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## Main Question or Discussion Point

Consider a copper disk rotating with constant angular velocity, as shown in the figure. A bar magnet is directed normal to the surface of the disk, as shown in the following figure:

If a galvanometer is used to measure the induced current in the outer rim of the disk, a nonzero induced current is detected. Using the differential form of Faraday's law and the Lorentz force law, this is easily explainable. Consider a small charge dq on the outer rim of the disk. At any given time, it has a nonzero velocity because of the rotation of the disk. For this reason, the bar magnet exerts a magnetic force on the charge. This makes the charge move differently than the disk itself. Some of these charge will go through the galvanometer, and thus the galvanometer will indicate the existence of a current.

But wait a minute. Let's try applying the integral form of Faraday's Law: the induced emf along a closed loop is equal to the rate of change of the magnetic flux through the loop. But the magnetic flux through surface of the copper disk is constant, and thus the rate of change of magnetic flux is zero. This is because both the magnetic field and the area are both constant. So we have a strange situation in which the induced emf is nonzero even though the rate of change of magnetic flux is zero.

Feynman gives this example in the Feynman Lectures on Physics Volume 2 Chapter 17-2. He states that you shouldn't worry about this, and that when the integral form or "flux rule" doesn't seem to work, you should just "go back to the fundamental equations," the Lorentz force law and the differential form of Faraday's Law as I did above. But what's wrong with using the integral form in this case?

If a galvanometer is used to measure the induced current in the outer rim of the disk, a nonzero induced current is detected. Using the differential form of Faraday's law and the Lorentz force law, this is easily explainable. Consider a small charge dq on the outer rim of the disk. At any given time, it has a nonzero velocity because of the rotation of the disk. For this reason, the bar magnet exerts a magnetic force on the charge. This makes the charge move differently than the disk itself. Some of these charge will go through the galvanometer, and thus the galvanometer will indicate the existence of a current.

But wait a minute. Let's try applying the integral form of Faraday's Law: the induced emf along a closed loop is equal to the rate of change of the magnetic flux through the loop. But the magnetic flux through surface of the copper disk is constant, and thus the rate of change of magnetic flux is zero. This is because both the magnetic field and the area are both constant. So we have a strange situation in which the induced emf is nonzero even though the rate of change of magnetic flux is zero.

Feynman gives this example in the Feynman Lectures on Physics Volume 2 Chapter 17-2. He states that you shouldn't worry about this, and that when the integral form or "flux rule" doesn't seem to work, you should just "go back to the fundamental equations," the Lorentz force law and the differential form of Faraday's Law as I did above. But what's wrong with using the integral form in this case?