The description: Skin depth is a property of a material which dictates to what extent an electromagnetic wave can penetrate the material. Think about what happens when you look into a mirror. The back of the mirror is coated with a conducting material, which reflects light. This is essentially due to the fact that electrons are free to move inside a conductor, but for our purposes we can just say that it has to do with some special properties of the material. However, in order for these special properties of the material to have any effect on the incident light, the light must penetrate into the material to some extent. Exactly the depth that the light will penetrate is dependent upon its frequency, and the conductivity of the material in question, but it is this depth that we call the "skin depth".

The derivation: We assume for x < 0 we are in free space, and for x > 0 there exists a conductor with conductivity σ. Maxwell’s equations on the free-space side look like:

∇ x B = (1/c2)dE/dt
∇ x E = -dB/dt
∇ • B = 0
∇ • E = 0
And on the conductor side, they look like:
∇ x B = μJ
∇ x E = -dB/dt
∇ • B = 0
∇ • E = 0
We are implicitly assuming that the dE/dt term on the right-hand side of the equation for ∇ x B is much smaller than the induced current μJ, since this term goes like 1/c2. This is known as the quasistatic approximation. It is valid for low frequencies and/or high conductivities, since the current is an induced current (meaning it is larger for high conductivities), and saying that the speed of light is fast is equivalent to saying that everything else in our system is slow (hence low frequencies).

The current is induced by the electric field via Ohm’s law, J = σE, meaning:

∇ x B = μσE

The simplest way to solve this is to introduce a vector potential A(x,t):

B = ∇ x A
E = -dA/dt.

And we are in Coulomb Gauge, ∇ • A = 0.

Then the equations for ∇ • B, ∇ x E, and ∇ • E are automatically satisfied, and we now have:

∇ x (∇ x A) = -(1/c2)d2A/dt2,        x < 0
∇ x (∇ x A) = - μσdA/dt,                x > 0

Where ∇ x (∇ x A) = ∇(∇ • A) – ∇2A = -∇2A, i.e.

2A = (1/c2)d2A/dt2,           x < 0
2A = μσdA/dt,                    x > 0

Plugging in the steady-state wave solution A = A0ei(kx - ωt), we get:

k2 = ω2/c2,     x < 0
k2 = iμσω       x > 0

So, in free space we get our usual relationship between wave vector and frequency, but within the conductor, the wave vector has both real and imaginary parts:

k = (1/√2)(i + 1)√(μσω) = (1 + i)/δ

The imaginary part of k results in exponential damping that looks like:

A0ei(kx - ωt) = e-x/δ • A0ei(x/δ - ωt)

δ is the skin depth, and is simply the inverse of the imaginary part of k given in the equation above:

δ = √(2/μσω).

For example, in copper, μ = μ0 = 4π x10-7 to within something like five orders of magnitude. σ = 5.95 x 107 Ω-1m-1, and ω = 2πf.

Thus, δ = .0652 /√f (in meters).

For a frequency of 1200 THz (I think that's probably yellow light), the skin depth of copper would be 1.88 nm (1.88 x 10-9 meters).

The end: So, we see that for most common materials and frequencies, the skin depth is quite small. Given this, you might wonder why anyone would care what the skin depth of a material might be. Well, if you were a manufacturer that wanted to design the cheapest mirror possible, you might want to know what is the minimum amount of conducting material you need to use for the back of the mirror. If you're a young theoretical physicist getting his or her first taste of electromagnetic theory, you might enjoy the fact that this derivation made such handy use of imaginary numbers. As cereed pointed out, it's useful for submarine captains who need to know how close they need to get to the ocean's surface to receive radio waves. Apparently for this reason, the US navy uses extremely low frequency (ELF) radio waves for submarine communication (lower frequencies give higher skin depth, by the equation above).

If anyone else can think of applications of skin depth, please let me know.

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