Hi, I'm looking for a proof of faraday's law. This is the only page which gives a formal proof of it, but I do not understand the transformation from the first to the second step. It someone could help me out, I would appreciate it. Also, if someone could explain the opposite as well: how do you go from maxwell and faraday's differential equation to the actual equation for emf? http://en.wikipedia.org/wiki/Faraday's_law_of_induction#Proof_of_Faraday.27s_law sorry, i posted this in the calculus forum as well. was not sure which to put it in. thanks
The first step is the chain rule so one variable is kept constant while another gets differentiated and vice versa and the two are summed together. Since the surface the integral is integrating over can vary with time, the surface is kept constant and the magnetic field is differentiated to get the first term and then the second term is obtained by differentiating the integral where the surface can vary with time but where the the magnetic field inside the integrand is kept constant. The proof you linked pretty much does that already. It assumes one of the Maxwell's laws and derives an expression for the rate of change of flux. If you wanted to derive the formula for emf you would then proceed to assume Faraday's law for emf and substitute the derived formula for the rate of change of magnetic flux into it to get the formula for emf. The proof you linked does it the over way around so it assumes the law for emf first and substitues it in to get Faraday's law.