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 September 2012: 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, 3e8 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 3e8 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 3e8 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 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 3e8 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 3e8 m/s. But it is of course still the case that the first detection downstream occurs at 3e8 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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15 Responses to The speed of light in a material

  1. pungsumbam Sachin Singh says:

    generaly we consider that E=hv (v=frequency of light)
    what I mean is there any loss in energy of light when it travel via glass or other material having heigher optical density .if it does then how?

  2. When light travels through air, then through glass, and then reemerges into air, the final frequency is the same as before it entered the glass (and the color is unchanged). Hence the photon energy has not changed. I’m not sure what the proper description is for the photon when inside the glass, but clearly there is no energy change.

  3. pungsumbam sachin singh says:

    if there is no variation in energy
    there will be defect in Einstine
    because,we have so far consider that speed of light does not hold constant for different medium of different optical density,so by above equation there will be variation in energy E
    in other case,speed of light will be same for every medium.

  4. Please read my blog article again. Its subject is that the speed of light is always c = 3e8 m/s. Also, note that in E = hf, the frequency f is the same in the air and in the glass. As for E = mc^2, that famous equation is not a general one: it just describes the energy content of an object of nonzero mass m that is at rest, and a photon is never at rest. For objects with mass, the correct relationship between mass and energy is E = gamma*mc^2, where gamma is 1/sqrt(1-v^2/c^2), and even this more general equation is irrelevant for massless particles (m=0) such as the photon, for which the most one can say is that E = pc, where p is the momentum. For particles with nonzero mass, one can say E^2 – (pc)^2 = (mc^2)^2.

    • sachin pungsumbam says:

      thank u sir for your kind information.if you don’t mind,can you please discribe me about the phenomina change and about permitivity of a particular substance wrt light speed

  5. No, sorry; I haven’t the time to reply, as this is a very complex subject. See an advanced E&M textbook for such discussions.

  6. Joel Rauber says:

    The latest issue (March 2013) of the American Journal of Physics has a very good article that has tangential bearing on this topic. “There are no particles, there are only fields”, is about as accessible a description I’ve seen that outlines the lessons of quantum field theory on the question of field/particle duality. The title almost says it all, but in the above discussion it fits with the field descriptors of the situation implying that thinking of this propagation in terms of photons may be a bit of a red herring.

  7. Anton says:

    It puzzles me that you don’t mention the group velocity of light when discussing such a fundamental topic. It can be equal to the phase velocity if the medium is not dispersive. So it is c/n , in this case, which means the energy of the light pulse is delayed compared to the free space propagation case. And, clearly, the measurement of the time delay in the experiment you proposed must prove that. So it seems “the speed of light in a medium with refractive index n is 3e8/n m/s” is true or pretty close to that. Correct me if I’m wrong.

    • If you suddenly turn on a single-frequency sine wave, downstream of the material you will first see light arrive at the speed of c = 3e8 m/s. It is only after many cycles that a steady state is established in which there is a phase shift at the observation location that is consistent with what would happen if light moved through the material at a speed of c/n. The speed of light is everywhere and always 3e8 m/s, but the superposition of all the fields produced by all the accelerated charges, the original ones and those in the material, makes in the steady state a situation in which the phase advances through the material with a speed of c/n. The first energy to arrive downstream, before the steady state is established, arrives at the speed of 3e8 m/s.

  8. Anton says:

    Well I agree with you on the question of phase speed. But it is still unclear what you mean by the the first energy to arrive downstream. If we have a short pulse which is convinient to use for measuring time delay, then it has a certain shape, an envelope of the wave which propagates through space at a group velocity. And this velocity can easily be equal to c/n or close to that, which means the slab of glass actually delays the light pulse comparing with its propagation in free space. Sorry, I still cannot see where I’m wrong.

  9. Consider a very detailed mechanistic view. You want to make a short pulse, which we can do by accelerating a point charge upwards from rest and then decelerate it to rest. This pulse will propagate downstream where it encounters a slab of glass, placed so that a normal to the glass passes through the source charge location. The electric field accelerates charges in the glass, mainly the electrons, because the nuclei are very massive. These accelerated electrons (re)radiate. At an observation location downstream (along a line to the source charge that is perpendicular to the slab) we observe an electric field that is the superposition of the radiation from the original source charge and the radiation from the electrons in the glass. With the exception of the small number of electrons lying exactly on the line connecting source and observation locations, all re-radiation from electrons in the glass is retarded compared to the original radiation, because the distance from source to off-axis electron in the glass to the observation location is longer than the direct path. This means that the first nonzero electric field is observed at a time determined by 3e8 m/s.

    Continuing to observe after first noting a nonzero electric field, the shape of the pulse will be different from the original pulse because there is a transient behavior of the bound electrons in the glass which is often modeled by an electron bound to the atom by a spring-like force, with damping. Of course the usual term for this is dispersion, that the phase velocity depends on the frequency.

    I’ve deliberately avoided using the terms “phase velocity” or “group velocity” to make the point as clearly as I know how. This doesn’t mean that these concepts aren’t useful, but the fundamental physics of the situation is that the speed of light is 3e8 m/s.

  10. Anton says:

    Thanks for your extensive reply. I think I begin to understand your point. Theses are just two different ways to look at the same situation. One way is to say that a light pulse travels at a group velocity an is therefore slowed down by a slab of glass, because this velocity is less than c in the glass. This is the conception I had in mind initially. But your point is that light always travels at 3e8 m/s and the reason why we observe the delay after it passed through a slab of glass is that at the observation location we have a superposition of the original wave and the wave from the accelerated charges in the glass. These two waves interfere destructively or constructively at different moments so that it appears to us that the light pulse was delayed.

    In order for these two approaches to give the same result we must suppose that the original and the secondary wave are in antiphase at the first moment when they arrive at the observation location. And they stop canceling out each other after a delay which corresponds to the delay we can obtain from the group velocity conception. Makes sense to me. Is this what you meant?

  11. No, that’s definitely not what I meant. I say again that in the case of the momentarily accelerated/decelerated charge, the first appearance of a nonzero field at the observation location takes place at a time that is d/3e8 seconds after the start of the acceleration, where d is the distance between source charge and observation location, for the reasons I gave. There is NO delay in the first appearance of a nonzero field. With or without the glass slab, the time when you first notice a field is d/3e8 seconds. What the glass slab does is to change E(t) at the observation location (the pulse shape), due to reradiation, to be different from what it would be without the slab.

    In the case of turning on pure sinusoidal acceleration of the source charge, after a transient, in the steady state, the timing of maxima of E at the observation location is the same that you would get if light traveled at a speed c/n through the glass, and this effect is due to the superposition of the field you would get without the glass and the field due to reradiation.

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