Magnetic Dipole Moment & Torque

Equivalent forms

\vec{\tau} = \vec{\mu}\times\vec{B}

U = -\vec{\mu}\cdot\vec{B}

\vec{\mu} = I\vec{A}

The cross product τ = μ×B encodes both magnitude and the restoring direction in one stroke — it's the magnetic twin of the electric dipole torque p×E.

Symbol	Name	SI unit	Dimension	Range
$τ$	Torqueoutput Twisting moment on the loop	N·m	M·L²·T⁻²	0 – 10
$μ$	Magnetic moment Strength and axis of the loop's moment, NIA	A·m²	I·L²	0.1 – 5
$B$	Magnetic field Uniform external field	T	M·T⁻²·I⁻¹	0.1 – 2
$θ$	Angle Angle between μ and B	deg	1	0 – 180

Unit systems

Symbol	SI	CGS	Imperial
$τ$	N·m	dyn·cm	ft·lb
$μ$	A·m²	erg/G	A·m²
$B$	T	G	T
$θ$	deg	deg	deg

Where it holds

Exact for a small rigid loop in a uniform field. In a non-uniform field the loop also feels a net force

F = \nabla (\mu \cdot B)

; for spins and atoms the quantum moment is gµ_B m_j rather than NIA.

Dimensional analysis

$[\mu ][B]$ = $(A\cdot m^{2})(kg\cdot s^{-2}\cdot A^{-1}) = kg\cdot m^{2}\cdot s^{-2} = N\cdot m \checkmark (\sin \theta$ dimensionless)

Discovery

André-Marie Ampère · 1820

Within weeks of Oersted's discovery, Ampère proposed that magnetism is electric current in disguise — every magnet is a swarm of circulating 'Ampèrian' loops, each with a magnetic moment. The torque law is the quantitative form of that insight.

Try this

A compass needle, an electric motor, and a spinning electron obey the same rule — what makes a current loop twist in a field?

A 50-turn coil of area 4 cm² carries 2 A in a 0.5 T field, its normal at 30° to B. Find its magnetic moment and the torque on it.

Research status: stable

Real-world applications

Electric motors and moving-coil meters
Compass needles and MEMS magnetometers
NMR/MRI spin dynamics
Magnetic nanoparticle hyperthermia

Common misconceptions

A uniform field pushes the loop — it only twists it; net force needs a gradient
Torque is maximal when $\mu is$ along B — it's actually zero there and maximal at $90^{\circ}$
Magnetic moment depends on the loop's shape — only its enclosed area and current matter

Experimental verification

A pivoted current loop in a known field deflects against a calibrated spring; the measured torque tracks

\sin \theta and

scales linearly with N, I, A, and B — the operating principle of the moving-coil (d'Arsonval) galvanometer.

Derivation

Each side of a rectangular loop carries force

IL\times B

.

The pair parallel to the rotation axis gives equal-and-opposite forces separated by the loop width, forming a couple of moment

\tau = (BIL)\cdot w\cdot \sin \theta = BI

A

\sin \theta per

turn.

For N turns,

\tau = NIA\cdot B\cdot \sin \theta = \mu B \sin \theta

, i.e.

\tau = \mu \times B

.

Limiting cases

\theta = 0^{\circ}

⟶

\tau = 0

(stable)

\mu

aligned with B is the minimum-energy, stable equilibrium.

\theta = 90^{\circ}

⟶

\tau = \mu B (\max )

Maximum twisting when the loop's face is along the field.

\theta = 180^{\circ}

⟶

\tau = 0

(unstable)Anti-aligned is an unstable equilibrium — a nudge flips it around.

What if…

What if you double the number of turns?

$\mu and$ therefore $\tau$ double — the loop area is effectively used N times.

What if the field is non-uniform?

Besides the torque, a net force $F = \nabla (\mu \cdot B)$ appears — this is how magnets attract and how Stern-Gerlach splits beams.

What

if \theta

starts at

180^{\circ}

?

It's an unstable equilibrium: $\tau = 0$ , but the slightest perturbation makes it swing all the $way to \theta = 0^{\circ}$ .

1

50-turn coil at 30°

Given ·

N:: 50
I:: 2
A:: 0.0004
B:: 0.5
θ:: 30

Find ·

\mu and \tau

Steps

$\mu = NIA = 50 \times 2 \times 4\times 10^{-4} = 0.04$ $A\cdot m^{2}$
$\sin 30^{\circ} = 0.5$
$\tau = 0.04 \times 0.5 \times 0.5 = 0.01 N\cdot m$

Result ·

\mu = NIA = 50\times 2\times 4\times 10^{-4} = 0.04

A\cdot m^{2}

;

\tau = \mu B \sin 30^{\circ} = 0.04\times 0.5\times 0.5 = 0.01 N\cdot m

Lorentz Force Law→Magnetic Force on a Current-Carrying Wire→Magnetic Field Inside a Solenoid→