Formula of Magnetic Moment: m = NIA, Units and Torque

The formula of magnetic moment for a coil is $m = NIA$ : multiply the number of turns N by the current I and the area A of one turn. The product tells you how strongly the coil behaves like a small magnet and how much it will twist when placed in a magnetic field.

Many books write the moment as μ or M instead of m. For a bar magnet the matching result is pole strength times the distance between the poles. Either way, the moment is a vector pointing from the south pole to the north pole, and for a loop it points along the normal given by the right hand rule. It is a Class 12 topic and a frequent JEE and NEET question.

Quick facts

Current loop
$m = NIA$
Bar magnet
$m = q_m \times 2l$
Torque in a field
$\tau = mB\sin\theta$
Potential energy
$U = -mB\cos\theta$
SI unit
A m², equal to J/T
Dimensions
[M⁰ L² T⁰ I¹]
Type
Vector, along the loop's axis

What the formula of magnetic moment contains

Factors in m = NIA
Symbol	Meaning	SI unit
$m$	Magnetic dipole moment	A m²
$N$	Turns in the coil	pure number
$I$	Current in each turn	A
$A$	Area of one turn (πr² for a circle)	m²

The shape of the loop does not matter, only its area. A square and a circle with the same area and current have the same moment. To find the direction, wrap your right hand fingers along the current; your thumb gives the direction of m.

Magnetic moment formula in physics problems: torque and energy

In a uniform field B, a loop feels no net force but does feel a torque that tries to line m up with B:

$\tau = mB\sin\theta$

Here θ is the angle between m and B. The torque is largest at 90° and zero when the two are aligned. The stored energy is $U = -mB\cos\theta$ , lowest at θ = 0 (stable) and highest at 180° (unstable). This is how the coil of a moving coil galvanometer and an electric motor turns. The work done to rotate it from θ₁ to θ₂ is $W = mB(\cos\theta_1 - \cos\theta_2)$ .

Worked problems

Problem 1: a rectangular coil

A coil of 100 turns, each of area 4.0 cm², carries 0.50 A. Find its magnetic moment.

A = 4.0 × 10⁻⁴ m².

$m = (100)(0.50)(4.0\times10^{-4}) = 0.020\ \text{A m}^2$

Problem 2: torque in a field

The coil in Problem 1 sits in a 0.30 T field with its axis at 30° to the field. Find the torque.

$\tau = (0.020)(0.30)\sin 30^\circ = 3.0\times10^{-3}\ \text{N m}$

Problem 3: a single circular loop

A single loop of radius 4.0 cm carries 2.0 A. Find m.

A = π(0.040)² = 5.03 × 10⁻³ m².

$m = (1)(2.0)(5.03\times10^{-3}) \approx 1.0\times10^{-2}\ \text{A m}^2$

Bar magnets and atoms

A bar magnet behaves like a current loop. If each pole has strength q_m and the poles are 2l apart, its moment is $m = q_m(2l)$ , directed from south to north. Its field far along the axis is $B = \frac{\mu_0}{4\pi}\frac{2m}{r^3}$ , which has the same shape as the electric field of a dipole.

An electron orbiting a nucleus is also a tiny current loop. Its natural unit of moment, the Bohr magneton, is about 9.27 × 10⁻²⁴ J/T. You do not need to derive it at school level, but it explains why some materials are magnetic.

Unit and dimension check

Amperes times square metres give A m². Because $U = -mB\cos\theta$ is an energy, joules per tesla is an equally valid unit. Current has dimension I and area L², which gives [M⁰ L² T⁰ I¹]. Quick check: m times B should give torque, and multiplying [L² I] by [M T⁻² I⁻¹] leaves [M L² T⁻²], the dimensions of a newton metre.

Torque and energy have the same dimensions, but torque is a vector and energy is a scalar. Write N m for torque and J for energy.

Slips that cost marks

Leaving the area in cm². Multiply by 10⁻⁴ first.
Using cos θ for torque or sin θ for energy. Torque goes with sin, energy with cos.
Forgetting N for a coil of many turns.
Measuring θ from the plane of the coil instead of its axis.

Fields that act on the moment are found on the magnetic field formula page. See also flux density and the electricity formulas overview.

Frequently asked questions

What is the formula of magnetic moment of a current loop?

m = NIA, the number of turns times the current times the area of one turn.

Which unit is used for magnetic moment?

A m² in SI. An equal form is joule per tesla (J/T).

Does magnetic moment have a direction?

It does, since magnetic moment is a vector. For a loop it points along the axis given by the right hand rule; for a magnet it points from south to north.

What is the torque on a magnetic dipole in a field?

τ = mB sin θ, where θ is the angle between the moment and the field. It is greatest when they are at right angles.

What is the magnetic moment of a bar magnet?

It is pole strength multiplied by the magnetic length, the distance between the two poles: m = q_m × 2l.