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.
We have Faraday's law $$\oint \vec E\cdot \vec {dl} = - \frac d{dt}\int \vec B.\vec {dA}$$
On the LHS is a line integral that can be evaluated clockwise or anticlockwise; on the RHS a surface integral that has a choice of two normals to the differential surface $\vec {dA}$. By convention, we use the right Hand Rule for choosing a direction of the normal we can all agree upon that is consistent for a chosen direction of the line integral. So if we choose the LHS to be evaluated clockwise, the surface normals on the RHS will point into the screen.
Now suppose $\vec B$ points into the screen and increases over time. To keep things straightforward, we choose the normals to the surface pointing most closely in the same direction as the magnetic field there, so that upon evaluating the surface integral we'll end up with a positive number. Because of the minus sign in front of the surface integral, we'll therefore get a negative number on the RHS.
We now know that the LHS must also be negative to match the RHS, and the line integral is evaluated clockwise. Therefore the electric field $\vec E$ is generally pointing in the opposite direction to $\vec dl$ in an anti-clockwise sense which drives an induced current in an anticlockwise direction, generating a magnetic field that's in the opposite direction to the changing magnetic field that caused it.
So the negative sign in Faraday's law together with the right hand rule convention is consistent with what is physically observed, including the conservation of energy. If there was no minus sign, this would reverse the induced electric field in a clockwise direction, driving a current in a clockwise direction. The direction of its changing magnetic field would add to the original changing magnetic field, producing a positive feedback effect and the creation of energy from nothing.
The textbook answer is that the minus sign in Faraday's law ensures the induced EMF works against the change causing it, summarized in Lenz's law, and that energy is conserved. However, it's important to bear in mind that if we used a left handed screw convention instead, there wouldn't be a need for the minus sign on the RHS in Faraday's law.
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.
