Maxwell's equations: 150 years of light

The chances are that you are reading this article on some kind of electronic technology. You are definitely seeing it via visible light, unless you have a braille or audio converter. And it probably got to you via wifi or a mobile phone signal. All of those things are understood in terms of the relationships between electric charges and electric and magnetic fields summarised in Maxwell’s equations, published by the Royal Society in 1865, 150 years ago.

Verbally, the equations can be summarised as something like:

Electric and magnetic fields make electric charges move. Electric charges cause electric fields, but there are no magnetic charges. Changes in magnetic fields cause electric fields, and vice versa.

The equations specify precisely how it all happens, but that is the gist of it.

Last week I was at a meeting celebrating the anniversary at the Royal Society in London, and was privileged to see the original manuscript, which is not generally on public view.

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It was submitted in 1864 but, in a situation familiar to scientists everywhere, was held up in peer review. There’s a letter, dated March 1865, from William Thomson (later Lord Kelvin) saying he was sorry for being slow, that he’d read most of it and it seemed pretty good (“decidely suitable for publication”).

The equations seem to have been very much a bottom-up affair, in that Maxwell collected together a number of known laws which were used to describe various experimental results, and (with a little extra ingredient of his own) fitted them into a unified framework. What is amazing is how much that framework then reveals, both in terms of deep physical principles, and rich physical phenomena.

Fields and Waves

The equations show that electric and magnetic fields can exist even in the absence of electric charges. A changing electric field causes a changing magnetic field, which will cause more changes in the electric field, and so on. Mathematically this is expressed in the fact that the equations can be rearranged and combined to get a new kind of equation, that describes a travelling wave. So not only do the fields become real physical objects - something that Faraday was the first to propose - but they can carry travelling waves. Those waves are electromagnetic radiation. That is, visible light, radio, wifi, X-rays and the rest, depending on the wavelength.

Relativity

The equations work in three dimensions, and relate fields pointing in different directions to each other. So the electric field in north-south direction depends upon what the magnetic field in the east-west direction is doing, for example. Maxwell wrote it all out component-by-component, direction-by-direction, in twenty seperate equations. These days we use vectors (objects with a length and an orientation, like an arrow) to condense the equations down to four. This makes a symmetry of the equations apparent. Like a sphere, they are the same from any angle. If I rotate the directions so that north becomes east, or southwest, or whatever, so long as I rotate all the axes together, nothing changes and the same equations still work.

Even more than this rotational symmetry, the equations stay the same if I boost my speed. And in particular, the speed of those waves described above stays the same. That is, the speed of light is the same for me, you and everyone, even if we are moving at different speeds relative to each other. This violates Newtonian mechanics, and it required Einstein and his relativity (for which the universality of the speed of light in a vacuum is a founding principle) to sort it out.

Conservation of charge

One of the things that is built into Maxwell’s equations is the conservation of electric charge. The equations can be rearranged to show that the only way to change the amount of electric charge in a given volume is to have an electric current take it away. You can’t just “vanish” the charge. Or create it. That’s what a conservation law means.

Now there is a theorem due to the mathematician Emmy Noether which is well-known to physicists, and which states a deep relationship between conservation laws and symmetries. The conservation of charge should be associated with a symmetry, but what symmetry is it?

The symmetry is a little obscured in the usual form of Maxwell’s equations, which uses electric and magnetic fields. But if, instead of the electric field, we use the voltage, and if we do a similar thing with the magnetic field, we get a new, equivalent set of equations which now do have a more obvious symmetry, in that only voltage differences matter. The absolute voltage has no meaning. This is why birds can sit on high-voltage electric cables without turning into tasty fried snacks. The wires are at a high voltage, but as long as the birds are at the same voltage, no electric current flows and no harm is done.

Changing the voltages everywhere in the world at once makes no difference to anything. In fact for all you know, I just did it then, while you were reading that sentence. Invariance under changes of voltage is a symmetry of the equations, which has important consequences, especially once quantum mechanics comes along.

The Standard Model

Now, one thing Maxwell’s equations don’t contain is quantum mechanics. They are classical equations. But if you take the quantum mechnical description of an electron, and you enforce the same charge conservation law/voltage symmetry that was contained in the classical Maxwell’s equations, something marvellous happens. The symmetry is denoted “U(1)”, and if you enforce it locally - that it, you say that you have to be allowed make different U(1) type changes to electrons at different points in space, you actually generate the quantum mechanical version of Maxwell’s equations out of nowhere. You produce the equations that describe the photon, and the whole of quantum electrodynamics.

For historical reasons, this local U(1) symmetry is called a gauge invariance. Enforcing similar invariances for other symmetries, known as SU(2) and SU(3), generates the weak force (the W and Z bosons) and the strong force (the gluon) respectively. Together, and with the Higgs boson thrown in to cope with the masses, they constitute the “Standard Model”, the best theory we have of fundamental particle physics so far.

These days, when building up a new theory, such gauge symmetries would be imposed as a general principle from the beginning, and we’d study their consequences. But remember, Maxwell’s theory was built up piece-by-piece from the painstaking observations of Faraday and others, and the general principles were discovered in his equations later.

Maxwell didn’t know all of the above, he couldn’t have. As was pointed out several times this week, both at the Royal Society meetings and at a Light and Dark Matters event at the Tate Modern on Friday, the equations of nature often seem to know much more than the people who discover them.

The physical phenomena arising from Maxwell’s equations are far too diverse and numerous to cover in one article. I already mentioned some of them in the opening paragraph. The art of Liliane Lijn, which she was discussing with Robert Dijkgraaf at the Tate, constitutes another, as does the Durham light show I wrote about last week. Many more were covered in the Royal Society meeting, including complicated but beautiful interactions with many different materials.

I have an animated illustration to finish off. The image at the start of the article shows a polarised dipole, a tiny structure which might be set spinning by a polarised beam of light. The dipole is near the surface of a metal. Waves of charge-density ripple outwards. But crucially, solving Maxwell’s equations for this system shows that they will head off in a direction that depends upon the orientation of the dipole. This is important as it offers a quick and efficient way of directing light around the insides of an optical, perhaps even quantum, computer or communications system. You can see a rather beautiful simulation of this happening in the video below.

The work is in this paper, and was presented by Francisco J. Rodríguez-Fortuño at the Royal Society meeting - thanks to him for providing the video.

Appropriately for this 150th anniversary, 2015 is the International Year of Light, and the event at the Tate Modern, sponsored by the Institute of Physics, was part of those celebrations.

It seems likely Maxwell’s equations contain plenty of treasure still to be found.

Jon Butterworth’s book Smashing Physics is available as “Most Wanted Particle” in Canada & the US and was shortlisted for the Royal Society Winton Prize for Science Books.