# Maxwell’s Equations #3

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.

1. #1 The Science Pundit
August 12, 2009

Magnets, transformers, and (Faraday) cages, oh my!

2. #2 SimplyHarmonic
August 12, 2009

My favorite of Maxwell’s Equations are Faraday’s and Ampere’s laws (with or without the correction).

After all, induction is basically magic. Every time I see it I get a little freaked out and more than a little bit awed.

3. #3 Jon Claerbout
August 12, 2009

Coming from years of computer experience, I like to think of a checker board, with a current around the rim of each square. Notice with just four squares making a 2×2 square, there is no current in the interior — because of equal currents in opposite directions.

4. #4 FeatheredFrog
August 12, 2009

Umm. When will you be getting to the “Silver Hammer” part?

5. #5 guthrie
August 12, 2009

Ironically the advert at the top of this page is telling me “never pay for electricity again” and leads to a website trying to sell you a “zero point magnetic power generator”.

6. #6 CCPhysicist
August 12, 2009

I love teaching Faraday for the reason mentioned @2. Generating a (small) voltage from the earth’s magnetic field is always entertaining.

Solar cells and batteries are the exceptions that work by “piling up charge” rather than Faraday’s law.

7. #7 Uncle Al
August 12, 2009

Curl as pictured defines a privileged axis. Chirality in chemistry overall assumes a privileged axis. However, one can create chiral centers with perfect T (not Td or Th), O (not Oh), or I (not Ih) symmetry. ACS, CAS, IUPAC, NIST, and a journal publisher now dislike Uncle Al (paper accepted by a math journal).

It would be interesting to do such in physics – curl in 3-D with no privileged direction.

8. #8 Jason A.
August 13, 2009

So you describe curl by picturing circulating water (I’m picturing an eddy behind a rock in a stream). But we’re really talking about fields here. So would I be wrong to think of curl as the potential that would induce circulation?

9. #9 rob
August 13, 2009

Jason A:

the analogy to circulating water can be misleading. when you talk about rotation of a vector field such as the B or E field, there is nothing actually rotating. there is no motion. when people say field lines “flow” there is no motion either. so it is incorrect to think of the curl as a potential that causes circulation.

10. #10 Nathan Myers
August 13, 2009

Yes, when he says a whirlpool has lots of circulation, he’s not talking about the paths followed by the water molecules, but about the velocities of those molecules at a moment in time. It’s a fine distinction, but essential.

11. #11 TheDude
August 14, 2009

Would this mean a geomagnetic reversal would cause an electric field? What would be the nature of that electric field?

12. #12 wholesale jordan shoes
August 16, 2009

its hard to understand

August 17, 2009

Note: EMF = Electro Motive Force, something that tries to move charges through a volume (of wire, space, or whatever).

he’s not talking about the paths followed by the water molecules, but about the velocities of those molecules at a moment in time.

Or, as I prefer to think of it, curl is a differential operator in a point (behavior of velocities), not an integral over a volume (geometry of paths).

Would this mean a geomagnetic reversal would cause an electric field?

In principle. But as I understand it these reversals are so slow [Wikipedia mentions 6 degrees/day maximum observed], and with a period of zero magnetic field between, that the electric field would be minuscule.

What would be the nature of that electric field?

That seems to be a complicated question.

A fast dynamic atmospheric field, say resulting from lightning, would force measurable currents in unshielded wires subjected to it. This not so much, see above.

Also, geomagnetism is thought to be induced by moving charges in the convecting outer Earth core. But the electric field feedback from the magnetic reversals would in principle be shielded by that very mobility. (In principle, because the mobility of charges may be low.) So the induced field would possibly only be noticed, if at all, outside that region.

The most obvious effect from a field reversal would be loss of the protection from radiation and solar wind that the field provides us. More radiation, less satellites (as the increased atmospheric loss drags them down in a larger volume).

14. #14 Anonymous
November 11, 2009

Your paddle wheel analogy is extremely misleading. As a counterexample, consider the vector potential outside of an infinitely long solenoid. The field lines of the vector potential will be circular, and if you slipped a bored-out paddle wheel over your solenoid and put it in this field, it would definitely “rotate”.

However, you would be incorrect in thinking that this field has a nonzero curl. In fact, the curl must be zero since there is no magnetic field outside the solenoid.

This means that the imaginary paddle wheel is not the way to intuit whether or not a field has nonzero curl. In the case of the solenoid, the curl is zero because of the way that the field magnitude drops off with distance (inversely).

In short, “curl” is just a very poor name for the mathematical concept it represents. A field that looks circular may have nonzero curl, and a field with nonzero curl may not look circular at all.

15. #15 Anonymous
November 11, 2009

er, I mean “a field that looks circular may have zero curl, and a field with nonzero curl may not look circular at all”