A force is exerted on a current carrying wire - you see this if you put it between two ends of a U shaped magnet with a current through it. Depending on the direction of the current, it will either move up or down.
In speakers (link to BBC bitesize), the current is constantly changing direction. This means the poles of the electromagnet change, and so the wire varies between moving up and down - the current varies with the music. The electromagnet moves back and forth, creating vibrations, generating sound waves whose frequency changes with the frequency of the current.
In simple DC motors (link to BBC bitesize) , as we know, the wire experiences a force when it is within a magnetic field, and it moves. The current needs to be reversed every half term, which is done by a split ring commutator, to keep it spinning. The momentum it gains after it has started spinning helps keep it going, too.
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