Ruben Glueck

Let's talk about Maxwell's Equations. I think anyone who has ever come into contact with Maxwell's Equations - scientists and non-scientists, mathematicians and experimentalists alike, all think they are cool. I am a mathematician so I don't think the math in Maxwell's Equations is the most important part. It is the physics. Nature is symmetric and beautiful and nowhere will you find a better expression of this fact than through Maxwell's Equations. I have not yet found a good reason why nature is so deeply connected with mathematics but it is and that's one of the many reasons I became a mathematician.

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So let's not waste any time and list Maxwell's "Equations" without using any math:

• Gauss' Law - The surface integral of the boundary of a region over E is proportional to the enclosed charge

• There are no magnetic monopoles

• Faraday's Law of induction - A changing magnetic field induces a current and conversely

• Ampere's Law - The line integral around a closed curve over B equals the sum of two terms, one proportional to current through the loop and the other a so-called displacement current density (compliments of Maxwell himself)

 

We will discuss these equations one at a time, but first let's talk about the math. Maxwell's equations cannot even be written down much less understood without a deep understanding of calculus. In a way, they are really Newton's equations because Newton invented calculus. Calculus has two main components: derivatives and integrals. They are connected. Mixing calculus, linear algebra and geometry we get divergence, gradient and curl which are generalizations of the simple derivative that is found in single-variable freshman calculus. On the integral side, we have many types of integrals in many dimensions. There are line integrals, surface integrals and volume integrals among others. But fear not. If you get an A in first year calculus, you really understand what's at the heart of all these things. The devil is merely in the details. That's how important calculus is! Keeping this in mind, there are two forms of Maxwell's Equations, one involving derivatives and one involving integrals. The integral form is geometrical and gives more understanding, but the differential form is better for calculation. As everyone knows it is easier to take a derivative than to grind though an integral numerically.

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Let's begin by talking about my personal tastes. There are electric and magnetic fields and they are called the E field and the B field. These are the only two letters I want and giving separate names to things that are proportional to them is confusing. What harm is there in having a few constants floating around? It's the same thing with just a change of units.

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Gauss' Law is merely an expression of what a point charge really is. Electrons or protons have an E field emanating in all directions. Or put another way, the E field stops short at a point charge. So if we have a closed surface and integrate the E field through it, all the lines that go in go out and cancel unless there is a source of charge within the surface that originates an E field that will cut the surface unopposed. By the way, positive charge cancels negative charge giving rise to a net charge and this is a fact invented by Benjamin Franklin.

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However, the B field is not this way. Try as they might, physicists have not been able to find a magnetic analog of the point charge. There are no magnetic monopoles.

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Faraday's Law - Quite by accident, it was discovered that when you throw a switch controlling current, a needle measuring magnetism will jump. A changing current induces a moving magnetic field. But the story does not end there. Nature is beautifully symmetric. When magnetic field lines cut a wire, a current is induced. This is the principle behind an electric generator that powers modern society.

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Ampere's Law - The line integral around a loop over the B field equals the current flowing through the loop plus Maxwell's displacement current density.

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Now a few words about the displacement current density. This idea represents the triumph of theoretical physics throughout the history of science. One by one Gauss' Law, Faraday's Law and Ampere's Law were discovered empirically and expressed in an elegant way using mathematics. Then Maxwell unified these separate discoveries into a collection of equations and upon examining them deduced a new idea. The symmetry of the mathematics alone implied that there might be a second term in Ampere's equation. Experimentalists went out and looked for it and lo and behold there it was! The same series of events happened with Einstein's relativity. Maxwell's equations were an elegant mathematical expression of a collection of empirical discoveries that were ripe at the end of the 19th century. 25 years later, Einstein reasoned out his theory of relativity and the fact that Maxwell's Equations are relativistically correct, gives us a lot of confidence that physicists are on the right track.

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I am a mathematician. Consequently, I worked out the mathematics behind vector calculus long ago and I am not overwhelmed by the mass of ideas and details behind it. Honestly, mathematician that I am, I am more impressed by the physics and history of science buried within Maxwell's Equations than I am by the math! Really.