Skin Effect in Conductors

Calculating Skin Depth and Attenuation

For a homogeneous, nonmagnetic, good conductor, the skin depth δ (delta) is defined as the depth at which the current density drops to 1/e (about 37%) of its surface value. The formula is:

δ = √(2 / (μ · σ · ω))

where μ is the permeability (for nonmagnetic materials, μ = 4π × 10^-7 H/m), σ is the electrical conductivity, and ω = 2πf is the angular frequency.

Let's calculate the skin depth for Copper (σ = 5.8 × 10⁷ S/m) at a frequency of 1 MHz (f = 10⁶ Hz), and find the relative amplitude and phase lag at a depth of 0.1 mm (x = 10^-4 m).

1. Calculate ω = 2π × 10⁶ ≈ 6.2831853 × 10⁶ rad/s.
2. Calculate the product μ · σ · ω = (4π × 10^-7) × (5.8 × 10⁷) × (2π × 10⁶) ≈ 4.5796 × 10⁸.
3. Find the skin depth δ = √(2 / 4.5796 × 10⁸) ≈ 66.0855 μm.
4. The complex current relative to the surface is exp(-(1+i)x/δ). At depth x = 0.1 mm (100 μm), the ratio x/δ is 100 / 66.0855 ≈ 1.5132.
5. The relative amplitude is exp(-x/δ) = exp(-1.5132) ≈ 0.2202.
6. The phase lag is x/δ radians, which is approximately 1.5132 radians.

Interactive Attenuation Profile

Open the Skin Effect in Conductors module to observe how the current profile decays into the material.

1. Select the "Copper" Conductor and set Frequency Hz to 1000000. Set Depth mm to 0.1.
2. Observe the Skin depth micrometers is ~66.08, and the Amplitude ratio matches our calculated ~0.2202.
3. Change the Conductor to "Aluminum" and note how the lower conductivity increases the skin depth to ~85.07 μm, resulting in less attenuation (ratio ~0.3087) at the same 0.1 mm depth.
4. Set Frequency Hz to 0 to observe the "Fixed-depth DC limit". Without oscillation, the current penetrates fully without exponential decay or phase lag.

Semi-Infinite Plane Approximation

The formulas used here assume a semi-infinite plane of a good conductor, meaning the surface is flat and the material extends infinitely inward.

This is a valid approximation when the conductor's radius of curvature and thickness are much larger than the skin depth. It does not accurately calculate the full impedance of thin wires at low frequencies, nor does it account for the high-frequency material dispersion (where permittivity and conductivity become complex and frequency-dependent) that occurs at optical frequencies.

Further Reading

The derivation of the wave equation in good conductors and the definition of skin depth can be found in standard electromagnetic theory texts.

• Electromagnetic Waves in Conductors - Richard Fitzpatrick, University of Texas at Austin (Supports the derivation of δ and the exponential decay form).