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According to Wikipedia, Faraday's law leads to an experimental situation called the Faraday paradox:

An often overlooked fact is that Faraday's law is based on the total derivative, not the partial derivative, of the magnetic flux. This means that an EMF may be generated even if total flux through the surface is constant.

The formula is

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

where $\mathcal{E}$ is the EMF (electro-motive force as we called it in university).

I do not understand how there can be a difference between partial and total differentiation. Does this difference only exist in cases with multiple variables of $\Phi$?

Furthermore, and more importantly, how can this derivative be equal to $0$ - and how not if it was partial?

Alfred Centauri
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Kutsubato
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The term 'electromagnetic induction' covers two rather different ways in which an emf (electromotive force, not electromagnetic force) can arise.

The first is by means of the magnetic Lorentz force that acts on charge-carriers in a conductor when the conductor is moved so that it cuts lines of flux. The emf in a short length $\Delta \mathscr l$ moving with velocity $\vec v$ is $$\mathscr{E}=(\vec{v} \times \vec{B}).\vec{\Delta \mathscr l}.$$

This is the fundamental equation for emf in a conductor due to its motion through a magnetic field. It works even in the 'paradoxical' cases referred to in the Wikipedia article. However, it's easy to show that, for a circuit consisting of a loop of thin wire which moves or changes shape in a magnetic field, the equation (applied all round the loop) leads simply to a total emf of $$\mathscr{E}=-\frac {d\Phi}{dt}\ \ \ \ \ \ \ \ [\Phi = \int\int \vec{B}.\vec{dA}]$$ Note that this is not partial differentiation. We're not holding the position or shape of the loop constant as time goes on.

The second type of electromagnetic induction occurs when (in our frame of reference) we have a stationary loop or circuit, through which there is a changing magnetic flux. The relevant fundamental law in this case is the Maxwell-Faraday equation $$\vec{\nabla} \times \vec{E} =-\frac{\partial \vec{B}}{\partial t}$$ Integrating this over the area of the loop (details omitted) gives $$\mathscr{E}=-\frac {\partial\Phi}{\partial t}.$$ The partial differentiation is appropriate to this type of e-m induction, which is due to a magnetic field that changes with time, not to movement in space.

The question now arises: suppose that the loop is moving or changing shape at the same time that the magnetic field is changing with time. The emfs due to the two processes can be added together. How would we represent this mathematically? I'm not sure whether or not there's a simple notation...

Philip Wood
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  • So what you say is, you can consider two effects of induction, the Lorentz formula (not as a force $F$ but as an EMF $\mathcal{E}$) and the Max.-Far. law which both lead to a similar fomula, $\mathcal{E}=-\frac{d\Phi}{dt}$ and $\mathcal{E}=-\frac{\partial \Phi}{\partial t}$, only differing in the kind of differentiation?

    I have to say, that is very interesting. I knew about the second fomula and Gauss's law but I have never seen the step from "Lorentz to $\Phi$"! Can you link an article showing the proof for that?

    Thanks!

     – Kutsubato Jan 01 '18 at 17:30
  • Sorry – I can't understand your comment. – Philip Wood Jan 01 '18 at 17:30
  • Now? I edited it. – Kutsubato Jan 01 '18 at 17:31
  • Yes – sorry about that. I'm not saying that the emf is equal to the Lorentz force; it's equal to that force dotted with a vector lying along the conductor, of length equal to the bit of conductor in which we want the emf. The step you want is in the Wiki article on e-m induction. There's a box in the article which it tells you to click on if you want details of the derivation. Good luck! – Philip Wood Jan 01 '18 at 17:42