Question about the Flux Rule

[lugita15]

The flux rule states that the emf induced in any loop is given by the rate of change of magnetic flux through that loop. In other words, $$\epsilon=-\frac{\partial}{\partial t}\int\int\vec{B}\cdot d\vec{A}.$$ The thing that troubles me is the coincidence that the flux rule works in ALL cases. If a loop rotates at a constant angular velocity in the presence of a uniform constant magnetic field, the emf in the loop is given by the flux rule. If the magnitude of a uniform magnetic field perpendicular to a loop increases at a constant rate, the emf is still given by the flux rule. What is the connection between these four cases, that the flux rule works in all four?:
1. The circuit is stationary, and the magnetic flux increases (or decreases) due to a region of magnetic field moving into (or out of) the the area of the circuit.
2. The region of the magnetic field is stationary, and the magnetic flux increases (or decreases) due to (part of) the circuit moving moving into (or out of) a region of magnetic field.
3. There is a constant, uniform magnetic field, and the magnetic flux is sinusoidal due to the loop rotating with constant angular velocity.
4. The circuit is stationary, and the magnetic flux increases due to increase in the magnitude of the uniform magnetic field directed perpendicular to the circuit.
In all cases, emf=-(time derivative of magnetic flux).
Einstein famously proved in his "On the Electrodynamics of Moving Bodies," his first paper on the special theory of relativity, that due to the Lorentz transformations of the electric and magnetic fields, case 1 and case 2 are equivalent. How are case 3 and case 4 related to the each other, and to the first two cases?

Any help would be greatly appreciated.
Thank You in Advance.
P.S. One of the reasons I am asking this is that Feynman raises similar questions in his Lectures on Physics. He says in Volume II Page 17-2:
"We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena. Usually such a beautiful generalization is found to stem from a single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication."

 

[Vailanor]

The answer to your question lies in the fact that all four cases are based on Faraday's Law of Induction. This law states that the electromotive force (emf) induced in an electric circuit is equal to the rate of change of magnetic flux linked with the circuit. This is why the flux rule works in all four cases: they all involve a changing magnetic flux. In the first two cases, the changing magnetic flux is due to either a region of magnetic field moving into or out of the area of the circuit (case 1), or (part of) the circuit moving into or out of a region of magnetic field (case 2). In the third case, the changing magnetic flux is due to the loop rotating with a constant angular velocity in the presence of a uniform constant magnetic field. In the fourth case, the changing magnetic flux is due to an increase in the magnitude of the uniform magnetic field directed perpendicular to the circuit. So even though each case may seem unrelated, they are all related by Faraday's Law of Induction. This is why the flux rule works in all four cases.

 

[astrokreng]

the concept of the flux rule may seem coincidental at first glance, but upon further analysis, it can be understood as a fundamental principle in electromagnetism. The key connection between the four cases is the concept of magnetic flux, which is defined as the measure of the number of magnetic field lines passing through a given area. This flux can change in different situations, but the flux rule remains constant.

In cases 1 and 2, the change in magnetic flux is due to either the circuit or the magnetic field moving relative to each other. This is where Einstein's theory of relativity comes into play, as it explains how the observed flux change can differ depending on the observer's frame of reference. However, the flux rule remains the same in both cases, showing its universality.

In cases 3 and 4, the change in magnetic flux is due to the rotation of the circuit or the increase in the magnitude of the magnetic field. These may seem like different scenarios, but they both involve a change in the magnetic flux through the circuit. The flux rule, again, remains consistent in both cases.

Therefore, the connection between all four cases is that they all involve a change in magnetic flux, which is directly related to the induced emf through the flux rule. This principle is not coincidental, but rather a fundamental law that governs electromagnetic phenomena. As for Feynman's statement, it is important to remember that the simplicity and accuracy of the flux rule make it a beautiful generalization, even if it requires an analysis in terms of two different phenomena.