Does Ampere's law (for steady currents) somehow assume
the magnetic field of an infinitely long line of current as opposed to
just a current element? Specifically, the magnetic field produced by just
one infinitesimal current element alone (with no other current elements
present in space) drops off in strength as 1/r^2, so it seems
incompatible with Ampere's law if one chooses certain paths in a plane
perpendicular to the current element. Why is this so?
No. Ampere's law relates the current passing through an area to the
line integral of the magnetic field over the perimeter of the area.
Sure, the field drops off as 1/r^2, but the distance over which the
line integral is taken increases as r^2. So it doesn't matter where
you draw the perimeter. You'll always get the same answer if the
current is completely bounded by the perimeter.
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Update: June 2012