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How it works
What you are seeing
A ferrofluid is oil loaded with magnetite grains about ten nanometres across, each coated so they never clump. In a vertical magnetic field the liquid magnetises, and any bump on its surface concentrates the field lines at its tip, which pulls the bump further up. Gravity and surface tension push back. Below a critical field they win and the surface stays a flat black mirror. Above it the flat state is unstable and the surface settles into a hexagonal lattice of spikes.
The bench
The dish sits on an electromagnet: 860 turns of enamelled copper around a soft iron core, fed in constant-current mode by a bench supply. The coil makes a nearly uniform vertical field over the dish, about 8.3 mT/A . The NdFeB magnet on the swivel arm adds a strong local field that you can move around.
The exact part
The spacing and the threshold come from the linear stability analysis of Cowley and Rosensweig (1967). The critical wavelength is λc = 2π√(σ / ρg) and the critical magnetisation obeys Mc ² = (2/μ0 )(1 + 1/μr )√(ρgσ). With ρ = 1210 kg/m³, σ = 0.025 N/m and χ = 1.6 (a light oil ferrofluid close to Ferrotec EFH1, with a linear magnetisation law) that gives 9.1 mm and an applied field of 12.6 mT . The spikes of a hexagonal pattern sit 2λc /√3 apart. The lift of the whole surface under the magnet is ferrohydrostatics: ρgΔh equals the change of the magnetic normal traction χB²/(2μ0 μr ). The field on the axis of the magnet is the exact formula for a cylindrical magnet.
The simplified part
How fast and how tall the spikes grow is not solved from the full nonlinear equations. The amplitude on each patch of the dish follows a Ginzburg–Landau type equation, dA/dt = (εA + βA² − A³)/τ + D∇²A, with ε = (B² − Bc ²)/Bc ². The quadratic term is the standard one for hexagons and it is why the transition is hysteretic: spikes that exist survive a few percent below the threshold. β, τ, D and the spike height scale are chosen, not measured. The spike shape sharpens with amplitude, which is qualitatively right. The radial profile of the magnet field is a point-dipole approximation, and the mound follows the moving magnet with a chosen viscous response time of 0.28 s.
The AC setting reverses the coil current at 0.8 Hz, slowed so each collapse and regrowth is visible. A tap on the fluid sends a transient ripple through the pattern; its speed and decay are chosen for the eye.
The cutaway view opens a wedge of the coil and draws its field lines. They are traced exactly for the winding as a stack of circular loops, but without the iron core, which makes the field stronger and bends the lines inside the core. They stop at the base plate. The pulses along them show the direction of the field and reverse with the current; they are not something that moves.
History
Ferrofluids were made at NASA in the early 1960s by Steve Papell, who wanted a rocket fuel that could be pulled to the pump in zero gravity. Ronald Rosensweig turned them into an engineering material, and the spike pattern carries his name. Today the same liquid seals the shafts of hard disks and cools the voice coils of loudspeakers.