The magnetic dipole moment calculator above computes the magnetic moment of a current loop or multi-turn coil from three quantities — the current, the number of turns and the area each turn encloses — and then the torque and potential energy that moment experiences in a uniform external magnetic field. It handles circular and rectangular loops or an area entered directly, since only the enclosed area matters.
Arb Digital publishes free physics calculators. This page is the magnetic counterpart of the live electric dipole moment calculator, which states explicitly that it does not cover the magnetic case. The two are mathematically parallel — both give a torque as a cross product with a field and an energy as a dot product — but they are built from different ingredients. An electric dipole is a charge times a separation; a magnetic dipole is a current times an enclosed area. Their units are different and they are not interchangeable.
What This Magnetic Dipole Moment Calculator Does
A current going round a closed loop produces a magnetic field that, seen from far enough away, looks exactly like the field of a tiny bar magnet. The magnetic dipole moment is the single vector that captures everything about that equivalence: its magnitude sets how strongly the loop responds to an external field, and its direction, given by the right-hand rule around the current, sets which way the loop tries to turn.
The tool computes that moment, the area it is built from, the torque the loop feels at the angle you specify, the maximum torque available when the moment is perpendicular to the field, and the orientation energy. It assumes the external field is uniform over the loop.
It does not compute the field the loop itself produces — that is a different question, handled by the solenoid magnetic field calculator for a coil and the magnetic field of a wire calculator for a straight conductor. It also does not compute the net force in a non-uniform field, which requires the field gradient rather than the field.
How to Use It
- Enter the current in amperes and the number of turns. Every turn contributes equally, so a hundred-turn coil has a hundred times the moment of a single loop.
- Choose the geometry — a circular loop by radius, a rectangle by its two sides, or the area entered directly.
- Enter the dimensions and pick the length unit. Millimetres are convenient for small coils and the tool converts internally.
- Enter the external flux density in tesla and the angle between the moment vector and the field.
- Read the moment, the torque at that angle, the maximum torque and the potential energy.
The Formula: How Magnetic Moment and Torque Are Calculated
For a planar loop of area A carrying current I, wound with N turns, the magnitude of the magnetic dipole moment is
m = N I A, in ampere square metres (A m²)
Its direction is perpendicular to the plane of the loop, given by curling the right hand in the direction of the current so the thumb points along the moment. In a uniform external field of flux density B, the torque is the cross product of the two vectors, so its magnitude is
τ = m B sin θ
and the orientation potential energy is the negative dot product,
U = −m B cos θ
Those two together describe the whole behaviour. The energy is lowest, at −mB, when the moment is aligned with the field, and highest, at +mB, when it is anti-aligned. The torque is zero at both of those orientations and largest at 90 degrees, where the energy is passing through zero. The energy required to flip a dipole from aligned to anti-aligned is therefore 2mB. HyperPhysics at Georgia State University sets out the same relations and notes that the torque acts perpendicular to the moment, which is why a moment with angular momentum precesses rather than simply swinging into line.
Work the defaults through by hand. A hundred-turn circular coil of radius 25 mm carrying 2 A. The area per turn is π × (0.025 m)² = 1.9635 × 10⁻³ m². The moment is 100 × 2 × 1.9635 × 10⁻³ = 0.3927 A m². In a 0.5 T field at 30 degrees, the torque is 0.3927 × 0.5 × sin 30° = 0.3927 × 0.5 × 0.5 = 0.09817 N m. The maximum torque, at 90 degrees, is 0.1963 N m. The potential energy is −0.3927 × 0.5 × cos 30° = −0.1700 J, and flipping the coil end for end would cost 0.3927 J.
Why Only the Enclosed Area Matters
The shape of the loop is irrelevant. A circle, a square, a triangle and a ragged outline all give the same moment if they enclose the same area and carry the same current. This is not an approximation for far-field behaviour; it is exact for the moment itself.
The reason is that any planar loop can be subdivided into a mesh of small loops whose internal boundaries carry equal and opposite currents that cancel. What survives is the outer boundary, and the total moment is the sum of the small areas — which is just the enclosed area, regardless of the outline's shape.
Two useful consequences follow. First, a figure-of-eight loop has almost no net moment, because the two lobes are traversed in opposite senses and their contributions subtract. That is why a twisted pair carries current with negligible magnetic signature and why non-inductive resistors are wound bifilar. Second, a loop that is not planar has to be treated as the vector sum of the projected areas, so a coil with a wobble in it has a slightly smaller effective moment than its nominal geometry suggests.
Where the Torque Relation Shows Up in Real Instruments
The moving-coil meter is the clearest example. A multi-turn coil sits in the radial field of a permanent magnet and is restrained by a hairspring. Current produces a moment, the moment produces a torque, and the pointer settles where that torque balances the spring. The radial pole pieces exist to keep the moment perpendicular to the field at every angle, so the sine term stays at one and the scale comes out linear instead of crowding at the ends.
An electric motor is the same physics run continuously. The rotor winding has a moment, the stator field exerts a torque on it, and the commutator or the drive electronics reverses the current every half turn so the torque never changes sign. Torque per ampere in a motor is, in essence, the moment per ampere multiplied by the field.
A compass needle is a permanent moment rather than a current loop, but it obeys the identical relations. It oscillates about alignment because torque and energy are out of step: the restoring torque is largest where the energy is passing through zero, so the needle overshoots and swings back until damping absorbs the difference.
At atomic scale the same quantity governs magnetic resonance. Nuclear and electron moments precess about an applied field at a rate set by the field strength, and the energy difference between aligned and anti-aligned states is the 2mB above. That splitting is what magnetic resonance imaging and electron spin resonance measure. The natural unit there is the Bohr magneton rather than the ampere square metre, and its value is one of the fundamental constants published in the NIST CODATA reference on constants, units and uncertainty, alongside the vacuum magnetic permeability that links B and H.
Uniform Field Gives Torque, Non-Uniform Field Gives Force
This distinction causes real confusion. In a genuinely uniform field, a dipole feels a torque but no net force. The forces on opposite sides of the loop are equal and opposite, so they rotate it without translating it.
Attraction and repulsion require a field gradient. The net force on a dipole is proportional to the rate at which the field changes across it, not to the field itself, which is why a magnet is pulled towards a pole piece where the field is converging and feels almost nothing in the middle of a large uniform gap.
This page computes the uniform-field case only. If you need the force in a gradient, you need the gradient, and the geometry of the source matters as much as its strength.
Where This Sits Next to the Other Magnetism Tools
This page gives the moment of a loop and what a field does to it. The magnetic force on a wire calculator gives the force on a straight current-carrying conductor, which is the same physics applied to one side of the loop rather than to the loop as a whole. The live Lorentz force calculator is a different case again: the force on a single moving charge, with a cyclotron radius and period rather than a torque.
For the fields themselves, the solenoid magnetic field calculator and the magnetic field of a wire calculator compute what a coil and a wire produce, the magnetic field converter handles tesla, gauss and oersted, and the magnetic permeability calculator derives permeability from B and H. The Faraday's law calculator covers the induced voltage when the flux through that same loop changes.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Forgetting the number of turns — the moment scales linearly with N, so a hundred-turn coil is a hundred times a single loop and omitting it is a two-order-of-magnitude error.
- Using the wire length instead of the enclosed area — the moment depends on the area the loop encloses, and two loops with the same perimeter can enclose very different areas.
- Measuring the angle from the loop plane — it is measured from the moment vector, which is perpendicular to the plane. Confusing the two swaps sine for cosine.
- Expecting a net force in a uniform field — there is none. Attraction and repulsion come from a field gradient, not from the field itself.
- Confusing magnetic and electric dipole moments — they behave in parallel ways but have different units and different ingredients, and the numbers are not interchangeable.
Related Free Tools From Arb Digital
Pair this with the magnetic force on a wire calculator for the force on a straight conductor and the live Lorentz force calculator for a moving charge. The solenoid magnetic field calculator and magnetic field of a wire calculator give the fields those currents produce, the magnetic field converter and magnetic permeability calculator handle units and material response, and the Faraday's law calculator covers induction. The electric dipole moment calculator is the electrostatic counterpart. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the current multiplied by the enclosed area, multiplied by the number of turns, with units of ampere square metres. Its direction is perpendicular to the loop, given by curling the right hand in the direction of the current. It summarises everything about how the loop behaves in an external field.
No, only the area it encloses. A circle, a square and an irregular outline of equal area carry identical moments for the same current. This is exact rather than approximate, because any planar loop can be decomposed into small loops whose internal currents cancel, leaving only the enclosed area.
Because the torque is a cross product, and it depends on the sine of the angle between the moment and the field. At alignment that sine is zero. Alignment is also where the potential energy is lowest, which is the stable equilibrium a compass needle settles into after its oscillations are damped.
Twice the product of the moment and the flux density. The energy is minus mB when aligned and plus mB when anti-aligned, so the difference between the two states is 2mB. That splitting is the basis of magnetic resonance techniques, where the energy gap is set by the applied field strength.
Only in a non-uniform one. In a uniform field the forces on opposite sides of the loop cancel exactly, so it rotates without translating. A net force requires a field gradient, which is why magnets are pulled towards pole pieces where the field converges and feel little in the middle of a wide uniform gap.
The relations are mathematically parallel — torque from a cross product, energy from a dot product — but the quantities are built differently. An electric dipole moment is a charge multiplied by a separation, in coulomb metres. A magnetic dipole moment is a current multiplied by an enclosed area, in ampere square metres. They are not interchangeable.
Because its pole pieces are shaped to produce a radial field, which keeps the moment perpendicular to the field at every angle of rotation. The sine term therefore stays at one, torque is proportional to current alone, and the deflection against a linear spring is proportional to current. Without the radial field the scale would crowd badly towards full deflection.
They subtract. Each turn contributes a moment whose direction follows its own current sense, so a reversed turn cancels a forward one. A figure-of-eight loop has nearly zero net moment for this reason, which is exactly why bifilar winding is used to make non-inductive resistors and why twisted pairs have almost no magnetic signature.
This tool is provided for educational and preliminary engineering use only. It evaluates published relations for an idealised planar current loop in a uniform external field and does not account for field non-uniformity, net translational force, coil self-inductance, saturation of nearby magnetic material or the mechanical design of any device. Strong magnetic fields and the equipment that produces them present real hazards to people with implanted medical devices and to loose ferromagnetic objects. Confirm any practical design with a qualified engineer.
