James Clerk Maxwell and his beautiful equations!

Today is the 175th anniversary of the birthday of James Clerk Maxwell, without doubt one of the top echelon of physicists, and certainly the greatest physicist of his era. Working across a number of areas, he is now probably chiefly remembered for his complete set of field equations describing electromagnetic theory. While this may have been his crowning achievement it was not the only work which still bears his name, with him doing important work in statistical mechanics, and even providing some of the foundational work on explaining colourblindness and colour perception in the human eye. Strangely while most physicists would likely rank him up with or just behind Newton and Einstein he is generally vastly less well known.

Seventy five years ago, Einstein celebrated the hundredth anniversary of Maxwell’s birth by saying Maxwell’s work was the “most profound and the most fruitful that physics has experienced since the time of Newton.” Maxwell’s equations described electromagnetic theory in short set of equations in terms of fields, a description which was later to inspire Einstein to achieve the same thing with gravity and resulted in the General Theory of Relativity. His discovery that light travelled at a constant speed, led ultimately to the development of the Special Theory of Relativity.

Most physics students get to know Maxwell first via his four famous equations. Maxwell never wrote them quite like this as the notation has changed since, but what he did was equivalent if not in such a condensed notation. Even in modern notation expressed in four equations there are several ways of writing them. I’ve always preferred the aesthetics of the integral form.

1. Gauss’s Law for Electromagnetism
GaussLawEM
I’ve always loved the closed integral signs, to indicate integrating over a closed surface or loop. Technically I think this should have a double integral sign, but the meaning is clear regardless. The equation describes how the amount of electric flux passing through a closed surface is always equal to the charge enclosed divided by the electric permittivity. You can think of this as water flowing out of the end of a pipe in a pool. If we enclose the source of water in any kind of surface the total amount of water flowing out through the surface will always be the same regardless of whatever surface I put around it. The thing I love about this equation in this form is that, for a number of problems by using simple arguments of symmetry you can construct clever surfaces that let you calculate electric fields very easily, and certainly much easier than trying to integrate Coulomb’s law to get the same result.

2. Gauss’s law for magnetism
GaussMagnetic
You can express this one simply as “there are no magnetic monopoles”. As you’ll notice if you compare the magnetism version with the electric version, they are the same with the exception that the net flux through the surface is always zero. Unlike a static electric charge which can be isolated and positive (a monopole), magnetic poles are always in pairs. Magnetic field lines form closed loops, and using our water analogy in any closed surface just as much water must flow in as flows out, because we don’t ever have a source emitting from one point. This law is a classic target for people investigating “alternative” physics. Forbid something and everyone wants to do it, but so far no one has and nor are they likely to.

3. Ampere’s Law
AmpereLaw
Ampere’s law describes how loops of magnetic field are formed by currents or changing electric field through a loop. In a similar way to Gauss’s law many problems can be solved simply by looking at situations of symmetry. Quite simply this describes how to make an electromagnet, and indeed calculating the field strength in a solenoid, is a classic example of a easy use of the symmetry to get a result.

4. Faraday’s Law
FaradayLaw
Faraday’s law is basically the converse of Ampere’s law, this describes how an electric field will be created around a time varying magnetic field. This describes how we can use magnets to generate current, in such devices as a transformer, or a dynamo in a power station.

The most important result Maxwell was able to derive from these equations was that perpendicular varying electric and magnetic field could form a wave, and he was able to derive the speed of that wave, which turned out to be very close to the speed of light. For the first time it was hypothesised that light was an electromagnetic phenomena, and remarkably it seemed to have a constant speed that was without reference to anything. Indeed it seemed that these waves would propagate in a vacuum. This was a concerning as waves were known to propagate in a medium – water waves require water, sound waves require air. For many years to alleviate this terror of the void, “the ether” was postulated, an invisible material that EM waves travelled through. Eventually as a result of Michelson and Morley’s famous experiment to measure the “ether wind”, which found instead that the speed of light was constant in all directions, Einstein was able to develop his special theory of relativity.

The crucial role that Maxwell, in his short life of 48 years played in revolutionising physics is nothing short of incredible, and it is a shame that his fame isn’t greater. Perhaps we need invent an apocryphal story involving that moment of sheer inspiration, or perhaps he just needed wild hair and the ability to arrive seemingly from nowhere.

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6 Responses to James Clerk Maxwell and his beautiful equations!

  1. mick says:

    Maxwell’s equations are beautiful. I’m reminded of a t-shirt that is popular among physics undergrads the world over that said “God said, [insert Maxwell’s equations] and then there was light”.

    I think Maxwell never received the fame he deserved because his equations didn’t fundamentally alter the way the general public perceived the universe. At least that’s my theory.

  2. mike says:

    Where do I find the shirt that Mick describes?

  3. Steve says:

    I haven’t seen one in years, but a quick google search for “physics T-Shirts” reveals the following site

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