Skin Depth

Alternating current does not use the middle of a conductor. The changing field inside induces eddy currents that oppose flow at the centre and reinforce it at the surface, so the current crowds into a skin whose characteristic thickness is $\delta = \sqrt{2\rho/\omega\mu}$. For copper at 60 Hz that comes to about 8.5 mm — so any conductor thicker than roughly 17 mm is carrying nothing useful in its core, which is why 1000 kcmil cable has such a disappointing AC resistance for its mass.

The frequency dependence is a square root, and it bites quickly. Copper's skin depth is 8.5 mm at 60 Hz, 0.66 mm at 10 kHz, 66 µm at 1 MHz, and about 2 µm at 1 GHz. That is why RF coils are wound with litz wire (many thin insulated strands, each thinner than a skin depth), why waveguide is silver-plated rather than solid silver, and why a printed circuit trace at microwave frequencies is really just a gold-plated surface with copper along for structural support.

The permeability term is the one people forget, and it is the reason steel behaves so differently from copper. Steel is a worse conductor and has a relative permeability in the hundreds, so its skin depth at 60 Hz is under a millimetre — which is why steel conduit and armour heat up around unbalanced circuits, and why induction heating works at all. Enter µ in henries per metre here: 1.2566e-6 for copper, aluminium and any non-magnetic metal, and a hundred to a thousand times more for magnetic steel.