Eddy currents and magnetic diffusion

Assume a stationary homogeneous conductor with constant positive conductivity and a linear isotropic, time-independent permeability. Ohm's law gives current density equal to conductivity times E. Neglect displacement current relative to conduction current, so curl B equals permeability times current density. For a sinusoidal process a basic material check is $\omega\epsilon/\sigma\ll1$; the geometry and frequencies must also justify a magnetoquasistatic description rather than significant radiation or propagation delay.

Taking the curl of Ohm's law combined with Ampère's law, then substituting Faraday's law, yields field diffusion. Zero magnetic divergence turns the curl of a curl into minus the vector Laplacian. Conductivity or permeability variations cannot simply be pulled through these derivatives. Motion would add an induction term, and nonlinear hysteretic magnetization would require a different constitutive model. Listing these exclusions identifies exactly which physical feedback the simple equation captures.

For an axial field depending on one transverse coordinate, a sine spatial mode is consistent with zero divergence. Two spatial derivatives return minus the squared wavenumber times that mode. Its amplitude therefore decreases exponentially with a time constant inversely proportional to wavenumber squared. Doubling the wavenumber shortens the decay time by a factor of four. The diffusion time over a length scale is proportional to permeability times conductivity times length squared, with numerical factors fixed by the actual boundary conditions.

A slab whose transient axial field vanishes at both faces has sine modes with wavenumbers equal to integer multiples of pi divided by slab thickness. A driven boundary field must first be separated from this homogeneous-boundary transient; otherwise the chosen sine expansion will violate the boundary data. Diffusion does not generally erase an imposed steady field. It erases departures from the steady solution while currents dissipate energy as heat.

A sinusoidal boundary drive produces a field with amplitude decaying over the skin depth and phase changing with depth. Raising frequency reduces penetration; raising conductivity also reduces penetration because induced currents oppose changing interior flux more effectively. This does not make a normal conductor a universal shield against static magnetic fields. In the zero-frequency limit its ideal skin depth grows without bound, and transient shielding eventually decays at finite conductivity.

In this linear model the magnetic energy density is B squared divided by twice permeability. Poynting's balance relates its change to boundary energy flux and positive Joule heating, with density current squared divided by conductivity. Thin insulated laminations interrupt large transverse current loops and reduce eddy-current loss in a suitable core geometry. This geometrical suppression differs from reducing hysteresis: an electrically resistive material can still dissipate energy through irreversible magnetization processes.