Most of the electricity you use comes from moving magnets around coils of wire. Electrical power plants either spin HUGE coils of wire around very powerful magnets or they spin very powerful magnets around HUGE coils of wire. The electricity to power your computer, your lights, your air conditioning, your radio or whatever, comes from spinning magnets or wires!
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An electromagnet is a magnet you can turn on and off using electricity. By hooking up a coil of wire up to a battery, you will create an electromagnet. When you disconnect it, it turns back into a coil of wire. Since moving electrons cause a magnetic field, when connecting the two ends of your wire up to the battery, you caused the electrons in the wire to move through the wire in one direction. Since many electrons are moving in one direction, you get a magnetic field!
The nail helps to focus the field and strengthen it. In fact, if you could see the atoms inside the nail, you would be able to see them turn to align themselves with the magnetic field created by the electrons moving through the wire. You can test the nail by itself (with the wire removed) after you’ve done the experiment, because you may have caused it to become a permanent magnet.