# Why is emf equal to the rate of change of magnetic flux?

I don't understand how Faraday figured out that the emf induced when a magnet is moved in a coil would be equal to the rate of change of magnetic flux. Yes, I have that formula drilled in my head like any high schooler but only yesterday did I ask myself whether there is any intuitive reason as to why the emf is equal to the rate of change of magnetic flux

$$\varepsilon = - \dfrac {d\phi}{dt}$$

On the LHS you got work done per unit charge across two points. On the RHS you got the magnetic flux derivative with respect to time. How are they intuitively connected?

I don't understand how Faraday figured out that the emf induced when a magnet is moved in a coil would be equal to the rate of change of magnetic flux.
Faraday did many experiments of his own and considered the experiments of others and from that he produced a theory which in its modern form we call Faraday's Law.

Michael Faraday· Discovery of Electromagnetic Induction is a paper which gives you a detailed explanation of what Faraday did and here are a few extracts to give you a flavour of the events which produced the theory.

The discovery of electro-magnetic induction took place through a series of experiments that he conducted from August 29 to November 4, 1831.

Have an iron ring made, iron round and 7/8 inches thick and the ring 6 inches in external diameter. Wound many coils of copper wire round one half, the coils being separated by twine and calico. There were 3 lengths of wire each about 24 feet long and they could be connected as one length or used as separate lengths .... Will call this side of the ring A. On the other side but separated by an interval was wound wire in two pieces together amounting to about 60 feet in length, the direction being as with the former coils; this side call B

Connected the ends of one of the pieces on A side with battery; immediately a sensible effect on the needle. It oscillated and settled at last in original position. On breaking connection of A side with battery again a disturbance of the needle ... Continued the contact of A side with battery but broke and closed alternately contact of B side. No effect at such times on the needle - depends upon the change in the battery side. Hence is no permanent or peculiar state of wire from B but effect due to a wave of electricity caused at moments of breaking and completing contacts at A side.

The [Mathematical] theory needed a new kind of concept, the question of the electromagnetic field, and also the question of lines of force. Faraday introduced these concepts at various times between 1822 and 1836 but they remained unacceptable to the community of physicists during Faraday's time.

The first support of Faraday's field concept came from William Thomson, later Lord Kelvin. Thomson showed that lines of force could be used to explain the mathematical theory of electrostatic action. This approach was brought to completion by James Clark Maxwell in 1855. Maxwell used Faraday's concepts about changing magnetic flux to lay the foundations of a complete theory of electromagnetism.

• Oh so it's an experimental observation, I see Oct 22, 2021 at 10:04

The result follows from Maxwell's equations and Stokes' theorem, though I hesitate to describe this as intuitive. However we can argue that the result makes sense on dimensional grounds.

The unit of the magnetic field is the Tesla, which has units of (Newton × second)/(Coulomb × metre). If we integrate this over an area to get the flux this introduces extra units of $$\textrm{m}^2$$ so we end up with $$\textrm{Nms/C}$$. Differentiate to get $$d\Phi/dt$$ and the units become $$\textrm{Nm/C}$$. Newton metres is the unit of work so this is the work per unit charge, which is of course just how we define the electrical potential.

This follows directly from Maxwell's equations. The Faraday-Maxwell equation states that a changing magnetic field leads to an electric field:

$$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

This electric field causes a voltage difference that we call electromotive force. Integrating the Faraday-Maxwell equation and applying Stokes' theorem gives the form in your question.

It's hard to tie intuition to this equation, if you ask me. Maxwell's equations are ultimately empirical and it takes a leap of faith to accept them, but it should be one of the very few assumptions you should need to make in electromagnetics.

• Faraday figured out his law well before Maxwell's equations were developed, or even the mathematical and physical concepts behind them were developed. Oct 22, 2021 at 10:04