Meanwhile I will sum-up our discussion about "1V AC box + resistor":

While measuring 1V AC voltage coming out of the box with voltmeter/scope, we cannot discern - source is transformer, piezo device or just AC generator powered by chemical battery (we agreed that electrons are the same long ago in this thread). Kirchoff's circuit law (KVL) holds for AC voltage source + resistor *only* when AC voltage is generated by anything but transformer or other kind contraption ruled by Faraday's law. When we have transformer in a box, KVL does not hold anymore. To know - KVL holds or not, we have to look inside the box, otherwise we may be mistaken. This is what you are claiming? Anybody else agreeing?

Let's try to explain. In the "circuit" below V

_{1} is whatever while V

_{2} is literally zero. Why? Because the net flux in the area bounded by the path determined by the terminals of V

_{2} is zero. So this is not a lumped circuit. Simply encasing this loop of wire that is under a varying magnetic field in a box won't lump it.

If you analyze other Maxwell's equations, there is one that shows that you can't have a magnetic monopole. If

**B** is going into the screen, where does it come out? The answer: elsewhere. How could Lewin then create a magnetic monopole when this is not possible? The answer: he showed it a little earlier in the same video where he relegated Kirchhoff to the birds. The answer: solenoids. Solenoids generate a strong magnetic field inside their cores, but very weak outside. In practice we can consider it zero. This happens because outside of a solenoid the lines of magnetic field are very sparse, while in the core they're highly concentrated.

Therefore we need to find a way to lump our loop, so that it can be considered the secondary of a transformer. To do that, we need to gather all the return lines of

**B** and make them cross the area between the loop and V

_{2}, like below. Now, bingo! V

_{1} = V

_{2}. Now we can box up the loop with the returning lines of

**B**.

Another way of lumping our loop is to make its area perpendicular to the area "seen" by V

_{2}. In that case the scalar product

**B ยท** d

**A** will be zero, once scalar products can be thought as the "shadow" one vector casts on another. If they are perpendicular, the "shadow" is zero.

You'll recognize these practical arrangements on real inductors. In the picture below, you have a solenoid (a). The area defined by the voltmeter is roughly perpendicular to the area of the solenoid. If you measure like in (b), the voltage will be the real voltage minus one turn. If N is not much grater than one, you can use a toroid as in (c) or other core with a closed magnetic path.