The induced electric field has closed field lines, so its circulation along a closed path is no longer zero as in electrostatics: it is determined by how quickly the linked magnetic flux changes. This is the fundamental law of induction.
Faraday-Neumann-Lenz law
The circulation of along a closed path equals minus the time derivative of the flux of through a surface bounded by :
The minus sign is not a formal detail: it encodes Lenz’s law, that is, the fact that the current induced in a circuit always opposes the change in flux that generated it.
Why the minus sign?
If the flux is increasing, the induced current creates a magnetic field that opposes the increase: the circuit reacts like an inertial system. If it were not so, the system would amplify the initial disturbance without limit, violating energy conservation.
In other words, the direction of the induced current is the one that brakes the change: nature prefers the status quo, and to change the flux someone must do work.
Links
Topics: Electromagnetic induction Concepts: Faraday-Neumann-Lenz law · Lenz’s law · Magnetic flux
Related exercises: Concentric rings with variable flux · Worked exercise — loop entering and leaving a field · Problem — Ranking the EMF of four square loops