Why do we need a coil to make a magnetic field?

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Why do electromagnets always need a coil to generate a magnetic field, and why do you need a AC current to generate inductance?

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You do not need a coil to create a magnetic field. Any conductor with an alternating current will create a magnetic field. When it is setup as a coil this current flow causes an amplification of the magnetic field due to superposition of magnetic field lines in corresponding turns giving you a stronger field.

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