AC current produces a mutual current in closely placed coils but not DC because the varying magnetic field generated by the constantly changing direction of the AC current induces a changing magnetic flux in the nearby coils, which in turn induces an electromotive force that generates a mutual current. This phenomenon is governed by Faraday's law of electromagnetic induction, where a changing magnetic field induces an electromotive force in a nearby conductor. In contrast, with DC, the magnetic field remains constant, resulting in a steady magnetic flux that does not induce a mutual current in nearby coils.