# Why does the flux rule work in all cases?

• lugita15
In summary, the flux rule states that the emf induced in any loop is given by the rate of change of magnetic flux through that loop.

#### 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.
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."

Very simple,
It is a general equation valid in all situations-Thia means there should be a general principle used for DERIVING this equation...INDEED IT IS THE CONSERVATION OF ENERGY(given by lenz's law) used...LETS SEE HOW.

We are interested to find INDUCED EMF,V...

Now V= work done on unit charge by magnetic force...

magnetic force =q(v.B.sinθ)

now , work done = F.dy = q(V.B.sinθ).dy

V= [q(v.B.sinθ).dy]/q

so,V=v.B.dy.sinθ.cosØ

now v=dx/dt

so,V=dx.dy.B.sinθ.cosØ/dt

V= -B.dA/dt = B.dA.(-cosω)/dt ... where [-cosω = sinθ.cosØ]

So.Mr.Feynman has said right about "DEEP UNDERLYNG PRINCIPLE"

i hope u will like it...and apply in all your situations undoubtfuly because it is derived from GENERAL LAW OF CONSERVATION OF ENERGY...

mr.survive said:
Very simple,
It is a general equation valid in all situations-Thia means there should be a general principle used for DERIVING this equation...INDEED IT IS THE CONSERVATION OF ENERGY(given by lenz's law) used...LETS SEE HOW.

We are interested to find INDUCED EMF,V...

Now V= work done on unit charge by magnetic force...

magnetic force =q(v.B.sinθ)

now , work done = F.dy = q(V.B.sinθ).dy

V= [q(v.B.sinθ).dy]/q

so,V=v.B.dy.sinθ.cosØ

now v=dx/dt

so,V=dx.dy.B.sinθ.cosØ/dt

V= -B.dA/dt = B.dA.(-cosω)/dt ... where [-cosω = sinθ.cosØ]

So.Mr.Feynman has said right about "DEEP UNDERLYNG PRINCIPLE"

i hope u will like it...and apply in all your situations undoubtfuly because it is derived from GENERAL LAW OF CONSERVATION OF ENERGY...

Could you please elaborate on your response? I had some difficulty understanding the equations you wrote. If possible, could you use Latex to write the equations.

Any further help would be greatly appreciated.

well time to bring in old topics
lugita hope u remeber our discussion and ofcourse we didn't reach any possible conclusion in our discussion here

well after lots of discussion finally i and my friends got http://www.mathlinks.ro/viewtopic.php?p=1036135#1036135 astounding( and i mean it) result!
look at the last post of mine over there...if u have anything to add surely do it it's one of the most interesting things that i have seen in Electromagnetics.

Last edited:
pardesi said:
well time to bring in old topics
lugita hope u remeber our discussion and ofcourse we didn't reach any possible conclusion in our discussion here
Yes, that thread is precisely the reason I posted this question.

As to the other link you posted, I will have to examine it later in greater detail.

## 1. What is the Flux Rule?

The Flux Rule, also known as Faraday's Law of Electromagnetic Induction, states that the induced electromotive force (EMF) in a closed loop is equal to the negative of the time rate of change of the magnetic flux through the loop.

## 2. How is the Flux Rule calculated?

The Flux Rule can be calculated by taking the negative derivative of the magnetic flux with respect to time. This can be expressed as EMF = -dΦ/dt, where EMF is the induced electromotive force and Φ is the magnetic flux.

## 3. What is the significance of the Flux Rule?

The Flux Rule is significant because it helps explain the relationship between changing magnetic fields and induced currents. It is a fundamental principle in electromagnetism and has many practical applications, such as in generators and transformers.

## 4. Can the Flux Rule be applied to all types of electromagnetic flux?

Yes, the Flux Rule can be applied to all types of electromagnetic flux, including magnetic flux and electric flux. It is a general rule that governs the relationship between changes in flux and induced EMF.

## 5. What are some real-world examples of the Flux Rule in action?

The Flux Rule can be observed in many everyday devices, such as generators, motors, and transformers. It is also used in technologies like wireless charging and electromagnetic induction stovetops. The Flux Rule also plays a crucial role in understanding the behavior of electromagnetic waves and their interaction with matter.