The speed of light in a material

This is in response to the question, “So the speed of light differs depending on medium, right? Is this also true for neutrinos?” This question from a friend was prompted by the recent measurements at CERN suggesting that neutrinos might travel very slightly faster than light. (Note added later: There turned out to be a problem with the experiment, and there is no evidence for neutrinos traveling faster than light.)

Actually, there is an important sense in which one can (and should) say that the speed of light does NOT depend on the medium! On my home page, see my article “Refraction and the speed of light”. If you accelerate charges, they radiate light. Light consists of traveling waves of electric and magnetic fields: see What is Light? What are Radio Waves?.

There is an extremely important though underrated property of charges and fields called the “superposition principle”: The value of the electric or magnetic field at a location in space is the vector sum of all the fields contributed by all the charges in the Universe, AND THE CONTRIBUTION OF ANY PARTICULAR CHARGE IS UNAFFECTED BY THE PRESENCE OF OTHER CHARGES.

It is the capitalized portion of the principle that despite its innocent-sounding content leads to quite counterintuitive consequences. For example, you’ve probably heard that a metal container shields out electric fields made by charges outside the container. False! There is no such thing as “shielding”. By the well validated superposition principle, the field at any location inside the metal container includes the field contributed by external charges. However, it LOOKS as though the metal prevents the field from getting in, because the external charges “polarize” the metal by shifting the mobile electrons in the metal, and the polarized metal contributes an additional electric field inside the container that is equal in magnitude but opposite in direction to the field contributed by the external charges. The effect is indeed as though the metal “shielded” the interior, but the actual mechanism has nothing to do with “shielding”, and the field due to the external charges is most definitely present inside the container.

Consider a cubical box with metal walls, and there’s a positive charge to the right of the box. That positive charge makes an electric field through the region, and that field causes (negatively charged) mobile electrons in the metal to move to the right, toward the external positive charge. That makes the right side of the box have an excess negative charge, and it leaves the left side with a deficiency of electrons, hence a positive charge.

By convention, the direction of electric field is said to be in the direction that a positive charge would be pushed, so the electric field inside the box due to the external positive charge is to the left. Note that the “polarization” charges, negative on the right side of the box and positive on the left side of the box, contribute a field inside the box to the right. The 1/r squared character of the electric field of point charges leads to the surprising result that the field inside the box contributed by the polarization charges is exactly equal in magnitude and opposite in direction to the field contributed by the external charge, so the vector sum of the field contributions of all the charges is in fact zero inside the box, as though the metal “shielded” the interior.

Back to the case of light, which is produced by accelerated charges. If you accelerate charges for a short time, they radiate a short pulse of light. Let’s accelerate some charges somewhere off to the left, for a short time. Light (electric and magnetic fields) propagates in all directions, but we’re interested in the light traveling to the right, toward a detector (which could be a camera) some known distance from the “source” (the accelerated charges). We measure the time from when we briefly accelerated the charges to when we detect the light a known distance away. Divide distance by time and get the speed of light in air, c = 3\times 10^8 m/s.

Now let’s repeat the experiment, except that there’s a thick slab of glass between the source and the detector. You’ve surely heard that “light travels much slower in glass than in air”, so you would expect the light to take significantly longer to reach the detector now that the glass is in place. But that’s not what happens! You find the same time interval between the emission and the first light reaching the detector, and you determine the same c = 3\times 10^8 m/s speed as before! And you must, because the field at any location in space is the vector sum of the field contributions of all the charges in the Universe, unaffected by the presence of other charges (in this case, the electrons and protons in the glass). The fields radiated by the accelerated charges are unaffected and reach the detector in the same amount of time as before.

However, there is an effect. As the electric field passes through the glass, it accelerates the electrons and protons (it accelerates the electrons much more than the protons, due to their very low mass). These accelerated electrons radiate electromagnetic radiation, like any accelerated charges. The traveling fields of this re-radiation also come to our detector, so that the shape of the pulse we receive is altered from what we saw without the glass, because there are now additional field contributions that were not present in the absence of the electron-containing glass. The first bit of light shows up on time, but then the situation becomes quite complicated.

An important special case is that where the source charges off to the left are accelerated not for a short time, but continuously, sinusoidally up and down (which involves accelerations as the charges move faster and slower and turn around). If you turn on this sinusoidal radiation abruptly, of course you’ll first see some light at the detector on time, with or without the glass being present. But let the sinusoidal acceleration of those source charges continue for a long long time. It can be shown that the vector sum of this radiation and the re-radiation from electrons accelerated in the glass leads to a detection of sinusoidal radiation, and that sinusoidal radiation has a phase which is shifted. That is, the peaks come at a different time than they did without the glass. In fact, in the “steady state”, the peaks come later than they used to, and the lateness is proportional to how thick the glass is. It is a useful shorthand to say that the “light travels more slowly in the glass”, as that description is consistent with the phase delay of peaks in the sinusoid, in the steady state, even though the speed of light in the glass is the usual c = 3\times 10^8 m/s. (The initial transient is messy, and not a simple sinusoid.)

Richard Feynman in the famous Feynman Lectures on Physics discusses this quantitatively in Chapter 31-1, Vol. 1 on “The Origin of the Refractive Index”. The “refractive index” is usually denoted by n, and it is common practice to say that “the speed of light in a medium with refractive index n is c/n, where c = 3\times 10^8 m/s”. But in fact the speed of light is a universal quantity. Although it is very often convenient to pretend that the speed of light is slower in glass, that’s just a calculational convenience — it’s a misleading description of what’s really going on. In fact, the refractive index and “speed of light” in glass is different for different frequencies of the sinusoidal radiation, because different frequencies of electric field affect the motion of the electrons differently in the glass.

The interaction of the electric field of the light with the matter (glass or whatever) can be (for nonobvious reasons) well modeled by the electric field exerting a force on an outer electron in an atom in an insulator such as glass as though the electron were bound to the atom by a spring-like force, with damping. The details of the spring stiffness and damping depend on the material and on the frequency of the electric field. In some materials this works out in such a way that in the downstream electric field (the sum of the field contributed by the accelerated source charges and the re-radiation by the accelerated electrons in the material) the peaks can actually be earlier than in the absence of the intervening material, in which case it looks as though the speed of transmission is actually faster than c = 3\times 10^8 m/s. But it is of course still the case that the first detection downstream occurs at c = 3\times 10^8 m/s.

Incidentally, when in the steady state light is traveling through glass, the frequency of the light in the glass (how many cycles of the sine function occur per second) is the same as the frequency of the light in the air. The speed with which a crest of the sine wave advances (the phase speed) is the distance between crests (the wavelength) divided by the time for one cycle, which is 1/frequency. Because the phase speed is slower in the glass, the wavelength is shorter in the glass than in the air: the crests are pushed closer together.

As to whether the (apparent) speed of propagation of neutrinos would differ in different materials, I think not. The change in phase speed for light is due to the rather strong interaction of light with matter, leading to re-radiation. Neutrinos have an amazingly small probability of interacting with matter, which is why one can detect them after they’ve traveled hundreds of kilometers through solid rock. So I wouldn’t expect matter to have any effect on the speed of neutrinos.

Bruce Sherwood

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12 Responses to The speed of light in a material

  1. Gavin Butcher says:

    Very well explained and extremely interesting, however I’m still puzzled by one aspect of the light through glass. You say that the photons arrive at the detector at the same time as no glass, and it appears that this must apply to every individual photon.
    However you also say that the photons interact with the glass particles on the way. Do they do so without slowing, or does this give rise to the phase change that you mention? Do some photons make it all the way through the glass unchanged, or is every photon that exits the glass (classically) a new photon?

    Many thanks, Gavin.

  2. OLIVER says:

    I enjoy your posts

  3. BruceSherwood says:

    I was careful not to use the word “photon” here. The model is the classical model of the interaction of light and matter, in which the electric field in the light accelerates electrons in the glass which reradiate, with the reradiation adding to the original light with the result that that there is a phase shift that makes it appear (in the steady state) that the light moves more slowly in the glass. I don’t feel competent to attempt a quantum-mechanical (photon) analysis of the situation. Note that the Feynman analysis upon which my comments are based is a classical analysis.

  4. Gavin Butcher says:

    Well, thanks for that. I understand you don’t want to get involved with quantum mechanics (although I’m sure you’re competent). And yet… I can’t help thinking that all we have to do is attenuate that light more and more until just one ‘chunk’ makes it through the glass, and we have to think of it completely differently. I did a little research and it appears that the one chunk must interact with every electron in the glass at the same time, no matter what size it is! Truly mindblowing.

    All the best.

  5. Daniel says:

    Hi professor,

    Why does a charge needs to accelerate to emit radiation? Won’t a charge moving at a constant velocity emit radiation since it should also create a “kink” in the electric field in space (the information about the change in the charge’s position wont reach very far yet)?

    Also why is the “kink” field perpendicular (transverse) when the charge’s field would be radial at any time? (Referring from “Matter and Interactions” book)

    • BruceSherwood says:

      Good question. Start with a charge at rest, far away, and suppose it is quickly accelerated to some speed v that then remains constant. There would be a kink and radiation during that short acceleration, but thereafter there will be no more kinks. During this later time the pattern of electric field is actually not what you see around a stationary charge; see p. 846 in the 4th edition of M&I. The full treatment of fast-moving (“relativistic”) charges is beyond the level of the intro physics course.

    • BruceSherwood says:

      Not sure what to say about your 2nd question, other than to repeat that there is an abrupt change in direction of the electric field. A full analysis is found starting on p. 983.

      • Daniel says:

        Thanks professor. Reading the part on p. 983 helped clear things a lot. Sorry to ask a dumb and unrelated question but why does the potential difference across 2 batteries in series equal to 2 emf?

        Suppose we connect 2 identical batteries in series.
        Battery 1 pumps the electron to a higher potential and this electron goes to the low potential plate of battery 2 (dropping in potential) but then again gets pumped to a high potential. Shouldn’t the potential difference across the 2 batteries be just 1 emf?

  6. Bruce Sherwood says:

    You are making the mistake of thinking that the negative end of a battery always has zero potential, which is wrong. The electric field E in each battery is the same, and the length L of each battery is the same, so the potential difference across the two batteries is 2EL, and EL is equal to one battery’s “emf”.

  7. Daniel says:

    So it’s sort of like a charge distribution going like
    (- – – >> – >> + ) with the distance between each lump of charge being L , giving a uniform E across L. Is it correct?

  8. BruceSherwood says:

    The electric field within a battery presumably isn’t actually uniform, so I deliberately simplified the description. The actual potential difference across a battery is actually the integral of vec(E) dot dl, where dl is an infinitesimal step along the path. The “E” in my “EL” is the magnitude of that integral, divided by L.

  9. Daniel says:

    Ahh so that’s how. Thanks again professor. Have a nice day! 😀

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