I was just wondering why Ampere's Law can't be applied
to a wire of finite length and still have the integral come out to be
simple enough to be useful.
For an infinitely long wire, we can assume that whatever "powers" the wire
is far enough away that it will not affect the magnetic field where we are
working. We can assume circular symmetry of the magnetic field.
For a finite length of wire, the magnetic field is affected by currents
entering and exiting the wire, by the current flowing through the power
source. We can no longer assume that the magnetic field is symmetric. When
doing the integral, the magnetic field can no longer be treated as a
constant. It may be a little stronger on one side than on the other. It
may not form perfectly circular field lines. As a result, the integral is
no longer simple.
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Update: June 2012