Magnetic Dipoles & Materials

A loop carrying current I around an enclosed area A, wound N times, has dipole moment μ = N I A. Curl the right hand along the current and the thumb gives the direction of μ, perpendicular to the loop's plane. The claim worth making is that this single vector is enough: at distances large compared with the loop, the field depends on the current and the area only through their product, so a circular loop, a square loop and a 200-turn pancake of the same μ are indistinguishable. A 200-turn coil of radius 1.5 cm carrying 0.40 A encloses A = π(0.015 m)² = 7.07 × 10⁻⁴ m² and so has μ = 200 × 0.40 A × 7.07 × 10⁻⁴ m² = 5.65 × 10⁻² A m². Its far field falls as 1/r³ — faster than a straight wire's 1/r, because the two sides of the loop very nearly cancel.

Put that loop in a uniform field. Opposite sides carry current in opposite directions, so the forces on them are equal and opposite: the net force is zero and the loop's centre of mass does not move. What survives is a couple, τ = μ × B, of magnitude μB sin θ, greatest when μ lies across the field and zero when it lies along it. Integrating that torque through an angle gives the orientation energy U = −μ⋅B = −μB cos θ, a minimum when μ is aligned and a maximum when it is reversed, so turning an aligned dipole end for end costs 2μB. For the coil above in B = 0.35 T the maximum torque is 5.65 × 10⁻² A m² × 0.35 T = 1.98 × 10⁻² N m and the flip costs 3.96 × 10⁻² J. This is what a compass does, and what a moving-coil meter does against a hairspring.

Since U = −μ_z B_z along an axis, the force is F_z = −dU/dz = μ_z (dB_z/dz) for a rigid moment: no gradient, no force. That single line explains the demonstration in which samples are hung near a pole piece. A paramagnetic sample develops μ parallel to B, so it is pulled toward the stronger field; a diamagnetic sample develops μ antiparallel, so it is pushed away. Numbers: a 1.0 cm³ sample with χₘ = 2.0 × 10⁻⁴ sitting in B = 1.2 T has M = χₘ B/μ₀ = 191 A m⁻¹, hence μ = M V = 1.91 × 10⁻⁴ A m², and in a gradient of 15 T m⁻¹ it feels F = 2.9 × 10⁻³ N — about a seventh of the weight of a 2 g sample. Water, with χₘ = −9.0 × 10⁻⁶, needs B(dB/dz) = μ₀ρg/|χₘ| = 1.37 × 10³ T² m⁻¹ to levitate, which is why the feat takes a 16 T magnet.

An electron circulating in an atom is a current loop of atomic size, and its moment comes in multiples of the Bohr magneton μ_B = eħ/(2mₑ) = 9.274 × 10⁻²⁴ A m². Electron spin contributes a moment of about the same size; nuclear moments are smaller by roughly the electron-to-proton mass ratio and can be ignored here. In a filled shell the moments pair off and cancel, which is why most substances carry no permanent moment at all. Add the survivors per unit volume and you have the magnetization M = (Σ μᵢ)/V in A m⁻¹. Iron packs n = 8.49 × 10²⁸ atoms m⁻³ and saturates at Mₛ = 1.71 × 10⁶ A m⁻¹, so each atom contributes 1.71 × 10⁶ ÷ 8.49 × 10²⁸ = 2.01 × 10⁻²³ A m², which is 2.17 μ_B — about two unpaired spins' worth, and well short of the four unpaired 3d electrons an isolated iron atom carries, because in the metal those electrons are shared through the crystal rather than owned by one atom.

Every material is diamagnetic. Switching on a field changes the flux through each atomic orbit, and by Lenz's law the induced change in that orbital motion opposes the change, so M points against B. The effect is tiny and temperature-independent: χₘ = −9.0 × 10⁻⁶ for water, −1.66 × 10⁻⁴ for bismuth. Where atoms keep a permanent moment a much larger alignment is available, but thermal agitation fights it. Compare the energies: at B = 1.0 T and T = 300 K, μ_B B/k_BT = 9.274 × 10⁻²⁴ ÷ (1.381 × 10⁻²³ × 300) = 2.2 × 10⁻³. The alignment energy is smaller than the thermal energy by a factor of about 450, so only a whisper of net order survives — which is why aluminium manages no more than χₘ = +2.2 × 10⁻⁵, a separate measured number and not that ratio. Because the competition is with k_BT, paramagnetic susceptibility obeys Curie's law, χₘ = C/T: cool the sample and the response grows.

In iron, cobalt, nickel and some alloys a quantum exchange interaction locks neighbouring spins parallel over regions from 10⁻² mm to a few mm — domains, each already magnetized close to saturation. An unmagnetized bar simply has its domains pointing every way. A modest applied field does not align single atoms; it moves the domain walls, letting favourably oriented domains grow, and the response is thousands of times larger than any paramagnet's. Three consequences break the linear model. The B-H curve bends over at saturation: once every domain points one way M can rise no further than Mₛ, and for iron μ₀Mₛ ≈ 2.15 T. It is not single-valued — B depends on the history of H, which is hysteresis, and what a permanent magnet lives on. And heating destroys the order: above the Curie temperature, 1043 K for iron and 627 K for nickel, the metal is an ordinary paramagnet.