

How to use Ampere's
Law to calculate B

Determine if there is any symmetry for the current distribution.
If there is any symmetry, determine if it is one of the special cases that
allows us to use Ampere's Law without having to integrate plane, cylinder
or solenoid (or torus). If there isn't one of these three types of
symmetry, then using Ampere's Law will be difficult and you might be better
off using a Biot Savart.

Draw an Amperian loop with the same symmetry as the
magnetic field. Choose the size of the loop such that the loop lies
at the location where you want to determine the magnetic field. With
the proper choice of an Ameprian loop, the magnetic field will be either
be parallel or perpendicular to the dl vector, making the integral doable.

Evaluate the line integral,
,which gives us the total magnetic field along our loop. Look for
ways to simplify the integral by examining each portion of our Amperian
loop separately. That way (with the correct symmetry) the magnetic
field should be constant over each portion of the loop. At this stage
the magnetic field is still unknown, so leave it as simply B

Determine the amount of current enclosed by the Amperian
loop. If the current distribution is continuous you may have to integrate
over the volume or area to find the total current.

Now with Ampere's Law use your line integral and enclosed
current to solve for the magnetic field.

