Proof of Induction: Induced Voltage in Any Rod or Circuit

[Kurret]

In my physicsbook they show that the induced voltage in a metal rod is e=lvB, and after that show that that can be written as $$e=-\frac{d\phi}{dt}$$

But how do you prove that this actually apply to any rod or circuit, not just to a metal rod moving through a uniform B-field?

 

[cepheid]

I think the short answer is : Maxwell's equations, more specifically the one particular Maxwell equation known as Faraday's Law of Induction

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

 

[kkrizka]

Try drawing your ciruit out as a rod sliding on two railings, which are connected at one end. Then the area to be used in your flux is the area enclosed by this new circuit.