Magnetic Field due to a Segment of Current

For example, to calculate the field in a rectangular current loop, find the field due to each of the four segments and add. It can be derived from the Biot-Savart law, and the result is

B_segment = (m _o/4p )(I/R)(cosa +cosb )

where the R is the ^ distance to the segment, and the angles are shown below.

at the center of a square loop for example, a and b are each 45^o, and B becomes

B = (2m _o/p 2^1/2)I/R.

What about just one plain segment? How do you get current to one end and away from the other end without contributing to the field due to the segment? That's easy, just have those two currents travel exactly toward and away from the point where the field is to be calculated.

What about a very long segment? The field is the familiar long straight wire formula,
B = m_oI/2pR. Check this, with a = b = 0, so cosa=cosb=1.

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