An alternating field at 50 Hz is applied to electrical steel sheet with an electrical resistivity of 4.8×10⁻⁷ Ω·m and a relative permeability of 5000. Which value is closest to the skin depth δ = √(2ρ/(ωµ))? Take ω = 2πf, µ = µᵣµ₀ and µ₀ = 4π×10⁻⁷ H/m.

Answer: A. About 0.70 mm

ωµ = 2π × 50 × 5000 × 4π×10⁻⁷ ≈ 1.97, so 2ρ/(ωµ) = 9.6×10⁻⁷/1.97 ≈ 4.9×10⁻⁷ m², and δ ≈ 7.0×10⁻⁴ m = 0.70 mm. About 0.49 mm drops the 2 in the formula, and about 1.7 mm substitutes the frequency f directly in place of ω. The skin depth is a measure of how deep alternating flux can penetrate into the sheet; unless the sheet is well thinner than this, the flux does not reach its center and the material is not used effectively. Electrical steel about 0.35 mm thick satisfies this condition at 50 Hz, but δ shrinks in inverse proportion to the square root of frequency, so at kHz and above, thin ribbons of amorphous alloys or high-resistivity ferrites are used.