Lenz's law: nature pushes back
That minus sign in Faraday's law hides a beautifully simple rule. Lenz's law says the induced current always flows in the direction that opposes the change that caused it. Push a magnet's north pole toward a coil and the coil's near face becomes a north pole too, shoving back to resist the intrusion. Pull the magnet away and the near face flips to a south pole, tugging to hold it. Whatever you do, the coil fights it.
Motional EMF: a rod moving through a field
The cleanest case of induction needs no magnet motion at all — just a conductor moving through a steady field. Slide a straight rod of length L at speed v along two rails, moving perpendicular to a field B. Each free charge inside the rod is now a moving charge in a magnetic field, so it feels the Lorentz force qvB pushing it along the rod. Charge piles up at the ends, building a voltage — a motional EMF.
Motional EMF of a rod of length L sliding at speed v perpendicular to field B.
This ties straight back to flux. As the rod slides, the circuit it closes with the rails grows in area at the rate $L v$, so the flux increases at d\Phi_B/dt = B L v. Faraday's law then gives \varepsilon = B L v — the very same answer. The motional EMF view and the flux view are two windows onto one physics.
The cost: force, current, and power
Close the circuit through a resistance R and a current I = \varepsilon/R = BLv/R flows. But that current sits in the same field B, so it feels a force F = BIL — and by Lenz's law this force points against the rod's motion. To keep the rod gliding at constant v you must push it with an equal and opposite force, doing mechanical work that reappears as electrical energy and finally as heat in R.
The induced current and the mechanical power you must supply — which equals the electrical power I²R dissipated. Energy is conserved.
Eddy currents
Induction is not confined to neat loops of wire. In a solid slab of metal moving through a changing field, the induced currents swirl in closed whirlpools called eddy currents. By Lenz's law they oppose the motion, giving a smooth, contact-free magnetic braking used on trains and roller coasters; and their resistive heating is exactly how an induction cooktop or induction furnace warms metal. Cut slots in the metal to break the loops and the drag largely vanishes — which is why transformer cores are built from thin, insulated sheets rather than a solid block.