Ampere's Law allows us to easily calculate magnetic fields in highly symmetric situations.

The Ampere's Law process:

• Draw a closed loop of any shape.
• Walk around the loop once. For each step you take, multiply the length of your step by the component of the magnetic field that is parallel to your step.
• Add up all those products around the entire loop.
• That sum is proportional to the net current passing through the loop.

Expressing Ampere's Law as an equation:

Around a closed loop SB_|| DL = m_o I_enc

What does the field from a long straight wire look like?

The field lines are circular loops centered on the wire.

What shape should we choose for our amperian loop?

1. a circle
2. a square
3. a triangle
4. it doesn't matter as long as the wire is inside the loop

In theory Ampere's Law can be applied for any loop, but the one that makes the calculation easy in this case is a circular loop. 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 sum in a counter-clockwise direction so all the terms will be positive.

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

SB_|| DL = B S DL = B 2pr = m_o I

B

=

mo I 2pr