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The term conductor is generally applied to a substance or material that has a lot of free electrons in it. The name conductor is applied because the free electrons are already there. A material does not have free electrons because it is a conductor, but is a conductor because it has a lot of free electrons. That said, let's look at what's going on.

These free electrons have energies that permit them to "wander" through the conductor; they're not "locked into" the structure of the material. And when a voltage (potential difference) is applied, current flows through the conductor because the free electrons are moving. They're made to move by the applied voltage.

If we take the case of a wire in a circuit, the wire is a conductor. This wire, say a copper one, has many free electrons in it, and when we apply a voltage, electrons move. The voltage forces electrons into one end of the wire, and the free electrons "shift over" and electrons emerge from the other end of the wire. This movement of free electrons in response to an applied voltage through an conductor is the essence of current flow in that conductor.

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