Induction by alternating voltage?

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The classic description of induction involves inducing a current in another conductor caused by the changing magnetic field caused by alternating current passing through a coil of wire around a metal core.

Nonetheless, wouldn't simply alternating the voltage in a coil cause induction in another conductor? My reasoning is based on a changing electric field causing a changing magnetic field. Isn't that the way an electromagnetic wave is produced? There is no actual current in an electromagnetic wave.

If I am wrong, what am I missing?
 
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If a coil has alternating current it also has alternating voltage, and vice-versa. With ac and a coil, voltage and current cannot exist independently. Inductive reactance, X is given by 2*pi*f*L. By Ohm's Law, V = I*X, I = V/X, X = V/I. They mutually co-exist.

Claude