So far we’ve seen that electric charges create electric fields. We’ve also seen that magnetic charges would create magnetic fields if there were any such things, but there aren’t. If you’re in the business of creating electric fields, as the entire electric power industry is, one way of doing so is to pile up a bunch of charge. This is a massive pain and is usually impractical to provide the EMF necessary to shove those electrons through our home electrical outlets. But since there is in fact current flowing through those wires, there must be another way to get electric fields.
There is. It’s described by the third Maxwell equation, which goes by the nom de physics of Faraday’s Law:
Now we have to translate that into English. The triangle and the x represent the curl of the electric field, which we’ll get to in a second. The right hand side of the equation represents the rate of change of the magnetic field with respect to time. If you have a changing magnetic field at a particular point in space, at that same point the equation tells you that the electric field will have curl at that location.
All right, what’s curl? I think it’s easiest to picture it as circulation, like water flow. Water flowing in a straight line has no curl, a whirlpool has lots of curl. The curl at a given point can be measured by putting a little paddle wheel in the current; if it spins, there’s curl. The always-useful Wikipedia has a diagram of a vector field with constant curl. Given suitable boundary conditions, it’s the electric field you’d see at the center of a circular region with a magnetic field increasing linearly with time:
Wrap a wire along the field direction and you’ve got yourself an EMF that can power the devices in your home. In fact this is exactly how most power plants work – coils of wire are subjected to a changing magnetic field via spinning magnets, where the magnets are made to spin by coal-burning engines or nuclear heat or something along those lines.
What’s particularly interesting about Faraday’s law from a theoretical standpoint is that it shows how a region of electric field curl can be produced entirely without reference to charges. All you need is a changing magnetic field to produce an electric field. If somehow we could think of a way for changing electric fields to produce magnetic fields, maybe we could get the fields to produce each other in empty space… but I get ahead of myself.