Magnetic Moment of Current Loop

Magnetic dipole moment for a planar loop of N turns, area A, carrying current I. Direction n̂ along the axis by right-hand rule. SI unit: A·m².

Definition

For a planar loop of $N$ turns, area $A$, carrying current $I$:

$$\boxed{\vec{m} = NIA\,\hat{n}}$$

$\hat{n}$ is the unit normal to the plane of the loop, given by the right-hand rule applied to the current direction.

SI Unit

$$[m] = \text{A·m}^2$$

Why This Form

The magnetic moment captures how strongly the loop interacts with an external field. From mc19, torque $\tau = mB\sin\theta$ — a larger $m$ means a stronger aligning torque for the same field.

Analogy with Electric Dipole

Far-Field Behaviour

At large distances along the axis ($x \gg R$), a current loop produces the same field as a magnetic dipole with moment $m = I\pi R^2$:

$$B_{\text{axis}} \approx \frac{\mu_0}{4\pi}\frac{2m}{x^3}$$