Why is it (in detail) that the more coils you add to an electromagnet, the stronger its magnetic field is?
If I go into detail, I will soon reach a point at which I can't do the math! But the answer to your question is simpler than you might think. First, picture the straight piece of wire that your magnet is made of before it's wrapped up in coils. Now, if you run a current through it, is it a magnet? The answer is "Yes." This is because (repeat after me): any electric charge in motion creates a magnetic field. The magnetic field lines run in concentric circles around the wire (the "direction" of the field follows the "right hand rule"). You can verify this by connecting a length of wire to a small battery, and placing a compass near the wire. The amount of magnetic field generated by the wire can be calculated if you know the length of wire and current. Now if you coil this wire around a core (let's assume you have a straight, "solenoid" magnet), you are reshaping the direction of the field lines (as well as "concentrating" them into a smaller size). Let's say you use 2 cm of wire for each turn. Depending on the thickness of the wire, you might be able to get a meter of wire wrapped in a single layer along a core only a few centimeters long. With each turn, you add the magnetic force associated with 2 cm "worth" of the straight wire. And you are coiling it such that the lines of magnetic force are parallel and pointing in the same direction. You can add more coils on top of the first row, and this just adds more field strength. In technical terms, every coil of wire increases the "magnetic flux density" (strength) of your magnet. The magnetic field on the outside of the coil resembles a bar magnet. Again, the right hand rule can be applied to determine the North pole: if you hold the coil in your right hand, and the current flow is in the direction your fingers are pointing, the North pole is the end where your thumb is.
Keith Welch, Radialogical Controls Group (Other answers by Keith Welch)