Faraday's law

Faraday's law is the quantitative statement of electromagnetic induction: the electromotive force induced in a closed loop equals the negative rate of change of the magnetic flux through the loop, EMF = −dΦ_B/dt, with Φ_B = ∫B·dA. It is the first of the two time-dependent equations in Maxwell's system (the other being the Ampère–Maxwell law with displacement current) and the one that couples electricity back to magnetism, closing the loop opened by Ørsted's 1820 discovery that currents create magnetic fields.

The equation appears in two forms. The integral form, ∮E·dℓ = −d/dt ∫B·dA, relates the circulation of the electric field around a closed loop to the time-derivative of the magnetic flux through any surface bounded by that loop. The differential form, ∇×E = −∂B/∂t, says the same thing pointwise: the curl of the electric field at any point equals the negative rate of change of the magnetic field at that point. The differential form is the one that survives in special relativity (Maxwell's equations are manifestly Lorentz-covariant when written with the four-potential); the integral form is more useful for engineering calculations with specific circuit geometries.

The sign convention is critical and sometimes confusing. Choose an orientation for the circuit loop (say, counterclockwise looking from the +z axis). The positive direction of the area vector dA follows from the right-hand rule (thumb along +z). The flux Φ_B is then positive if B points in the +z direction through the loop. A decreasing positive flux (Φ_B decreasing with time) gives a positive EMF, which drives current in the counterclockwise direction — in agreement with Lenz's law, since the induced current's field points in +z, opposing the decrease in the original flux. Every mistake students make in induction problems traces back to sloppy sign bookkeeping at this step. Get the signs right and the rest of the problem unwinds itself.