The bar magnet as a dipole. A bar magnet behaves like a magnetic dipole with two equal and opposite poles. Its magnetic dipole moment is $\vec{m}=m_p\,(2\vec{l})$, a vector pointing from the south to the north pole; its SI unit is $\text{A m}^2$. As with electric dipoles, isolated magnetic poles (monopoles) have never been found — cutting a magnet always yields smaller complete dipoles.
Field of a magnetic dipole. Treating the bar magnet like an electric dipole (with $\frac{\mu_0}{4\pi}$ replacing $\frac{1}{4\pi\varepsilon_0}$), the field at a point on the axial line at distance $r$ is $B_{axial}=\frac{\mu_0}{4\pi}\frac{2m}{r^3}$, and on the equatorial line it is $B_{eq}=\frac{\mu_0}{4\pi}\frac{m}{r^3}$. The axial field is therefore twice the equatorial field and points along $\vec{m}$, while the equatorial field opposes $\vec{m}$.
Torque and potential energy. In a uniform field a dipole feels a torque $\vec{\tau}=\vec{m}\times\vec{B}$ (magnitude $mB\sin\theta$) and has potential energy $U=-\vec{m}\cdot\vec{B}=-mB\cos\theta$. It is in stable equilibrium when aligned with the field ($\theta=0$) and unstable when anti-aligned ($\theta=180^\circ$).
The Earth's magnetism. The Earth behaves like a giant bar magnet tilted from its rotation axis, with its magnetic south near the geographic north. Three elements of terrestrial magnetism describe the field at a place:
- Declination ($D$): the angle between geographic north and magnetic north (the horizontal direction the compass points).
- Dip or inclination ($I$): the angle the total field makes with the horizontal. At the magnetic equator $I=0^\circ$; at the magnetic poles $I=90^\circ$.
- Horizontal component ($B_H$): $B_H=B\cos I$ and the vertical component $B_V=B\sin I$, so $\tan I=\frac{B_V}{B_H}$.
Magnetic intensity and material quantities. Inside a material the field is $\vec{B}=\mu_0(\vec{H}+\vec{M})$, where $\vec{H}$ is the magnetising field and $\vec{M}$ the magnetisation (moment per unit volume). The susceptibility $\chi=\frac{M}{H}$ and the relative permeability $\mu_r=1+\chi$ classify materials.
Three classes of magnetic materials.
- Diamagnetic (e.g. bismuth, copper, water): weakly repelled by a magnet; $\chi$ is small and negative; $\mu_r$ slightly less than 1. They move from strong to weak field regions.
- Paramagnetic (e.g. aluminium, sodium, oxygen): weakly attracted; $\chi$ is small and positive; $\mu_r$ slightly more than 1. Magnetisation follows Curie's law $\chi\propto\frac{1}{T}$.
- Ferromagnetic (e.g. iron, cobalt, nickel): strongly attracted; $\chi$ is large and positive; $\mu_r$ very large. Above the Curie temperature they become paramagnetic.
Hysteresis. When a ferromagnetic sample is taken through a full cycle of magnetising field, the $B$-$H$ graph forms a closed loop called the hysteresis loop. The lag of $B$ behind $H$ is hysteresis. The field $B$ left when $H=0$ is the retentivity, and the reverse $H$ needed to make $B=0$ is the coercivity. The loop area equals the energy lost per cycle. Soft ferromagnets (soft iron) have a thin loop, low coercivity and small loss — ideal for electromagnets and transformer cores; hard ferromagnets (steel, alnico) have a wide loop and high retentivity — ideal for permanent magnets.