Why a Current Loop Behaves Exactly Like a Tiny Bar Magnet

Break a bar magnet in half and you get two smaller magnets, never a lone north pole. The reason is that magnetism comes from circulating currents — from coils of wire and from electrons moving inside atoms.

This lesson covers the current loop as a magnetic dipole, the field of a bar magnet, and how different materials respond to a magnetic field.

**A current loop produces the same field pattern as a bar magnet, and its magnetic dipole moment is $m=NIA$, directed perpendicular to the loop by the right-hand rule.

Why it is a dipole.** Looking at the loop from one face, the current circulates anticlockwise and that face behaves as a north pole; the other face is a south pole. Far away, its field falls as $1/r^3$, just like a bar magnet's.

In a uniform field, the loop feels torque $\tau =mB\sin \theta$ and has energy $U=-mB\cos \theta$.

Worked example 1. A 100-turn coil of radius 0.040 m carries 0.50 A. Its area is $\pi \times 0.040^2=5.03\times 10^{-3}$ m${}^2$, so

$$m=100\times 0.50\times 5.03\times 10^{-3}=0.251{\text{ A m}}^2$$

In a 0.20 T field at $30^{\circ }$: $\tau =0.251\times 0.20\times \sin 30^{\circ }=0.025$ N m.

Worked example 2 — an atom. An electron in the lowest Bohr orbit has a magnetic moment of

$${\mu }_B=\frac{eh}{4\pi m_e}=\frac{1.6\times 10^{-19}\times 6.63\times 10^{-34}}{4\pi \times 9.1\times 10^{-31}}=9.27\times 10^{-24}{\text{ A m}}^2$$

An everyday example. A compass needle swings to line up with the axis of a current-carrying coil, just as it would with a bar magnet.

The substance. Every atom with orbiting or spinning electrons is a tiny current loop — which is how ordinary matter can become magnetic.

**For a short bar magnet of moment m, the field at distance r is $B=\frac{{\mu }_0}{4\pi }\frac{2m}{r^3}$ on the axial line, along m, and $B=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$ on the equatorial line, opposite to m.

Same form as the electric dipole.** These results match the electric dipole's, with $\frac{1}{4\pi {\epsilon }_0}$ replaced by $\frac{{\mu }_0}{4\pi }$ and p by m.

Worked example. A short magnet of moment 2.0 A m${}^2$, at 0.10 m:

$$B_{\text{axial}}=10^{-7}\times \frac{2\times 2.0}{0.10^3}=4.0\times 10^{-4}\text{ T}{B}_{\text{equatorial}}=10^{-7}\times \frac{2.0}{0.10^3}=2.0\times 10^{-4}\text{ T}$$

At 0.20 m, both fields fall by a factor of $2^3=8$.

Gauss's law for magnetism. The net magnetic flux through any closed surface is zero, because field lines always form closed loops, running from N to S outside the magnet and from S to N inside.

An everyday example. Iron filings sprinkled on paper over a bar magnet trace these closed field lines, crowding near the poles where the field is strongest.

The substance. Unlike electric field lines, magnetic field lines never begin or end — there are no isolated magnetic poles.

**Magnetic susceptibility $\chi =M/H$ measures how strongly a material magnetises, and relative permeability is ${\mu }_r=1+\chi$; diamagnets have small negative $\chi$, paramagnets small positive $\chi$, and ferromagnets very large positive $\chi$.

The three classes:

- Diamagnetic**, such as bismuth, copper and water: weakly repelled, with ${\mu }_r$ just below 1
- Paramagnetic, such as aluminium and oxygen: weakly attracted, with ${\mu }_r$ just above 1, following Curie's law $\chi =C/T$
- Ferromagnetic, such as iron, nickel and cobalt: strongly attracted, with domains that align, until heating above the Curie temperature turns them paramagnetic

Worked example 1 — an iron core. A solenoid with 500 turns per metre carries 2.0 A, so $H=nI=1.0\times 10^3$ A m${}^{-1}$. With an iron core of ${\mu }_r=800$:

$$B={\mu }_0{\mu }_{r}H=4\pi \times 10^{-7}\times 800\times 1.0\times 10^3=1.0\text{ T}$$

Worked example 2 — Curie's law. A paramagnet with $\chi =2.4\times 10^{-5}$ at 300 K has $\chi =2.4\times 10^{-5}\times \frac{300}{150}=4.8\times 10^{-5}$ at 150 K.

An everyday example. Transformer and motor cores are made of soft iron, because its huge permeability concentrates the field.

The substance. Diamagnetism is present in every material, but in paramagnets and ferromagnets it is swamped by the stronger effect.