Skin Depth⚠ unverified
Physics / Electromagnetics · Compute the skin depth of an alternating current in a conductor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| f | f | Hz | 1.0 | Frequency of the alternating current |
| mu | μ | H/m | 1.0 | Magnetic permeability of the conductor |
| sigma | σ | S/m | 1.0 | Electrical conductivity of the conductor |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | δ | m | Skin depth, in metres (m). Returns 0.0 when any input is not positive |
The science & history
Understanding the Parameters
- $f$ — $\delta \propto 1/\sqrt{f}$; RF current crowds to the surface.
- $\mu$ — ferromagnetic materials (high $\mu$) have smaller skin depth.
- $\sigma$ — copper ≈ $5.8\times 10^{7}$ S/m; higher $\sigma$ → smaller $\delta$.
- $\delta$ — at depth $\delta$, $|J|$ falls to $1/e$ of surface value (good-conductor limit).
Derivation (Approaching a Proof)
From the diffusion equation for $\mathbf{B}$ or $\mathbf{E}$ in a linear conductor ($\nabla^{2}\mathbf{E} = \mu\sigma\partial\mathbf{E}/\partial t$), a surface-parallel time-harmonic field decays as $e^{-z/\delta}$ with
$$\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f\mu\sigma}}.$$
(Assumes displacement current negligible vs conduction current — good conductors at moderate $f$.)
History
Skin effect was known in 19th-century telegraphy and is critical for RF coils, busbars, and induction heating.
Related Concepts: Wave Impedance, Inductance Solenoid, Radiation Resistance, Poynting Vector
Notes: Registry calculator skin-depth (unverified). Good-conductor, linear isotropic media.