Proving the consistency of Faraday's law of electromagnetic induction

Question

Here is a question which frequently occurs on my school exam paper:

"Prove that Faraday's law of electromagnetic induction is consistent with the law of principle of conservation of energy." What does this actually mean? Any help would be better...

P.S. I apologize if the question is too elementary.

Answer 1

Look at Faraday's law $$\oint \vec E\cdot \vec {dl} = - \frac d{dt}\int \vec B.\vec {dA}$$

We use the right Hand Rule to get the correct directions of the path and surface $\vec E$ and $\vec B$ are integrated over respectively.

Now suppose $\vec B$ points into the screen and increases over time. Using the Right Hand Rule, we integrate $\vec E$ around the path clockwise to to get an induced emf of V volts, say. If there was no minus sign in Faraday's Law, this would imply the induced current would be driven around the loop in a clockwise direction, and the direction of its changing magnetic field would add to the original changing magnetic field, producing a postive feedback effect and the creation of energy from nothing.

So the minus sign ensures the induced emf works against the change causing it, summarized in Lenz's law, and that energy is conserved.

Answer 2

OP wrote(v3):

Prove that Faraday's law of electromagnetic induction is consistent with the [...] principle of conservation of energy.

Since Faraday's law is part of Maxwell's equations, one may generalize OP's question as:

In the context of classical electromagnetism, prove that Maxwell's equations are consistent with the principle of conservation of energy.

In the case of Maxwell's equations, one can use Poynting's theorem (which follows from Maxwell's equations) to argue that there is a local energy balance for the electromagnetic field. Quoting Poynting's theorem in words from Wikipedia:

The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.