What is the proximity effect?

Nearby conductors carrying AC create magnetic fields that distort each other's current distribution. Current crowds toward or away from neighboring conductors, increasing AC resistance beyond skin effect alone.

Why does Rac/Rdc rise with frequency?

As frequency increases, skin depth shrinks. The same current uses a thinner effective region near the surface, and nearby conductors further crowd the current, so the effective resistance rises.

Why does the skin effect increase the effective resistance of a wire at high frequencies?

The skin effect confines current to a thinner cross-sectional area near the surface. Since resistance is inversely proportional to the effective cross-sectional area through which current flows, this confinement forces the same total current through a smaller area, increasing the effective AC resistance compared to the DC resistance. This is why high-frequency conductors are often plated or made hollow to save material without sacrificing performance.

Does the skin effect occur with direct current (DC)?

No, the skin effect is a direct consequence of time-varying electromagnetic fields. With steady DC (ω=0), the skin depth formula gives an infinite depth, meaning the current distributes uniformly across the conductor's cross-section (for a homogeneous material). The effect only becomes significant when the frequency is high enough that the skin depth is comparable to or smaller than the conductor's radius.

What is a key limitation of this 1D exponential model?

This model assumes a semi-infinite planar conductor, which is an excellent approximation for the flat surface of a large wire. However, for a cylindrical wire of finite radius, the current distribution is more complex, described by Bessel functions. The simple exponential decay is not perfectly accurate near the center of a round wire, especially when the wire radius is not much larger than the skin depth.

How does the choice of conductor material impact the skin depth?

Skin depth depends on both permeability (μ) and conductivity (σ). For a given frequency, a higher conductivity (like silver vs. copper) leads to a smaller skin depth, concentrating current more sharply. More significantly, ferromagnetic materials (like iron) have a much higher relative permeability (μᵣ >> 1), which dramatically reduces skin depth, making the skin effect extreme even at power-line frequencies.

Skin Depth & Proximity Effect

The skin effect describes the tendency of alternating current (AC) to flow primarily near the surface of a conductor, with current density decaying exponentially with depth. This simulator visualizes this phenomenon in one dimension, plotting the normalized current density J(z)/J₀ against depth z into a semi-infinite planar conductor. The core physics is governed by Maxwell's equations, specifically the diffusion of electromagnetic fields into a conductive material. For a good conductor, solving the wave equation leads to a complex propagation constant, whose real part defines the skin depth δ. The skin depth formula, δ = √(2/(ωμσ)), is central to the model. Here, ω is the angular frequency of the AC signal, μ is the permeability of the material (typically μ₀ for non-magnetic conductors), and σ is the electrical conductivity. The simulator plots the function J(z) = J₀ e^{-z/δ}, showing that at a depth of one skin depth, the current density falls to about 37% of its surface value. Key simplifications include assuming a uniform, linear, isotropic conductor with a perfectly flat surface and a plane wave excitation, neglecting edge effects and any temperature dependence of σ. By adjusting parameters like frequency and conductivity, users can explore how the skin depth changes, understanding why high-frequency circuits use hollow or surface-plated conductors and how power transmission at 50/60 Hz experiences only a mild skin effect in large cables.

Who it's for: Undergraduate physics and electrical engineering students studying electromagnetism, wave propagation in materials, and AC circuit design.

Key terms

Skin Effect
Skin Depth
Proximity Effect
Current Crowding
AC Resistance
Permeability
Electrical Conductivity

The cross-section and resistance ratio are educational approximations. Accurate round-wire AC resistance and proximity loss require Bessel/Dowell-style field solutions and winding geometry.

How it works

Skin depth and proximity-effect simulator: current crowding in a round conductor and an estimated AC resistance ratio versus frequency.

Key equations

δ = √(2/(ω μ σ)), ω = 2πf

teaching estimate: Rac/Rdc grows as conductor radius a becomes larger than δ; nearby conductors add proximity crowding

Frequently asked questions

What is the proximity effect?: Nearby conductors carrying AC create magnetic fields that distort each other's current distribution. Current crowds toward or away from neighboring conductors, increasing AC resistance beyond skin effect alone.
Why does Rac/Rdc rise with frequency?: As frequency increases, skin depth shrinks. The same current uses a thinner effective region near the surface, and nearby conductors further crowd the current, so the effective resistance rises.
Why does the skin effect increase the effective resistance of a wire at high frequencies?: The skin effect confines current to a thinner cross-sectional area near the surface. Since resistance is inversely proportional to the effective cross-sectional area through which current flows, this confinement forces the same total current through a smaller area, increasing the effective AC resistance compared to the DC resistance. This is why high-frequency conductors are often plated or made hollow to save material without sacrificing performance.
Does the skin effect occur with direct current (DC)?: No, the skin effect is a direct consequence of time-varying electromagnetic fields. With steady DC (ω=0), the skin depth formula gives an infinite depth, meaning the current distributes uniformly across the conductor's cross-section (for a homogeneous material). The effect only becomes significant when the frequency is high enough that the skin depth is comparable to or smaller than the conductor's radius.
What is a key limitation of this 1D exponential model?: This model assumes a semi-infinite planar conductor, which is an excellent approximation for the flat surface of a large wire. However, for a cylindrical wire of finite radius, the current distribution is more complex, described by Bessel functions. The simple exponential decay is not perfectly accurate near the center of a round wire, especially when the wire radius is not much larger than the skin depth.
How does the choice of conductor material impact the skin depth?: Skin depth depends on both permeability (μ) and conductivity (σ). For a given frequency, a higher conductivity (like silver vs. copper) leads to a smaller skin depth, concentrating current more sharply. More significantly, ferromagnetic materials (like iron) have a much higher relative permeability (μᵣ >> 1), which dramatically reduces skin depth, making the skin effect extreme even at power-line frequencies.

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