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  • According to Freeman Dyson, the two greatest advances in science in the 19th century were Charles Darwin's The Origin of Species (1859) and James Clerk Maxwell's A Dynamical Theory of the Electromagnetic Field(1865). In his 1999 essay, Why is Maxwell's Theory so Hard to Understand?, Dyson continues:

    "But the importance of Maxwell's work was not obvious to his contemporaries. For more than twenty years, his theory of electromagnetism was largely ignored. Physicists found it hard to understand because the equations were complicated. Mathematicians found it hard to understand because Maxwell used physical language to explain it. It was regarded as an obscure speculation with not much experimental evidence to support it."

    Dyson goes on to explain that although Maxwell was at the level of Newton, he was a very modest and nice person, in contrast with Newton, who had a giant ego and was a difficult person to deal with. Dyson claims that Maxwell delayed progress in physics by 20 years because he did not aggressively promote his own theory, as others might have.

    (It's interesting to note that Dyson just missed out on the 1965 Nobel prize in physics, which was awarded to Feynman, Schwinger, and Tomonaga. Dyson is 91 years old and still very much alive as of February 2015, and is said to be shy and modest himself.)

    Who knows if Dyson's claim about Maxwell's modesty delaying progress in physics is correct? Dyson himself mentions another important reason:

    Maxwell's Equations

    "There were other reasons, besides Maxwell's modesty, why his theory was hard to understand. He replaced the Newtonian universe of tangible objects interacting with one another at a

    Chapter 24 Electromagnetic WavesWednesday, March 24, 2010 3:16 PM

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  • Nearly two centuries of the spectacular successes of Newtonian mechanics had trained physicists to conceive of the universe in mechanical terms. Although Faraday introduced the field concept, it took a few generations for physicists to get the hang of the field concept, and begin to look at the world from a non-mechanical perspective. Certainly Einstein had fully absorbed the field concept by the first decade of the 20th century, and his special and general theories of relativity helped other physicists to appreciate the field concept.

    In many ways Maxwell's theory of electromagnetism is a paradigm for modern theories in physics. For example, when Einstein's special theory of relativity showed that Newtonian mechanics was in need of revision,

    universe of tangible objects interacting with one another at a distance by a universe of fields extending through space and only interacting locally with tangible objects. The notion of a field was hard to grasp because fields are intangible. The scientists of that time, including Maxwell himself tried to picture fields as mechanical structures composed of a multitude of little wheels and vortices extending throughout space. These structures were supposed to carry the mechanical stresses that electric and magnetic fields transmitted between electric charges and currents. To make the fields satisfy Maxwell's equations, the system of wheels and vortices had to be extremely complicated. If you try to visualise the Maxwell theory with such mechanical models, it looks like a throwback to Ptolemaic astronomy with planets riding on cycles and epicycles in the sky. It does not look like the elegant astronomy of Newton. Maxwell's equations, written in the clumsy notations that Maxwell used, were forbiddingly complicated, and his mechanical models were even worse. To his contemporaries. Maxwell's theory was only one of many theories of electricity and magnetism. It was difficult to visualise, and it did not have any clear advantage over other theories that described electric and magnetic forces in Newtonian style as direct action at a distance between charges and magnets. It is no wonder that few of Maxwell's contemporaries made the effort to learn it."

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  • Returning to a previous train of thought, Maxwell's equations were complicated in his time because Maxwell wrote them out in component form. Nowadays we use vector notation, which makes his equations easier to write, easier to understand, and easier to work with. This may be the true reason for the delay that Dyson mentions. Although Maxwell himself became a champion of vectors, and Clifford (starting in 1878), Gibbs (starting in 1881), and Heaviside (starting in 1883) worked hard to promote them, others, such as Tait (since 1867), promoted quaternions as a better alternative to vectors. Maxwell died young (only 47 years old) in 1879, so he didn't see the resolution of the battle

    of relativity showed that Newtonian mechanics was in need of revision, an assessment of Maxwell's theory showed that it was perfectly consistent with special relativity and needed no revision.

    As another example, Maxwell's theory provides, in a way, a model for our approach to quantum mechanics, which we will begin discussing next week. Maxwell's theory has "two layers," in Dyson's words; there is the primary layer, which consists of fields that satisfy partial differential equations, and which can be calculated but not measured. They give rise to tangible, measurable quantities, such as forces and energies, which form the secondary layer, and which are typically quadratic (or bilinear) functions of primary quantities. Quantities in the secondary layer are more directly experienced and can be directly measured, but they do not satisfy simple equations, and therefore are not as fundamental in the theory. Quantum mechanics is similar in that there is a primary layer consisting of what are called wave functions, which satisfy a partial differential equation, and which are not measurable, and a secondary layer of measurable quantities that are quadratic (or bilinear) functions of the primary quantities.

    As a third example, Maxwell's theory is now understood as a gauge theory, and all modern quantum field theories are gauge theories. As such, Maxwell's theory is a paradigm for all modern quantum field theories, and progress in the development of quantum field theories was aided by parallel studies and deepening understanding of Maxwell's theory of electromagnetism.

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  • 47 years old) in 1879, so he didn't see the resolution of the battle between vectors and quaternions, which by about 1910 had turned decisively in favour of vectors thanks to the promotional work of Gibbs and Heaviside.

    One of the main reasons for vectors winning out over quaternions is the heavy use by Gibbs, and especially Heaviside, in applying vectors to the teaching of Maxwell's equations. So let's get back to Maxwell's equations, which did so much to unify electrical and magnetic phenomena. Maxwell's equations are a system of four partial differential equations (or eight, actually, if you write them in component form). What do Maxwell's equations look like in modern notation? Behold:

    Here are Maxwell's equations written out in Cartesian components:

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  • Cartesian coordinates are not always the sensible ones to use; indeed, an important part of problem-solving in physics is to choose a coordinate system wisely, and in this case wisdom often amounts to choosing a coordinate system that is adapted to any symmetry present. In any case, one can see how much more complicated Maxwell's equations appear in component form than in the compact vector form. It makes Maxwell's achievements (the formulation of the equations,

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  • It makes Maxwell's achievements (the formulation of the equations, and the truly remarkable conclusions he was able to deduce from them, as we'll see shortly) all the more impressive.

    Part of what Maxwell did in formulating his equations was just to take all of the existing electrical and magnetic relationships discovered by others and either placing them in mathematical form, and/or extracting a logically minimal subset of them from which all the others can be derived, and finally correcting one of the equations with an ingenious discovery. Equation 1 is Gauss's law, Equation 2 is Gauss's law for magnetic fields, Equation 3 is Faraday's law of induction (placed in mathematical form by Maxwell), and Equation 4 is Ampere's law corrected/completed by Maxwell.

    And what do Maxwell's equations mean? Can we understand them in broad outline, intuitively? Yes we can; let's tackle them one by one:

    1: Gauss's law for electric fields says, in essence, that the source of an electric field (OK, let's say electrostatic field to be more precise) is electric charge. The equation gives a lot more specific mathematical detail: The left side of the equation describes the rate of change of the electric field as you move in space, and the relationship tells us that this rate of change of the electric field is proportional to the electric charge density.

    You can get a complementary interpretation of Gauss's law by considering an imaginary closed surface S in space and integrating both sides of the equation over the volume V enclosed by S. The integral of the right side is proportional to the total electric charge within S. The integral of the left side represents the total electric flux leaving V, the region enclosed by S. In other words, the density of electric field lines on S is proportional to the total charge enclosed by S. In this sense, each of Maxwell's equations is a precise mathematical expression of what we have already learned in less precise English phrases.

    2: Gauss's law for magnetic fields has a zero on its right side, which expresses the fact that there is no magnetic analogue of electric charge.

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  • expresses the fact that the