Faraday Paradox - An Additional Rule

An Additional Rule

Now that it has been proven that the magnetic field rotates with the magnet as discussed in the observation section, what is really causing the paradox? Before addressing this question, we will discuss when Faraday's Law is valid and when it breaks down as in the disk experiment. In the case when the disk alone spins there is no change in flux through the circuit, however, there is an electromotive force induced contrary to Faraday's law. We can also show an example when there is a change in flux, but no induced voltage. Figure 5 (near right) shows the setup used in Tilley's experiment. It is a circuit with two loops or meshes. There is a galvanometer connected in the righthand loop, a magnet in the center of the lefthand loop, a switch in the lefthand loop, and a switch between the loops. We start with the switch on the left open and that on the right closed. When the switch on the left is closed and the switch on the right is open there is no change in the field of the magnet, but there is a change in the area of the galvanometer circuit. This means that there is a change in flux. However the galvanometer did not deflect meaning there was no induced voltage, and Faraday's law does not work in this case. According to A. G. Kelly this suggests that an induced voltage in Faraday's experiment is due to the "cutting" of the circuit by the flux lines, and not by "flux linking" or the actual change in flux. This follows from the Tilley experiment because there is no movement of the lines of force across the circuit and therefore no current induced although there is a change in flux through the circuit. Nussbaum suggests that for Faraday's law to be valid work must be done in producing the change in flux.
To understand this idea, we will step through the argument given by Nussbaum. We start by calculating the force between two current carrying wires. The force on wire 1 due to wire 2 is given by:

The magnetic field from the second wire is given by:

So we can rewrite the force on wire 1 as:

Now consider a segment of a conductor displaced in a constant magnetic field. The work done is found from:

If we plug in what we previously found for we get:

The area covered by the displacement of the conductor is:

Therefore:

The differential work can also be given in terms of charge and potential difference V:

By setting the two equations for differential work equal to each other we arrive at Faraday's Law.

Furthermore, we now see that this is only true if is nonvanishing. Meaning, Faraday's Law is only valid if work is performed in bringing about the change in flux.