Introductory Physics Volume Two

102 Sources of Magnetic Field 5.2
Fact: Ampere’s Law
For any closed curve the line integral of the magnetic field around
the curve is equal to the µ 0 times the net current through the
surface enclosed by the curve.
∮
∫
⃗B · ⃗dl = µ 0 I through = µ 0
⃗J · dA ⃗
Example
It is not at all apparent that Ampere’s law is equivalent to the Biot-
Savart law, but it is. One can prove the Biot-Savart law from Ampere’s
Law and visa versa. While they are mathematically equivalent, they
are useful in different situations. Ampere’s law is useful for abstract
reasoning about fields, and for finding the field strength in highly symmetric
configurations.
It is important when applying Ampere’s law to keep in mind that
the amperian loop does not correspond to anything physical. There
does not need to be anything there in order for the law to work. You
are free to choose the amperian loop to be any shape you like. Of
course, as was the case with applying Gauss’s law, if you want to use
the law to find the field strength you need to pick the loop correctly.
In order to use Ampere’s Law to find the field strength, you need three
things from the loop.
• The loop must pass through the point at which you want to find the
field strength and be parallel to the field at that point.
• The loop must be either parallel or perpendicular to the field at all
points on the loop.
• In all regions where the loop is parallel to the field, the field must
have the same strength.
Ampere’s law can be used to find the field strength a distance r from
a long straight wire. We will take our loop to be a circle that wraps
around the wire and has a radius r.