tl;dr, any electricity flowing makes magnetic fields, coils make usefully shaped magnetic fields.
Electricity flowing in a straight line causes magnetic fields in circles curling around the line, kind of like saturn’s rings orbiting the wire. This is a fundamental physical relationship between electricity and magnetism. Change in one causes an amount of the other in a perpendicular way.
If you make a coil of wire, each tiny piece of the coil looks like a line if you zoom in far enough, which makes its own tiny ring of magnetic field around it. Because they’re facing different directions, if you add all of them together, most of the perpendicular parts cancel out with others in the opposite direction, but on the inside of the ring they’re facing the same direction and add together. So the total net magnetic force is a straight line following the inside of the coil. Which is useful for pushing things.
A DC current will just push in the same direction forever. This is useful if you have something like a railgun where you want to launch something in that direction and have it never come back. This is less useful if you have a motor that you want to move back and forth or in circles or something. A DC current is going to push it as far as it can in one direction and then it’ll get stuck when it goes as far as it can in whatever contraption you’ve built. An AC current cycles back and forth, back and forth, which is useful if you want to move your motor back and forth and then use that to turn a wheel or move a car or something.
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.
You actually do not need a coil. A straight piece of wire will also work. But the magnetic field will be weaker with only a wire. A coil is a way to have more wire in a single place so you can focus the magnetic field. Electromagnets do work with DC. But a magnetic field does not induce current in a wire, for that you need a changing magnetic field.
Any electric current going through a wire will generate a magnetic field. Turning the wire around an axis (a coil) will strengthen the field.
Induction of the magnetic field is directly proportional to the rate of change of voltage. If you put DC across the wire, there is only one change (the start of the current flow). If it’s AC however, then the flow is always changing which therefore generates a constant field. Fun fact: this also works in reverse – a moving conductor in a magnetic field will induce a current in the coil of the magnet. It’s how electric guitars work.
See Faraday’s Law of Induction.
You don’t. Any electricity creates a magnetic field. It’s just usually easier to reuse the same current over and over again to get a stronger field, by having the same wire wrapped up over and over again. The simplest inductors are [ferrite beads around a wire](https://i.stack.imgur.com/llejr.jpg) and you see them everywhere.
Inductors interact with a *change* in current. Switching a DC current on or off will cause it to interact with inductors. Technically any electricity that changes is AC, so you’re technically correct, but inductors are used in “DC” circuits all the time.
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