EM Waves in Conductors: Skin Depth Formula and Meaning

When an electromagnetic wave enters a good conductor such as copper, it does not travel far. The fields decay rapidly with depth, and the currents flow mainly near the surface. This is called the skin effect.

Skin depth explains why high-frequency cables are often hollow or silver-coated, why metal shields block radio waves, and why induction heating heats the surface of metal parts first.

Key Concepts

Skin depth δ is the distance inside a conductor over which the field amplitude falls to 1/e, about 37 percent, of its surface value. Inside a conductor, free charges respond to the wave and create currents that oppose and absorb the field. The higher the frequency, the shallower the penetration. The higher the conductivity σ and permeability μ, the shallower the penetration too. Most of the current is carried within a few skin depths of the surface, raising the effective resistance at high frequency compared with direct current.

Formulas

δ = √(2 / (μ σ ω)), where δ is in metres, μ is permeability in H/m, σ is conductivity in S/m, and ω = 2πf is angular frequency in rad/s with f in Hz. Field decay is roughly E(x) = E0 e^(-x/δ) for depth x. The formula assumes a good conductor where conduction current dominates displacement current, a flat surface, and uniform material properties. For copper, σ ≈ 5.8 × 10^7 S/m and μ ≈ μ0.

Worked Example

Find the skin depth in copper at 1 MHz. Step 1: f = 1 × 10^6 Hz, so ω = 2πf = 6.28 × 10^6 rad/s. Step 2: μσ ω = (4π × 10^-7) × (5.8 × 10^7) × (6.28 × 10^6). First μσ = 4π × 10^-7 × 5.8 × 10^7 = 72.9. Step 3: Multiply by ω: 72.9 × 6.28 × 10^6 = 4.58 × 10^8. Step 4: δ = √(2 / 4.58 × 10^8) = √(4.37 × 10^-9) = 6.6 × 10^-5 m = 0.066 mm. So at 1 MHz the current in copper is confined to a layer about a fifteenth of a millimetre thick.

Common Misconceptions

• Skin effect is not only for alternating current in wires. It is the general penetration behaviour of EM waves in conductors.
• Thicker solid wire does not always help at high frequency. The centre carries little current, so stranded or hollow conductors can be better.
• Metal shielding does not need to be very thick for radio frequencies. A few skin depths of good conductor attenuate fields strongly.

Real-World Applications

• Radio-frequency cables, waveguides and antenna conductors designed around surface current.
• Induction cookers and industrial induction hardening heat metal surfaces.
• Electromagnetic shielding for electronics, MRI rooms and secure enclosures.

In conductors, EM fields fade exponentially with skin depth δ = √(2/(μσω)). Higher frequency means thinner skin, which shapes how we build high-frequency conductors and shields.