electron circling the nucleus - a tiny current loop. Quantum mechanics restricts this to discrete values: mu_L = mu_B * sqrt(l*(l+1)), where l is the orbital quantum number (0 for s, 1 for p, 2 for d, 3 for f) and mu_B = 9.274e-24 J/T is the Bohr magneton. The spin moment comes from the electron's intrinsic spin and is given by mu_S = g * mu_B * sqrt(s*(s+1)), where s = 1/2 for an electron and g = 2.0023 is the electron g-factor. For iron-group transition metals it is the spin moment that dominates, giving rise to ferromagnetism. The total atomic moment is a vector sum of all electron orbital and spin contributions.
Applications and practical context
Magnetic moments underpin a remarkable range of technology. In electric motors the force on a current-carrying loop in a field (directly related to the loop's magnetic moment) produces rotation. In MRI machines, proton spin moments precess around a strong field, and radio-frequency pulses flip them: the signal they emit encodes tissue information. In data storage, the orientation of tiny magnetic domains (each a collection of aligned spin moments) represents bits. Superconducting quantum interference devices (SQUIDs) can detect moments as small as 10^-18 A·m², enabling brain magnetometry. Even the Earth's geomagnetic field is best described as a giant magnetic dipole moment of roughly 8 x 10^22 A·m².
Typical magnetic moments and magnetizations
| System | Magnetic moment | Notes |
|---|---|---|
| Electron spin (free) | 1.00 mu_B (1.73 effective) | s = 1/2, g ~ 2 |
| Hydrogen atom (ground state) | ~1 mu_B | orbital + spin |
| Iron atom | ~2.2 mu_B | ferromagnetic exchange |
| Small bar magnet (toy) | ~0.01-0.1 A·m² | varies by size |
| Compass needle | ~0.1-1 A·m² | low-carbon steel |
| MRI whole-body gradient coil | ~100-1000 A·m² | pulsed |
| Neodymium (NdFeB) magnet (typical) | M ~ 10^6 A/m | strong permanent magnet |
| Iron (magnetization at saturation) | M ~ 1.7 x 10^6 A/m | ferromagnetic saturation |
Reference values for common systems. Atomic moments are given in Bohr magnetons; macroscopic moments in A·m² or A/m.
Frequently asked questions
What is the unit of magnetic moment?
The SI unit is the ampere-square metre (A·m²), which is identical to the joule per tesla (J/T). In atomic and nuclear physics, moments are often quoted in Bohr magnetons (mu_B = 9.274e-24 J/T) for electrons, or nuclear magnetons (mu_N = 5.051e-27 J/T) for protons and neutrons.
How does adding more turns change the magnetic moment?
Each additional turn carries the same current around the same area, so it adds an equal contribution to the moment. The total moment is simply N * I * A, making the number of turns a powerful lever: doubling the turns doubles the moment without changing the current or coil size.
What is the Bohr magneton?
The Bohr magneton (mu_B = 9.274e-24 J/T) is the natural unit of magnetic moment at the atomic scale. It equals the magnetic moment of an electron orbiting a proton in the lowest Bohr orbit: mu_B = e * hbar / (2 * m_e), where e is the electron charge, hbar is the reduced Planck constant, and m_e is the electron mass.
Why is the electron g-factor approximately 2 and not 1?
The orbital g-factor of an electron is exactly 1 by analogy with a classical current loop. The spin g-factor is approximately 2 because of the relativistic structure of the Dirac equation, and its deviation from exactly 2 (the anomalous magnetic moment, g = 2.0023) arises from quantum electrodynamic corrections - loop interactions with virtual photons. This tiny deviation is one of the most precisely measured numbers in physics.
What is the difference between magnetic moment and magnetic field?
The magnetic moment is a property of the source (the current loop, the atom, the magnet) and describes how strong and oriented the source is. The magnetic field is what the source creates in the surrounding space. A larger magnetic moment produces a stronger field at a given distance, but the field also falls off with the cube of distance from the source (the dipole field pattern).
How is magnetization related to magnetic moment?
Magnetization (M, in A/m) is the magnetic moment per unit volume of a material: M = mu / V. It describes how densely packed the aligned moments are inside a bulk sample. A strongly magnetized material has many aligned atomic moments per cubic metre, giving a large M. In ferromagnets like iron, M can reach about 1.7 million A/m at saturation.