Magnetic perturbation due to a sheet current.
Magnetic perturbation due to a sheet current.

The geometry is shown in the figure in which the current sheet is made up of many line currents of elemental cross-section. Using the equations for a line current that extends to a finite length yo into and out of the plane of the figure, we have = - = tan-1(r/yo) and the magnetic perturbation at point P due to the elemental line current, dI, is:

But dI = jhdx where j is the current density and h is the height of the thin ionospheric current layer. Also and therefore

Hence we have:

Writing o = tan-1(H/yo) we get:

which upon integration gives:

For the case of an infinite plane current sheet we find:

This is the formula that relates a magnetic perturbation Bx to the equivalent current, jy.

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