Skin effect
Formula
Formula
-------

The AC current density  in a conductor [decreases exponentially](/wiki/Exponential_decay "Exponential decay") from its value at the surface 
according to the depth  from the surface, as follows:
J \= J\_\\mathrm{S} \\,e^{\-{(1\+j)d/\\delta 
where \\delta is called the *skin depth* which is defined as the depth below the surface of the conductor at which the current density has fallen to 1/[e](/wiki/E_%28mathematical_constant%29 "E (mathematical constant)") (about 0\.37\) of S. The imaginary part of the exponent indicates that the phase of the current density is [delayed](/wiki/Phase_delay "Phase delay") 1 radian for each skin depth of penetration. One full [wavelength](/wiki/Wavelength "Wavelength") in the conductor requires 2 skin depths, at which point the current density is attenuated to e−2 (1\.87×, or −54\.6 dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in [vacuum](/wiki/Vacuum "Vacuum"), or equivalently, the [phase velocity](/wiki/Phase_velocity "Phase velocity") in a conductor is very much slower than the [speed of light](/wiki/Speed_of_light "Speed of light") in a vacuum. For example, a 1 MHz radio wave has a wavelength in vacuum  of about 300 m, whereas in copper, the wavelength is reduced to only about 0\.5 mm with a phase velocity of only about 500 m/s. As a consequence of [Snell's law](/wiki/Snell%27s_law "Snell's law") and this very tiny phase velocity in a conductor, any wave entering a conductor, even at grazing incidence, refracts essentially in the direction perpendicular to the conductor's surface.

The general formula for skin depth when there is no dielectric or magnetic loss is:The formula as shown is algebraically equivalent to the formula found on page 130 
\\delta\= \\sqrt{ \\frac{\\, 2\\rho \\,}{\\omega\\mu }
\\left (\\sqrt{1 \+ \\left({\\rho\\omega\\varepsilon}\\right)^2 \\,}
\+ \\rho\\omega\\varepsilon \\right) \\,}

where

At frequencies much below 1/(\\rho \\varepsilon) the quantity inside the large radical is close to unity and the formula is more usually given as:
\\delta\=\\sqrt{\\frac{\\, 2\\rho \\,}{\\omega\\mu} \\,} \~.

This formula is valid at frequencies away from strong atomic or molecular resonances (where \\varepsilon would have a large imaginary part) and at frequencies that are much below both the material's [plasma frequency](/wiki/Plasma_frequency "Plasma frequency") (dependent on the density of free electrons in the material) and the reciprocal of the mean time between collisions involving the conduction electrons. In good conductors such as metals all of those conditions are ensured at least up to microwave frequencies, justifying this formula's validity.Note that the above equation for the current density inside the conductor as a function of depth applies to cases where the usual approximation for skin depth holds. In the extreme cases where it doesn't, the exponential decrease with respect to skin depth still applies to the *magnitude* of the induced currents, however the imaginary part of the exponent in that equation, and thus the phase velocity inside the material, are altered with respect to that equation. For example, in the case of copper, this would be true for frequencies much below .

However, in very poor conductors, at sufficiently high frequencies, the factor under the large radical increases. At frequencies much higher than 1/(\\rho \\varepsilon) it can be shown that skin depth, rather than continuing to decrease, approaches an asymptotic value:
\\delta \\approx {2 \\rho} \\sqrt{\\frac{\\, \\varepsilon \\,}{ \\mu }\\,} \~.

This departure from the usual formula only applies for materials of rather low conductivity and at frequencies where the vacuum wavelength is not much larger than the skin depth itself. For instance, bulk silicon (undoped) is a poor conductor and has a skin depth of about 40 meters at 100 kHz ( \= 3 km). However, as the frequency is increased well into the megahertz range, its skin depth never falls below the asymptotic value of 11 meters. The conclusion is that in poor solid conductors, such as undoped silicon, skin effect does not need to be taken into account in most practical situations: Any current is equally distributed throughout the material's cross\-section, regardless of its frequency.


Are there any materials that exhibit negative skin depth?