mutual inductance
Two nearby coils talk to each other magnetically. Change the current in one and it induces a voltage in the other, with no wire connecting them. Mutual inductance measures how strongly one coil's changing current drives its neighbour. It answers the question: how does a transformer pass energy across a gap?
Precisely, if coil 1's current I_1 changes, it changes the flux through coil 2 and induces EMF_2 = - M dI_1/dt, where M is the mutual inductance, measured in henries. The very same M works in both directions (EMF_1 = - M dI_2/dt). Its size depends on the geometry and on how well the two coils share their flux — it is largest when both are wound tightly on a common iron core.
Mutual inductance is the whole basis of transformers, wireless charging pads, induction cooktops, and contactless cards. Energy or a signal crosses empty space through a shared, changing magnetic field. The honest key point is that only a CHANGING current couples the coils: a steady DC in coil 1 induces nothing in coil 2, which is exactly why transformers and chargers need AC, not DC.
An electric toothbrush charges with no metal contacts: a coil in the base carries AC, and its changing field induces a current in a coil inside the brush right through the plastic — mutual inductance at work.
AC in the base coil induces charging current in the brush across a plastic gap.
Because only changing currents couple, a transformer or wireless charger does nothing with steady DC — mutual inductance needs the current to keep changing.