Skin depth
δ = √(2/(μσω)), the 1/e penetration depth of an EM wave into a conductor. Fields decay exponentially with depth, so high-frequency currents flow only in a thin surface layer — the skin effect.
Definition
The skin depth δ = √(2/(μσω)) is the characteristic 1/e decay length for electromagnetic field penetration into a good conductor at angular frequency ω, where μ is the permeability and σ the conductivity. Inside the metal, the plane-wave solution of the modified Maxwell equations (with the conductivity term σE included) gives E(z) = E₀ e^{−z/δ} e^{i(z/δ − ωt)} — a wave that attenuates exponentially and rotates in phase over the same length scale δ. Typical copper skin depths: 9 mm at 50 Hz (power grid); 66 μm at 1 MHz (AM radio); 2 μm at 1 GHz (microwave); 660 nm at 100 GHz.
The practical consequence is the skin effect: high-frequency currents flow only in a thin surface layer of a conductor, not through the bulk. This raises the effective AC resistance of a wire above its DC resistance — a 1 cm copper wire at 1 MHz has the same AC resistance as a 66-μm-thick hollow tube of the same outer diameter. Radio transmitters therefore use hollow pipe or Litz wire (many fine strands individually insulated) to reduce losses. The skin effect is also why microwave ovens can be built out of thin sheet metal (the fields never penetrate more than microns into the cavity walls) and why superconducting RF cavities for particle accelerators need only a micron of superconductor bonded to a copper support structure.