The EM Wave in a Conductor calculator helps describe the complex wavenumber of an EM wave traveling through a conducting medium.

EM waves in conductors attenuate (the amplitude slowly decreases). We can understand this qualitatively by recognizing that the better conductor is, the more charges (electrons) it has available to move.  As the wave travels through the conductor, the electric field exerts a force on the charges. In an insulator, these charges don't move anywhere, so no work is done. In a conductor, the charges move and a force exerted over a distance means that work is done. Therefore, the wave must be losing energy, this means that the wave is attenuating.

Plane waves in conducting media have complex wave numbers¹:

• `tilde k = k + ikappa`

Where:

• `k = omega sqrt((epsilon mu) /2) [sqrt(1 + (sigma/(epsilon omega))) + 1]^(1/2)`
• `kappa =  omega sqrt((epsilon mu) /2) [sqrt(1 + (sigma/(epsilon omega))) - 1]^(1/2)`

`kappa` describes how quickly the wave attenuates, how quickly its amplitude shrinks.  Higher `kappa` means the wave dies off faster. Lower, lasts longer.
  A useful measure of how quickly a wave attenuates is its skin depth² (`d`), the distance it can travel before its amplitude is reduced by a factor of e. 

• `d = 1/kappa`

For some more information on 'd', please see this page.

Parameters

• `sigma` is the conductivity of the conductor (how good of a conductor it is).  Conductivity is generally measured in siemens (S), which is equivalent to `Omega^-1`.
• `omega` is the angular frequency of the wave.  Note that angular frequency (rad/s) is not the same as frequency (Hz).  If you have the frequency (`nu`), then you can find the angular frequency (`omega`) by `omega = 2 pi nu`.
• `epsilon` is the permittivity of the conductor.  `epsilon` is assumed to have the same units as `epsilon_0`, the constant for the permittivity in vacuum.
• `mu` is the permeability of the conductor.  `mu` is assumed to have the same units as `mu_0`, the constant for the permeability in vacuum.

How to use the calculator

• ˜k gives an answer in the form of κ+ικ, but assumes units of rad/m.
• k just gives k, but you should be able to convert the units once vCalc updates the available units for spatial angular frequency (a pretty niche set of units, so it's understandable that they don't have a whole library of options).
• κ returns κ, but you can convert the units (again, once vCalc adds more options).
• d calculates and returns `1/kappa`.
• Ω simply computes angular frequency Ω = 2 π ν. 

If you want different options for spatial angular frequency (rad/m), then use the "contact us" button at the very bottom of this page to request the combination of units.  Examples of things you would request include rad/km or rad/ft or whatever you need.

See Also

Attenuation in Fiber Optic Cables - a different take on essentially the same topic

1. ^ Griffiths, David J. "Electromagnetic Waves in Matter." Introduction to Electrodynamics. 4th ed. N.p.: Prentice Hall, 2013. 413. Print.
2. ^ Griffiths, David J. "Electromagnetic Waves in Matter." Introduction to Electrodynamics. 4th ed. N.p.: Prentice Hall, 2013. 413. Print.