Understanding Griffiths' Flux Rule Derivation for Motional EMF

[zezima1]

Hi, the attached picture should show Griffiths derivation of the flux rule for motional emf. To refresh your memory it says that minus the rate of change of the magnetic flux equals the emf:

-d$\Phi$/dt = $\epsilon$

Now, I need some help in understanding the attached derivation. Griffiths starts by saying that the flux rule actually works for arbitrarily shaped current loops in non-uniform magnetic fields and even if the surface of the loop changes shape(*). I am however unsure in which of these he actually sets out to prove. Since he calculates the flux for a surface S and the same surface deformed, I would say he tries to prove (*) - is that correct?
And if so, are the two other properties also proven or are they simply something left for the rigorous mathematicians?

 

[eljose79]

Hello! Thank you for sharing this interesting topic. I am happy to help you understand Griffiths' derivation of the flux rule for motional emf.

Firstly, you are correct in saying that Griffiths is trying to prove the property (*) - that the flux rule works for arbitrarily shaped current loops in non-uniform magnetic fields and even if the surface of the loop changes shape. He starts by considering a current loop with a surface S, and then deforming the surface into another shape while keeping the current constant. By doing this, he shows that the flux through the loop remains the same, and thus the flux rule holds true for any shape.

As for the other two properties, they are not explicitly proven in Griffiths' derivation. The second property, that the flux rule holds true for non-moving loops in changing magnetic fields, can be easily proven using Faraday's law. The third property, that the flux rule holds true for moving loops in uniform magnetic fields, is a special case of the first property and can also be proven using Faraday's law.

However, it is important to note that while Griffiths' derivation is not a rigorous mathematical proof, it provides a clear and intuitive explanation for the flux rule and its properties. It also highlights the importance of understanding the concept of flux and its relationship with emf.

I hope this helps in your understanding of Griffiths' derivation. If you have any further questions, please do not hesitate to ask. Keep up the scientific curiosity!