Built on Facts

Maxwell’s Equations & Light

And God said, Let there be light: and there was light.
– Genesis 1:3

This post is going to need the standard math disclaimer. Though there’s math that looks horrible, you don’t actually have to know any math to follow the ideas. Give it a try!

Over this week we’ve looked at the four Maxwell equations that describe the entirety of classical electromagnetism. Let’s write them all down:

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These four are everything that it’s possible to know about the classical behavior of electromagnetic fields. This notation is a little more modern than what Maxwell originally wrote down in the 1860s, but when he put together this theory one of the first things he did was think along these lines: Faraday’s law says a changing magnetic field produces and electric field. Ampere’s law say a changing electric field produces a magnetic field. It just might be possible for one to produce the other which in turn brings you back to the beginning of the repeating cycle. You don’t need any kind of matter or charge, the mysterious intertwining dance of the fields themselves might be able to propagate itself out forever. Maxwell thought so, and with the machinery of his completed theory he investigated the situation mathematically.

Start with Maxwell’s equations, and let J and rho equal zero. We’re in empty space, so there’s no charge or current floating about. Take the equation for the curl of E, and find the curl of both sides of the equation. Yes, we’re finding the circulation of the circulation. Don’t worry, there’s a reason:

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We can apply a vector identity to simplify the left side of the equation, resulting in this:

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Some explanation is in order. The triangle is pronounced “del”, and applied to a scalar function it represents the gradient, or in essence the directional rate of change of the field with respect to position. The del squared is called the Laplacian, and roughly represents the rate of change of the gradient. The first term on the left is equal to zero, because the divergence of the electric field is zero by virtue of the fact that there’s no electric charge.

The right hand side simplifies as well. Nothing prevents us for interchanging the order of the space and time derivatives, so let’s do it. Applying all that and canceling the minus signs:

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All right, that’s an improvement. We can improve it yet more. What we have on the right is the curl of the magnetic field. And that’s already one of our Maxwell equations at the top of the post. Plug that equation in, remembering that in empty space J = 0, and you’ll get:

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To a physicist, I imagine being the first human to see that equation must have been like being struck by lightning. This equation is just the old and venerable wave equation. Maxwell’s equations imply that the electric field can travel along in space as a wave. The same steps lightly modified give the exact same expression for the magnetic field. The two waves are intimately interleaved, creating each other and traveling outward. There’s more. The wave equation says that the speed of the wave is 1 divided by the square root of the constants in front of the time derivative. Plugging in (and feel free to try it yourself!), Maxwell saw that the speed of these electromagnetic waves matched the value of the speed of light.

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This was an astonishing revelation. Light must actually be these waves. As Maxwell put it, “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” He was exactly right, except for the fact that there’s no medium actually vibrating. The fields themselves are all that’s needed.

These equations and this result are some of the greatest triumphs of science. It’s difficult to conceive of much more impressive physics, but I’m an optimist. Physics is not yet out of surprises.

Comments

  1. #1 Uncle Al
    August 14, 2009

    And God said, Let there be light: and there was light. And God said, “Let there be Maxwell,” and there was light.
    The triangle is pronounced “del” “Del” is the operation, “nabla” is the symbol – but we won’t harp on it.

    Finite lightspeed requires a non-empty vacuum possessing both non-zero permitivity and non-zero permeability. We can do some housekeeping to arbitrarily increase lightspeed,

    http://www.npl.washington.edu/AV/altvw43.html

  2. #2 Markk
    August 14, 2009

    This is called “classical” because it isn’t quantum, or overtly relativistic (although that is buried inside as you said), but I have always felt that creating these equations was really the true start of “modern” physics. Yes we have quantum mechanics and all, but the framework is still field equations or related mathematical structures.

  3. #3 The Science Pundit
    August 14, 2009

    “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” He was exactly right, except for the fact that there’s no medium actually vibrating. The fields themselves are all that’s needed.

    But we needed Michaelson and Morely to demonstrate that and Einstein to explain why.

  4. #4 Peter Morgan
    August 14, 2009

    Re #2, The comparable classical framework is not that of differentiable fields, which cannot accommodate even thermal fluctuations. The much more directly comparable (rigorous) classical framework is that of “continuous random fields”. When you say “related mathematical structures”, this is the first entry in the list. They are, inevitably, harder mathematics than differentiable fields, until you get used to them.

    Re #3, ZapperZ has a nice post on the relationship between the Michelson-Morley experiment and Einstein, from Wednesday, on an arXiv paper that is in press, http://physicsandphysicists.blogspot.com/2009/08/on-role-of-michelson-morley-experiment.html
    Einstein may, or may not, have known about the Michelson-Morley experiment before he constructed his “explanation”. He said different things about it at different points in his life. I recommend ZapperZ’s post, and even more so the arXiv preprint, http://arxiv.org/abs/0908.1545, which casts some interesting light.

  5. #5 Eric Lund
    August 14, 2009

    The wave equation says that the speed of the wave is 1 divided by the square root of the constants in front of the time derivative.

    Nowadays, that constitutes a de facto definition of ε_0. μ_0 is defined as 4π * 10^-7 H/m, and in the 1980s the meter was redefined so that c is exactly 299 792 458 m/s. Thus ε_0 is by definition 1/(μ_0 c^2), or 8.854… * 10^-12 F/m.

  6. #6 CCPhysicist
    August 14, 2009

    Maxwell also noticed “Faraday rotation” could be explained with this model.

    Apropos the discussion of fluid flow, it should be noted that Maxwell used the flow of that medium to figure out how to add that funky “displacement current” term to Ampere’s formula in #4 of the series. However, I often wonder if he saw the lack of mathematical symmetry before he saw the internal inconsistency that had to be fixed for this set of equations to function as a physical law.

    @3: We also needed Lorentz to put together a force law and try to apply Maxwell’s equations to dynamical problems. But Maxwell could have discovered Special Relativity if he had simply asked Einstein’s question: What does an electromagnetic wave look like if I travel next to it at the speed of light? (The answer is that it looks like something that violates Maxwell’s equations, forcing a choice between Maxwell and Newton’s galilean transforms.) I also wonder if Maxwell noticed this problem and pretended it wasn’t there because everything else worked so well.

  7. #7 John Ellis
    August 15, 2009

    Coincidently I was re-watching the Doctor Who episode ‘The Impossible Planet’ last night. In one scene in the background there is some graffiti like markings on a table that are actually Maxwell’s equations.

    As a small mathematical note, Maxwell didn’t use vector calculus which was invented/discovered some years later. Instead he had a fiendish blizzard of single component equations. How the hell anyone managed to work with that lot I’ll never know.

    I think Uncyclopedia summarises vector calculus nicely http://uncyclopedia.wikia.com/wiki/Vector_calculus

  8. #8 Randy P., MD
    August 15, 2009

    It’s been a couple decades since I used my EE degree for anything related to engineering, but I seem to recall reading at one point that Maxwell’s had expressed his equations in a different form, and that the form we use now is due to Heaviside. Is that correct?

  9. #9 wholesale jordan shoes
    August 16, 2009

    However, I often wonder if he saw the lack of mathematical symmetry before he saw the about the Michelson-Morley experiment before he constructed his “explanation”.

  10. #10 gibbon1
    August 16, 2009

    But we needed Michaelson and Morely to demonstrate that and Einstein to explain why.

    Meaning

    Maxwell picked the lock on the the door.
    Einstein found the light switch.

  11. #11 Torbjörn Larsson, OM
    August 17, 2009

    Re the earlier post on “profoundness”, I’ll give you “triumph”. Because it was a success, on any measure you care to define in and out of science.

    Let there be light: and there was light.

    Actually, as the post cover in some detail, it was Maxwell who said that. (o.O)

    Instead he had a fiendish blizzard of single component equations. How the hell anyone managed to work with that lot I’ll never know.

    Ah yes, I mentioned the history earlier in the series comments and why it can illuminate Maxwell’s results, and the CCPhysicist puts in the missing piece:

    IIRC Maxwell, not having the concept of field, first derived his equations from a clunky virtual mechanical model with a number of gear and axis. This explicit model is probably why he used explicit 3D coordinates, I can imagine that it forced him to. I ought to check all of this, of course…

    Later he switched model at least in parts, apparently to flows as closer to the field concept, but seems to have kept to his by then familiar mathematical apparatus.

  12. #12 ObsessiveMathsFreak
    August 17, 2009

    Maxwell’s original form of his equations was in fact a nightmare of about 20 equations in various forms. And I don’t mean it was just about components. Maxwell apparently used quaternions extensively and perhaps something like Clifford Algebra. It was a formidible presentation.

    The current formulation, is apparently due entirely to Heaviside, and was created in direct defiance to the insistence on mathematical propriety in equations. If I remember correctly, Heaviside extracted the “curl” operation from the quaternion algebra as the only useful one, beside the gradient and divergence operators.

    I can’t remember exactly, but Gibbs was also involved in promoting “vector calculus” as an alternative to Quaternion methods, particularly in electromagnetism. Vector-Calculus triumphed, as frankly it should have.

    But nothing is without cost. This like pseudo vectors don’t carry across into the vector calculus world very well. But the alternative is having only one engineer in 10 being able to work with and/or understand Maxwell’s equations. In that sense, vector calculus is almost as important as the equations themselves.

  13. #13 Boy Farrell
    August 18, 2009

    The last part is quite profound:
    This was an astonishing revelation. Light must actually be these waves. As Maxwell put it, “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” He was exactly right, except for the fact that there’s no medium actually vibrating. The fields themselves are all that’s needed.

    Did Maxwell think he was supporting the theory of the aether? Because the result implies the opposite, as you say, the “fields themselves are all that’s needed”. An interesting question is, what event surrounds the initial creation of the light-wave? Moreover, there must be an initial electric and magnetic field variations such that the wave can “self-propagate”? My guess is this must be caused by time varying charges. For example, the recombination of an electron and hole in a semiconductor.

  14. #14 Neil B ♪
    August 18, 2009

    Actually, I don’t think it’s fully correct that Maxwell’s equations tell you everything about the classical behavior of fields, at least not directly as given. The curl of E and B depending on variation in B and E respectively are given, but not the values themselves. Consider that the Coulomb law is merely constrained by, not given by, the MEs.

    Also, we need more to specify the magnitude of E in the vicinity of a changing B field etc. That can best be done by using auxiliary field concepts like the A field, the rate of change of which gives E in conjunction with the direct source projection. OK, maybe you can derive A indirectly etc. but it isn’t the same as seeing it right there. So my favorite is a pseudo-ME: the one that directly shows E and B in terms of sources behavior.

  15. #15 JP
    August 21, 2009

    @13

    When discussing electron-hole recombination, don’t forget that holes are a construct. The electron is simply switching energy bands and releasing the difference in energy as a photon.

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    February 11, 2010

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  17. #17 Robert R
    February 24, 2010

    What about this?

    Alcubierre is best known for his May 1994 paper “The Warp Drive: Hyper-fast travel within general relativity” (which appeared in the science journal Classical and Quantum Gravity). In this, he describes the Alcubierre drive, a theoretical means of traveling faster than light, that does not violate the physical principle and that nothing can locally travel faster than light.

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