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The skin effect equation was derived from Maxwell's equations for the case of a uniform electromagnetic field at low frequency penetrating a semi-infinite conducting slab with a flat face.
The amplitude decay is expressed as Bz(distance into the slab)/Bz(surface) = exp(-distance/skin depth)
The phase change in radians with distance is: Phase(Bz(distance into the slab) - Phase(Bz(surface)) = -distance/skin depth
where the electromagnetic skin depth = sqrt(2/(angular frequency x electrical conductivity x magnetic permeability))
angular frequency is frequency x 2pi
Although this equation is derived from a special case with a flat interface between free space and a semi-infinite slab with a flat face, it is often used as an approximation for other geometries.
TB-FIELD.EXE was used to calculate the change in axial magnetic field Bz as with respect to radial distance into a conducting rod. Results from TB-FIELD.EXE were compared to an estimate using the skin effect equation.
The radius of the rod was 6.35 mm (radius = 1/4" or diameter = 1/2")
Electrical Conductivity was 2.08E6 Siemens/m
Relative Magnetic Permeability = 20
Inner radius of the encircling coil Ri = 7 mm, outer radius of the coil Ro = 8 mm, Coil Length = 20 mm
Input screens for the TB-FIELD.EXE program are shown below:
Within TB-FIELD.EXE, the external tube was given the properties of free space and the inner "tube" was given a very small inner radius (1E-10 m or 1 angstrom). Field calculations were performed at various radial positions within the rod.
Results showing TB-FIELD.EXE calculated Bz amplitude decays as a function of radial distance from the outer surface of the rod is shown in the chart below as compared to the estimation made using the basic skin-effect equation. With higher magnetic permeability and much lower skin depth, a logarithmic scale of the amplitude attenuation was used in this case:
A chart of phase change with respect to distance from the surface of the rod is shown below:
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