THE MAGNETIC FIELD

Faraday invented the magnetic field B, and defined it to point in the direction a compass needle would point.  Naturally magnetic materials, called ferromagnets, have an intrinsic magnetic field, that turns out to be due to the fact that electrons themselves have an intrinsic magnetic field, just as they have an intrinsic electric field due to their charge, and in some materials, short-range forces align the “magnetic moments” of the individual electrons to produce a significant magnetic field external to the material.

All known sources of a magnetic field are “dipoles.” The magnetic field lines come out of a region called the North pole, and enter a region known as the South pole, but this behavior is deceptive, because if we follow the field lines where they lead, we find that they always form closed loops.

The other way to get a magnetic field is from a current. Surrounding every current is a magnetic field, forming closed loops centered on the current path. There does not seem to be any difference in the field due to a current versus the field due to a natural magnetic material, like a lodestone.

Doing cross products with unit vectors.

The Biot-Savart Law gives the field due to an arbitrarily-shaped wire carrying a current.





Typical simple problems that use the Biot-Savart Law for a quick solution.
The simplest possible Biot-Savart Law example, a single circular loop of wire.



Ampere's Law, the magnetic version of Gauss's Law








Famous Ampere Law example, infinite sheet of current.


Vector cross product animations!

Solutions to some examples from Chs. 29 and 30.
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