square dipole. This part of the proof can be most easily accom-
plished by the methods of section 11.4. It should, for example, be
plausible in the case illustrated here. The field on the left edge is
stronger than the field on the right, so the overall contribution of
these two edges to the circulation is slightly counterclockwise. How-
ever, the field is not quite perpendicular to the top and bottoms
edges, so they both make small clockwise contributions. The clock-
wise and counterclockwise parts of the circulation end up canceling
each other out. Once Ampere’s law is established for a square sur- face like f/4, it follows that it is true for an irregular surface like f/5, since we can build such a shape out of squares, and the circulations are additive when we paste the surfaces together this way. By pasting a square dipole onto the wire, f/6, like a flag attached to a flagpole, we can cancel out a segment of the wire’s current and create a detour. Ampere’s law is still true because, as shown
in the last step, the square dipole makes zero contribution to the
circulation. We can make as many detours as we like in this manner,
thereby morphing the wire into an arbitrary shape like f/7.
What about a wire like f/8? It doesn’t pierce the Amperian sur- face, so it doesn’t add anything toIthrough, and we need to show that it likewise doesn’t change the circulation. This wire, however, could be built by tiling the half-plane on its right with square dipoles, and we’ve already established that the field of a distant dipole doesn’t contribute to the circulation. (Note that we couldn’t have done this with a wire like f/7, because some of the dipoles would have been right on top of the Amperian surface.)
If Ampere’s law holds for cases like f/7 and f/8, then it holds for any complex bundle of wires, including some that pass through the Amperian surface and some that don’t. But we can build just
about any static current distribution we like using such a bundle of
wires, so it follows that Amp`ere’s law is valid for any static current
distribution.
11.3.3 Maxwell’s equations for static fields
Static electric fieldsdon’tcurl the way magnetic fields do, so we
can state a version of Amp`ere’s law for electric fields, which is that
the circulation of the electric field is zero. Summarizing what we
know so far about static fields, we have
ΦE= 4πkqin
ΦB= 0
ΓE= 0ΓB=4 πk
c^2
Ithrough.This set of equations is the static case of the more general relations
known as Maxwell’s equations. On the left side of each equation, we
Section 11.3 Magnetic fields by Ampere’s law` 703