Magnetic Dipole Moment Calculator
Enter the current and loop dimensions to find the magnetic dipole moment in ampere-square metres (A·m²). Switch between a single current loop, a multi-turn solenoid, or a permanent magnet. You can also supply an external magnetic field to compute the torque and potential energy your dipole experiences. Results update as you type, with step-by-step working shown below.
What is the magnetic dipole moment?
The magnetic dipole moment (symbol μ, unit A·m²) is a vector quantity that describes both the strength and the orientation of a magnetic source. Any current-carrying loop, solenoid, or permanently magnetised object produces a magnetic field that, far enough away, looks exactly like the field of an ideal magnetic dipole. The moment tells you how large a torque the source will experience when placed in an external magnetic field, and it determines the strength of the far field the source creates. The concept appears across atomic physics (electron spin is expressed in Bohr magnetons), engineering (motor coils, MRI magnets) and geophysics (Earth's field is modelled as a geocentric dipole with a moment of about 8 × 10²² A·m²).
How to calculate the magnetic dipole moment
For a single current loop the formula is μ = I × A, where I is the current in amperes and A is the area of the loop in square metres. If the loop is formed from a wire of length L bent into a circle, the area is A = L² / (4π). For a solenoid with N turns, each carrying current I through a cross-sectional area A, the moments of all turns add up so μ = N × I × A. For a permanent magnet with residual flux density Br (remanence, in tesla) and volume V (in m³), the equivalent dipole moment is μ = Br × V / μ₀, where μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space. This calculator handles all three cases and can also derive the area from a given loop radius.
Torque and potential energy in an external field
When a magnetic dipole is placed in a uniform external magnetic field B (in tesla), it experiences a torque τ = μ × B × sin(θ), where θ is the angle between the moment vector and the field. The torque is maximum (τ = μB) when the dipole is perpendicular to the field and zero when it is aligned. The corresponding potential energy is U = -μ × B × cos(θ). The lowest-energy state is U = -μB, when the dipole is fully aligned with the field. A compass needle seeking north is a direct demonstration: Earth's field exerts a restoring torque that rotates the needle until θ approaches zero and U is minimised.
Practical applications and the dipole approximation
The dipole model is used in the design of electric motors, speakers, and antenna systems, where the far-field radiation pattern depends on the dipole moment of the source. In MRI, the proton magnetic moments of hydrogen nuclei are aligned by a strong field and then perturbed by radio-frequency pulses; their subsequent precession generates the imaging signal. In materials science, the saturation magnetisation of a sample is reported as a moment per unit volume (A/m). The dipole approximation is valid when the observation point is at a distance much larger than the source size; a rule of thumb is to stay at least twice the largest dimension away from the source to keep errors below a few percent.
Typical magnetic dipole moments
| System | Approximate moment (A·m²) | Notes |
|---|---|---|
| Electron spin | 9.28 × 10⁻²⁴ | Bohr magneton (μB) |
| Proton | 1.41 × 10⁻²⁶ | Nuclear magneton |
| Single-turn loop, 1 cm radius, 1 A | 3.14 × 10⁻⁴ | μ = I × πr² |
| Small ferrite magnet | 0.01 - 0.1 | Typical fridge magnet |
| Neodymium disc magnet (1 cm dia, 2 mm thick) | 0.05 - 0.5 | Depends on grade |
| Bar magnet (typical demonstration) | 1 - 5 | |
| Earth's magnetic field source | 8 × 10²² | Equivalent geocentric dipole |
| MRI superconducting coil (1.5 T) | 1 × 10⁶ | Approximate |
Representative values for common physical systems. Permanent magnet values depend on material grade and geometry.
Frequently asked questions
What unit is the magnetic dipole moment measured in?
The SI unit is the ampere-square metre (A·m²), sometimes written Am². An equivalent unit is the joule per tesla (J/T), which makes the relation U = -μ · B dimensionally clear. Older literature uses the CGS unit erg/G (erg per gauss); 1 A·m² = 1000 erg/G.
How does adding more turns to a solenoid increase the magnetic moment?
Each turn of wire acts as its own current loop with moment I × A. Because the turns are stacked in parallel planes with the same winding direction, their individual moments all point the same way and simply add. A 100-turn solenoid with current I and cross-sectional area A therefore has a moment of 100 × I × A, exactly 100 times that of a single loop under the same conditions.
Can I calculate the magnetic moment of a bar magnet?
Yes. Use the permanent magnet mode and enter the residual flux density Br of the magnet material and the volume of the magnet. The calculator uses μ = Br × V / μ₀. Typical values of Br are about 1.0-1.3 T for standard neodymium magnets and 0.2-0.4 T for ferrite magnets. Make sure to convert volume to cubic metres (1 cm³ = 1 × 10⁻⁶ m³).
What is the difference between magnetic moment and magnetisation?
Magnetisation M (unit: A/m) is an intensive material property that is independent of the size or shape of the object. It equals Br / μ₀ for a uniformly magnetised body. The magnetic dipole moment μ (unit: A·m²) is an extensive property that depends on the total volume of the material: μ = M × V. Doubling the volume doubles the moment but leaves the magnetisation unchanged.
Why does torque become zero when the dipole aligns with the field?
The torque formula is τ = μ × B × sin(θ). When θ = 0° (dipole parallel to the field) or θ = 180° (antiparallel), sin(θ) = 0, so τ = 0. The 0° state is the stable equilibrium (lowest potential energy U = -μB); the 180° state is unstable. Any small perturbation from 180° produces a torque that rotates the dipole back towards 0°, which is why compass needles seek north.
Is the magnetic dipole model accurate close to a magnet?
No. The dipole approximation treats the source as a mathematical point and is only accurate when the observation distance is much larger than the size of the source, typically at least two to three times the largest dimension. Close to the magnet surface the actual field distribution depends on the shape, and finite-element modelling or multipole expansions are needed for accuracy within a few percent.