We have seen that electric currents create magnetic fields that circulate around the moving charges. This is the essence of Biot-Savart and Ampere's laws. But in certain cases of electromagnetic induction, magnetic fields can vary with time. The easiest example is simply moving a magnet relative to a solenoid or loop of wire. The trouble is, when one considers the physical reason for the induced currents that we find, there is no magnetic force on the charges of the wire, since the wire is at rest (i.e. qv x B = 0). So how does a current begin?
Think of a moving charge. At some fixed point in space, a moving charge would mean that the E-field at the point is changing...think dE/dt. What is the result of this changing E-field? A circulating magnetic field! Could it be that a changing magnetic field then induces a circulating electric field around the magnetic field? Absolutely! And we even know how to mathematically handle a circulating field from Ampere's law.
Turns out that whenever there is a changing magnetic field, dB/dt, an E-field is induced that circulates around the magnetic field! It is basically Ampere's law for electric fields, and therefore it is actually an electric force, F = qE, that pushes the current in the circuit. This video walks through the details of how induced currents physically form.