Skin depth and power in dissipative medium

Skin depth (δ) is the distance a wave travels in a conducting (dissipative) medium over which its amplitude falls to 1/e (about 36.8 %) of its value at the surface. It equals the reciprocal of the attenuation constant: δ = 1/α.

For a good conductor (σ ≫ ωε), α = β = √(πfμσ), so

δ = 1/√(πfμσ)

Skin depth decreases as frequency, permeability or conductivity increases. Example: copper at 1 MHz has δ ≈ 66 μm, so high-frequency current flows only in a thin surface layer.

Time-average power density in a dissipative medium

Let the wave travel in +z with E along a_x, in a medium with γ = α + jβ and intrinsic impedance η = |η|∠θ_η (θ_η is 0 for a lossless medium, up to 45° for a good conductor).

E = E₀ e^(−αz) cos(ωt − βz) a_x
H = (E₀/|η|) e^(−αz) cos(ωt − βz − θ_η) a_y

Instantaneous Poynting vector:

S = E × H
  = (E₀²/|η|) e^(−2αz) cos(ωt − βz) cos(ωt − βz − θ_η) a_z

Use cos A cos B = ½[cos(A − B) + cos(A + B)]:

S = (E₀²/2|η|) e^(−2αz) [cos θ_η + cos(2ωt − 2βz − θ_η)] a_z

The second term averages to zero over one period, so

S_av = (E₀² / 2|η|) e^(−2αz) cos θ_η a_z W/m²

The same result follows from the phasor form S_av = ½ Re(E_s × H_s*).

Remarks:

• Power density falls as e^(−2αz), twice as fast as the field amplitude.
• At z = δ the power density is e⁻² ≈ 13.5 % of its surface value.
• The factor cos θ_η is the "power factor" of the medium; for a lossless medium θ_η = 0 and S_av = E₀²/2η.
• The power lost per unit volume appears as ohmic heat (σE²).