Induced EMF from a Changing Magnetic Field

Ramp up the current in one coil and its magnetic field grows. Any nearby loop sits inside that growing field, so the flux through it is changing. Faraday's law says a changing flux always comes with an induced emf: emf = -N dPhi/dt, where N is the number of turns and Phi = B A for a uniform field through a flat loop. That emf pushes charge around the second coil even though the two coils never touch. Nothing moves mechanically here — the coupling is purely through the changing field.

The deeper mechanism is that a changing B does not just produce an emf in a wire that happens to be there. It generates an actual electric field in space, circling around the axis of symmetry like rings around a tree. Inside a long solenoid with B climbing steadily, this induced E field loops around the central axis, and its strength depends on how much flux is enclosed by the circular path you draw, not on the local value of B at that radius. That is why the field persists even outside the solenoid, where B itself is essentially zero: a circular path drawn out there still encloses all the solenoid's changing flux, so Faraday's law still demands a nonzero circulating E field there.

This is the bridge from static fields to induction. Up to now E fields came from charges and pointed radially; now a changing B produces an E field with closed loops and no starting or ending point. That distinction sets up Lenz's law next, which fixes the direction of this induced field and the current it drives.