Ampere's Law allows us to easily calculate magnetic fields in highly symmetric situations, much as Gauss' Law allowed us to determine electric fields.

Ampere's Law states that the line integral of B · dl around a closed (i.e., complete) loop is proportional to the current passing through the loop:

Around a closed loop ò B · dl = m_o I_enc

Things to keep in mind when applying Ampere's Law:

• You should have a good idea what the field looks like so you can draw an appropriate loop.
  
  
• Choose your loop so that, along all or part of the loop, the magnitude of the magnetic field is constant and the field is parallel to the loop. This turns B · dl into B dl and allows you to bring B out of the integral.
  
  
• Choose your loop so that, along all or part of the loop, the field is perpendicular to the loop. This makes B · dl equal to zero, so you don't have to worry about that part of the integral.
  
  
• Remember the right side of the equation. In other words, identify how much current passes through the loop.

Let's use Ampere's Law to find the magnetic field from a long straight wire.

What does the field look like? The field lines are circular loops centered on the wire.

To find the field a distance r from the wire, use a loop of radius r centered on the wire. The enclosed current is I directed out of the page, producing a counter-clockwise field. Carry out the integral in a counter-clockwise direction so the dot product will be positive.

Because the field is the same magnitude at all points on the loop, and the field is tangent to the loop everywhere:

ò B · dl = B ò dl = m_o I

ò dl is the length of the loop, which is 2pr.

This gives:

B

=

mo I 2pr