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?
 A: 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.
A: 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.
A: 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.
