To end this week, we wrap up electricity and magnetism with the fourth and final of Maxwell's equations. this one includes Maxwell's own personal contribution to these:
This is sort of the mirror image of Faraday's Law from yesterday, with the curl of the magnetic field on the left, and stuff related to a change in the electric field on the right. There are two terms instead of just one on the right, though, because when you're dealing with electric fields, you can get a change in the field in two ways: by changing the field directly, or by moving charged particles around.
The first term describes the moving-particles-around part, where J is the current density flowing through the point of interest. The second term, which looks a lot like the right-hand side of Faraday's Law, describes changes in the field at some point (which might be caused by the motion of charged particles a long way off, or might be caused by something else entirely, about which more later).
So, why is this important?
Since there are twom terms on the right, there are two important applications of this. The first has to do with the first term, which tells you that an electric current will produce a "curly" magnetic field. This is Ampère's Law, developed by André-Marie Ampère, and is the basis for electromagnets: If you run current through a wire, it creates a magnetic field in the vicinity of that current, and you can use that to make a controllable magnetic field. This has innumerable applications, from card locks on doors, to atom traps, to particle accelerators-- any time you want a magnetic field you can switch on and off, you use the first part of this equation.
The second term is the really fundamentally important part, though. This is the "displacement current," and is the one thing in Maxwell's equations that is originally by Maxwell himself. It tells you that an electric field that is changing in time produces a magnetic field, in the same way that a changing magnetic field in Faraday's Law produced an electric field.
Why does this matter? By itself, it's not incredibly important, but when you put it together with Faraday's Law, you get something amazing. A changing electric field in empty space (that is, with no currents around) produces a magnetic field, according to the Ampère-Maxwell equation above. But this magnetic field, by definition, must be changing in time (we started with just an electric field). So, by Faraday's Law, it must create an electric field, which combines with the electric field we started with. But that means the electric field is changing, which creates a changing magnetic field, which creates a changing electric field, and so on.
Faraday's Law and the Ampère-Maxwell Law together let us produce electromagnetic waves-- oscillating electric and magnetic fields, sustaining each other as they move along through space. The combination of the two equations also tells you the speed of these waves, which-- drumroll-- is the speed of light. Classically, then, light is an electromagnetic wave.
The wave nature of light was well known in Maxwell's time, having been demonstrated by Thomas Young around 1800. Maxwell's equations explained what it was that was waving, and around twenty years later Heinrich Hertz demonstrated that these waves did, in fact, exist exactly as Maxwell predicted, by generating and detecting electromagnetic waves created by an electrical spark.
So, as with Faraday's Law, you quite literally would not be able to read this without the Ampère-Maxwell Law. In this case, because without this equation, there wouldn't be any light for you to see the words, or to carry this blog post down the fiber-optic lines of the modern telecommunications network.
(Of course, we know now that the classical picture of light is incomplete, and we need to use a quantum model in which light has particle-like as well as wave-like properties. And the great success of Maxwell's equations at predicting light also precipitated another great crisis in physics, because they predict one and only one speed for light, independent of motion. But we'll get to those a little bit later...)
So, as you surf the Internet today, take a moment to appreciate all four of Maxwell's equations, without which modern life would be impossible. And come back tomorrow for the next equation of the season, from an entirely different sub-field of physics.
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Mathematically, J should be a vector. So, it's the current density flowing through a point, and the direction it's flowing?
That's right. The direction is important as well as the amount; the induced magnetic field curls around the current.
I think Maxwell figured the displacement current had to exist because of a consistency argument. Suppose that instead of a continuous electric current through a wire or something, you've just got a charged particle moving through space, and it passes through a certain imaginary surface, in which you consider a circular loop that the charge goes through the middle of. The charge is only intersecting the surface for a moment; before and after that moment, it's just its electric field that is interacting with the surface.
The extent to which the magnetic field curls around the loop depends on the curl of B, and (this is Stokes' Theorem) it's independent of the precise location of the surface inside the loop, as long as it stretches all the way across. But when the particle passes through the surface does depend on the position of the surface. So there has to be another term that produces curl B even when the particle isn't passing through the surface, and that's the displacement current term.
...When I posted that comment, I immediately suspected that I was historically wrong, and indeed I was historically wrong. Maxwell didn't use this general consistency argument; he was reasoning from his peculiar mechanical model of electromagnetism.
Are you sure about the sign on the second term? If I remember correctly, one important difference between this and Faraday's Law is the absence of the minus sign.
I think you're right, George; that minus sign shouldn't be there.
Faradays' Law and Ampere's Law in a vacuum are symmetric except for the negative sign.