Lenz's law
The direction of an induced current is always such that its own magnetic field opposes the change in flux that caused the induction. Equivalently: the minus sign in Faraday's law is a consequence of energy conservation.
Definition
Lenz's law is the qualitative statement that fixes the sign of every induced current: the induced current flows in whichever direction makes its own magnetic field oppose the change driving the induction. If the flux through a loop is increasing, the induced current creates a field pointing against the existing flux, trying to reduce it back. If the flux is decreasing, the induced current creates a field pointing with the existing flux, trying to hold it up. Either way, the induced response pushes back against the change.
The deep reason for the sign is energy conservation. Imagine for a moment that the induced current went the other way, reinforcing the flux change instead of opposing it. Pushing a bar magnet toward a loop would make a current flow that amplified the flux, attracting the magnet more strongly, which would make the flux grow faster, which would drive more current… a runaway positive feedback that produces unlimited energy from nothing. Lenz's sign is what prevents this. The induced current must extract energy from the process that's driving the change, converting the work done against the induced force into electrical (and eventually thermal) energy. The minus sign in EMF = −dΦ/dt is not an experimental convention; it is a direct consequence of the first law of thermodynamics applied to the induction process.
Engineering applications exploit the opposition literally. Eddy-current brakes on roller coasters and electric vehicles pass a conducting plate through a strong magnet; the induced currents oppose the motion by dissipation, slowing the vehicle without any mechanical contact. Induction cooktops dump kHz-frequency eddy currents into ferromagnetic cookware; the induced currents fight the changing field, dissipating energy as heat inside the pan. Generator back-torque — the reason it's physically hard to turn the shaft of a dynamo with its terminals shorted — is Lenz's law directly: the induced current in the armature produces a magnetic moment that pushes back against the magnet doing the turning.