An electromagnet is a device that uses electric current for the purpose of generating a magnetic field. Such devices are based upon Ampere's Law (or equivalently the Biot-Savart Law), which says basically that a flowing current will generate a magnetic field around it. For a straight wire, this field goes in concentric circles around the wire. Usually, if you make an electromagnet, you want to get a strong field aligned in one direction, so this is normally done by coiling the wire up in such a way that these circular fields around each small length of wire reinforce each other.

If you make the wire into a loop, then at the center the field due to each piece points in the same direction and they all add up to make a stronger field. Unfortunately, this gets weak very quickly as you move away in the direction perpendicular to the plane of the loop. You can coil the wire up in a long coil like a spring (forming a helix), so that it's like a series of loops, one on top of another. This means the field is even stronger and more uniform over the inside of the coil. This configuration is called a solenoid. It is usually made by wrapping wire around a cylinder. Still, even in this configuration, the field spreads out and gets weak quickly outside the ends of the coil, so you can improve it by taking the two ends of the coil and bending the coil until the two ends meet (so that it is now like the wire is wrapped around a torus). This is called a toroidal solenoid. In this configuration the fields from what used to be the ends reinforce one another. This is now just a closed doughnut shaped coil of wire that has a very strong field, which is completely uniform in the azimuthal direction. Also, if you left a little space between the ends when you bent the solenoid (so that it formed a doughnut with a bite out of it), in that region the field would be much stronger and more uniform than at the end of a cylindrical solenoid of equal number of turns per unit length. There are of course, even more complicated schemes, but these two form the basis of the vast majority of all electromagnets. Which one you use depends on what your materials are, where it needs to fit, and what properties are most important for the magnetic field to have.

One more way to increase the strength of electromagnets that is usually used is to put in a core, made out of some material that will amplify the magnetic field, inside the coils of the electromagnet. This core is made out of a material that has the property that its molecules have magnetic moments (are like little bar magnets) that tend to line up along the direction of an external magnetic field. These materials are either paramagnetic or ferromagnetic, though ferromagnets are in general much stronger and, therefore, a better choice. With a core, when current flows through the wires of the electromagnet it creates a magnetic field that magnetizes the core (temporarily or permanently), thus increasing the magnetic field. With a ferromagnetic core this can be a huge increase in strength.

The main limitations on an electromagnet are set by the properties of your core, your wire, and your power source. We already discussed the core, and I don't want to get into power sources (mainly because I know nothing about them, really), so let's talk briefly about wiring. You want to be able to wrap your wire tightly enough and in a well enough organized way to get the field you want, and you want to be able to pass enough current through it to get the strength you need without burning or melting the wire. This often means that you need to cool the wire, or get rid of the resistance. The former can be done with air in some cases but may require more elaborate cooling systems in others. The latter can be done using superconducting wire (which requires cooling for a different reason).

Finally, I just wanted to say that electromagnets are really, really important. They are integral parts of a ton of electrical devices. Some of those include:

More examples, thanks mricicle:

...and, well, almost any time you want to use electricity to move something (unless it's very highly charged, in which case electric fields can be used successfully). Well, there are a lot more examples, because they're in practically everything electronic. Please /msg me or write-up more examples if you think of them.

E*lec`tro*mag"net (?), n.

A mass, usually of soft iron, but sometimes of some other magnetic metal, as nickel or cobalt, rendered temporarily magnetic by being placed within a coil of wire through which a current of electricity is passing. The metal is generally in the form of a bar, either straight, or bent into the shape of a horseshoe.

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