A student asked me a question that I couldn't satisfactorily answer
Perhaps others don't agree, but I have not seen in this discussion
anything that Wolfgang can tell to his student. Here is my attempt
(which I am sure will require "refinement").
First, we have to be very careful to restrict our discussion to circular
paths. This has to do with the fact that knowing the curl of E is not
enough to completely specify E. But if we stick with circular paths
then we can talk about the special solution in which E is constant in
magnitude and purely tangential to the path. This also has the
advantage of being a field that is (relatively) easy to visualize.
Now we can talk about the split loop. Clearly no current flows, so
there is clearly no electric field inside the metal. But we haven't
made dB/dt vanish---that electric field is still there. So there must
be a _second_ electric field which exactly cancels the first (_inside_
the metal). This second field comes from electric charge on the surface
of the wire. In the first picosecond or so after dB/dt was turned on,
the field in the metal was not zero, so charge moved. Once it got to
the surface it could go no further. So it sits there with it's own
electric field acting to cancel the E field from Faraday's law. In the
steady state, this cancellation is complete.
If you think about it for a while, you realize that you need quite an
elaborate arrangement of charge all along the wire. Cancelling the
field at the ends of the wire is easy---it just requires a uniform
sheet. But all that curvature is hard to deal with.
Now let's open up the gap, wider and wider. We'll be careful keep the
remaining wire circular along the original path and always calculate E
from integrating all the way around that path. With a wider gap, the
wire is shorter. That means there is less of the curved part that is so
difficult, so there needs to be less charge on the surface. This
process continues smoothly until the surface charge vanishes at the same
moment the length of wire goes to zero.
[This argument will be more difficult to make to students if you haven't
been teaching them about surface charges on wires, but then my prejudice
is that you should be!]