# Question about the Flux Rule

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.