Moving along through our countdown to Newton’s birthday, we have an equation that combines two other titans of British science:
This is the third of Maxwell’s equations (named after the great Scottish physicist James Clerk Maxwell), but it originates with Michael Faraday, one of the greatest experimentalists of the day. Faraday was a fascinating guy, who came from humble origins– he was an apprentice bookbinder who managed to get a job as Humphrey Davey’s assistant– to become hugely influential in both chemistry and physics. He also played an important role in science communication and outreach, launching the Royal Institution Christmas Lectures in 1825, a series of science talks for young people that continues to this day.
Faraday had great physical insight, but wasn’t much of a mathematician. Maxwell formalized Faraday’s ideas, resulting in the above equation, which is possibly the most technologically important equation we’ll talk about.
So, what does this equation mean, and why is it important?
The right-hand side of this is the time derivative of the magnetic field B at some point, which expresses how the field is changing. The left-hand side involves the electric field E, and also that upside-down Delta symbol, which has to do with the variation of the electric field in space. Where the previous two Maxwell equations involved the dot product of this del and the field, though, this one involves the cross product.
This is a particularly mysterious-looking object, and kind of hard to make sense of. If you recall our earlier encounters with the cross product, though (angular momentum and torque), you’ll recall that the cross product of two vectors is perpendicular to both of the original vectors. So this funny cross product construction probably ought to involve fields at funny angles in some way.
This quantity known as the “curl” of a vector field, and the meaning of the name is fairly obvious: it expresses the degree to which a field tends to run in loops. If you think about the electric field as a collection of little arrows located at regularly spaced points on a grid in space, the curl of that field at some point represents the degree to which you can make a circular loop around that point by following the arrows. If all the little arrows point in the same direction, the curl will be zero, but if they point one way on one side, and bend around to point in the opposite direction on the other side, then the curl will be non-zero.
So, what does that mean? This equation tells you that if you have a magnetic field that changes in time (that is, one whose derivative in time is not zero), it will produce an electric field that has a “loop” sort of character to it (that is, one with a non-zero curl).This is a very strange sort of object, and not something you can make with static charges. (There are also some subtleties to the whole business– one of the local mathematicians once asked whether the direction of the induced field at a given point is really well-defined, since different ways of drawing a loop though the same point suggest different directions for the field. I never did get that one sorted out in my head.)
Why is this important? Because an electric field that makes loops will cause charged particles to move in circles. Charged particles such as, say, the electrons in a loop of copper wire. Moving charges in a loop of wire represent an electric current, so what this equation tells you is that a changing magnetic field can induce an electric current in a nearby bit of wire.
I called this the most technologically important equation we’ll talk about because without Faraday’s Law, you wouldn’t be able to read this. The vast majority of the electricity used to power modern technological society is generated through Faraday’s Law (the only exception being solar power). Whether it’s coal, gas, or nuclear fuel burning to produce steam to spin a turbine, or falling water spinning the turbine directly, the basic technology used to generate electricity involves spinning loops of wire inside a magnetic field (thus creating a situation where the field is constantly changing direction relative to the loop, which gives you a non-zero derivative).
So, as you sit in your office with electric lights, reading blogs on your computer, maybe sipping drinks that were heated or chilled using electricity, take a moment to appreciate Faraday’s Law, which makes it all possible. And come back tomorrow for the next equation of the season.
Even some solar power stations are turbine driven, and therefore, Faraday-based. The thermosolar plants use solar energy to heat water (directly or indirectly), making steam, and activating turbines.
“There are also some subtleties to the whole business– one of the local mathematicians once asked whether the direction of the induced field at a given point is really well-defined, since different ways of drawing a loop though the same point suggest different directions for the field. I never did get that one sorted out in my head.”
There has to be some wiggle room, because Gauss’s law for the divergence is a separate equation, and because you also have to be able to accommodate boundary conditions. After all, this could be happening in the presence of a static electric field, say inside a capacitor.
The same mathematical issue comes up with Ampere’s law, curl B proportional to the current density. Establish boundary conditions of no field at infinity, specify zero divergence as we were talking about before, and neglect the other thing you haven’t mentioned yet, and we get back the Biot-Savart law, which gives a definite value for the field due to any steady current distribution.
Pretty sweet, although the right-hand side should be in terms of partial derivatives, as the magnetic field generally depends not only on time, but on position (just a technical point). Apart from that, I discovered this by searching Newtonmas in Twitter – one word to describe all this: AWESOME!