Biot-Savart
law and Ampere's law
(See the bottom of the page if you just want the field due to a current segment. The derivation of it requires calculus, but you don't need calculus to use it.)
These laws deal with the magnetic field caused by a current and the distance
from the current, so they must really be the same law in different disguises. Biot-Savart is the differential version and Ampere's is the
integral version. We will show an example in which the result is the same. By
the way, in Biot and Savart,
the t's are silent. But I suppose you pronounce the
final t in
Consider an infinitely long wire with current going toward the right in the
sketch below. At distance R above the wire, the magnetic field B will be toward
you, and Ampere's law says that
B(2p R) = m oI, because you can
circle around the wire in a path with radius R, so B is constant on that path,
and the path length is 2p R. Now let us
find B by Biot-Savart. From an element of length dx at slant distance r,
dB = (m o/4p )Idxsinq /r2, where q is
the angle between dx and r. This is the Biot-Savart law.
But x = -Rcotq , so dx = Rcscq dq .
Also, r = Rcscq . Make these substitutions and your
only variable is q ,
and fortunately the csc drops out and you just have
the integral of sinq which is -cosq . The limits are zero and p , so the integral
of the sinq is 2, and we end up with
B = m oI/2p R, which agrees with the Ampere's law
equation.
x……dl
Note that if we wanted the field due to a segment, this would change the limits and we would have the difference of two cosines. One of them is negative so we can redefine the angles to make it the sum of two cosines:
B = (m o I/4p R)(cosa + cosb )
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