The geometry is shown in the figure. Using the Biot-Savart law the magnetic
perturbation at point P due to a current element is:
But l = rcot and therefore dl = -rcosec2d.
Hence we have:
Integration from to yields:
For a very long line current, -> 0, and -> and in the limit we have:
If this line current flows at a height H above the ground then the horizontal and vertical components
of the ground magnetic perturbation vary along a path at right angles to the current as shown in panel A
of the figure. Mathematically, this is given by:
where the magnetic perturbation that is directed at right angles to the current and the
position vector has been resolved parallel and perpendicular to the ground plane.