I think you miss the main point in all of this.

What 'sees' flux is not the wires

It's the surface of 'wire current loop' what sees it

Therefore there is voltage induced in the coil until flux is changing inside loop surface

And your example clearly shows it

I think you are the one who misses it. What you are referring to is amperian loop and i am well aware of Amper's laws.

And yes, it IS the wires that see the flux, be it single piece of wire or coil made of many turns.

The distinction is subtle. For a sufficiently rapid AC flux, the voltage induced in the surface of the wire causes a current flow which opposes the external magnetic field. Consequently, the magnetic field is attenuated into the depth of the wire -- skin effect. It is this surface which Danny is referring to.

The true underlying mechanisms, and the math which describes them, are very specific, very well known, and very very accurate (indeed perfect, as far as we can tell). Of particular note, when working a problem of this type, one may use a loop integral, which is perfectly equivalent to an area integral, for the loop (perimeter) and area being of the same continuous region. This is Green's theorem.

This may seem surprising -- the loop integral does not actually need to sample the field inside the region. We can construct arbitrary fields very easily which violate this; but as it happens, such fields cannot be magnetic or electric fields! It is a consequence of this property -- that these real fields must be continuous and analytic -- that allow such shortcuts as this to exist.

Also note that the loop must be closed. You can't enclose a region with an open path. That's meaningless. The closed loop path is exactly the path taken by the current flow -- as I said, and TimFox desmonstrates, voltage cannot be measured without forming a loop.

You can't cheat it with quirky physics, either. For example, suppose we put an open wire in an AC field, thus presumably inducing a voltage along its length, then shoot UV light at it, causing it to emit photoelectrons from whatever voltage the surface was at at that instant. Well, two things must happen before we can detect the emitted electrons: one, they must flow

*along some definite path** to the detector; and two, they must also flow in free space through the field, which as an AC field, has electric as well as magnetic components to it, and these act on the path taken (which, if we know the field, we can solve backwards for, of course). The only thing we've done here, is make the loop harder to figure out, than simply connecting wires to the experiment!

*Definite by the rules of quantum mechanical paths: the integral over all possible paths, times the probability of each path.

Tim