Wednesday, January 13, 2016

Why is "C" the speed of light?

We get all sorts of questions in our "Ask a Physicist" inbox, (including a positively disheartening number from people who seem to think it's "Ask a Psychic") but one topic that consistently seems to spark people's imagination and curiosity is the speed of light. What defines it, and why can't anything go faster than that? What happens if we try? Thinking about these questions and trying to find their answers is fascinating and fun in its own right, but more importantly it gives us insight into the rules underlying our universe. Today, we'll dig into one of these questions and its enlightening (no pun intended) answer: Why is the speed of light in a vacuum ~300,000,000 meters per second? Why c?

Regardless of wavelength and energy, all electromagnetic waves move at the same speed.
Imagine you've got a charged wire that extends infinitely in both directions. Since it's infinite, it's hard to talk about how much total charge is on the wire, the way we'd be able to if it were something like a sphere. However, by looking at a finite unit of length, we can talk about—for instance—the charge per meter, or charge density.

An infinite wire looks the same from any point along its length, so when you think about the strength of the electric field created by the charge in this wire—how strongly a charged particle would be attracted or repelled by it—it's going to depend solely on the wire's charge density and that particle's distance from the wire (as well as the permittivity of the medium you're in, which for our purposes is a vacuum.) The equation for the electric field around this wire is shown below:

Now, off in the infinite distance, someone begins reeling in this wire, pulling it along its axis. For all practical purposes, this motion creates a current; rather than moving the charges in the wire (as you would by changing the voltage at one end), we're moving the wire itself, along with the charges it contains. As for why, you'll hopefully see in a moment.

As you may know, a current in a wire creates a magnetic field that circles around that wire. The strength of that magnetic field will depend on your distance from the wire (d), but also on the strength of the current, which in this case is the product of the wire's charge density and the speed at which it's being pulled along.


Now imagine you've got a second one of these wires, parallel to the first, charged to the same voltage, and being pulled in the same direction at the same speed. Being of like charge, the two wires will repel one another, pushed apart by their electrostatic repulsion.
When calculating the force between two charged objects, their charges are multiplied together, leading to the lambda-squared term above (since each wire has a charge density of lambda).  
The static electric charge on these wires drives them to repel one another. However, since the wires are being pulled along in the same direction, there's effectively a current in each of them, and the magnetic field that accompanies those currents. When you've got two currents pointing the same direction in parallel wires, their magnetic fields create an attractive force between the two—the faster they're going, the stronger this attractive force becomes.

The equation for the magnetically-created attractive force between the wires.
If you're following closely, you'll see that we've set up a scenario where the attractive force of magnetism counteracts the repulsive electrical force between these wires. As you can see from the above equations, though, the strength of that magnetic force depends on how fast the wires are moving, while the repulsive electric force doesn't (hence the common physics term electrostatic). So how fast would the wires have to move for the electric repulsion to be cancelled out by the magnetic attraction? We can find out by setting the two force equations equal to each other, as below, and then solving for v.
A bit of algebra helps us get rid of the parentheses and reduce the fraction on the right side of the equation, yielding this:
One surprising result at this step is that the charge density term appears in the same place on both sides of the equation, and raised to the same power, meaning that it can be "cancelled out"—the speed that the wires have to move for their electric and magnetic forces to balance out doesn't depend at all on how strongly they're charged. The factor of 2*pi*d also cancels, meaning the distance between the wires is also irrelevant in this equation. Dividing out all the redundant terms turns the equation into:
and, finally, solving for v yields:


If you plug in the actual numerical values for the vacuum permittivity and permeability, it works out to 299,792,400 meters per second—precisely the speed of light!

So what does this mean? For one, it means that in reality you could never move the wires fast enough for their electric repulsion to be completely counteracted by their magnetic attraction, since no massive object can ever move at light speed. More importantly, though, it gives us a clue as to why the speed of light in vacuum is what it is; it's the speed where electric and magnetic forces balance out to create a stable electromagnetic wave packet that can travel indefinitely. Any slower and the photon would come undone, just as the wires would be pushed apart by the electric repulsion. Any faster, and the magnetism would overcome that repulsion and draw them together, collapsing the system. With nothing more than high school-level math, it's easy to show that the speed of light in a medium (or in the vacuum of space) inevitably arises as a consequence of that medium's electric permittivity and magnetic permeability.

I know this was awfully math-y for a blog post (we actually had to work all this out as a homework problem back in college), but hopefully it's given you a glimpse of one of the most exciting and addicting parts of physics—the potential to derive and discover literal universal truths with nothing but a bit of imagination and math.

13 comments:

  1. Short answer: because light to subject to the maximum speed of causality. Your answer may be more rigorous, but it's also circular.

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  2. Hmm, I'm not sure this thought works properly. If both wires are moving at the same speed, then they are at rest with respect to one another, and if they are at rest with respect to one another, you are not going to see the magnetic effect. Two wires sitting on a table next to each other don't attract each other!

    The problem with the thought experiment is that the magnetic effect of a current arises because the electrons are moving but the protons are not. @veritasium has a good video illustrating how this works:
    https://www.youtube.com/watch?v=1TKSfAkWWN0

    So you really can't say that reeling in a charged wire is the same as putting current through the wire. In one case the protons are moving along with the electrons, and in the other they're not.

    Still, I like the idea of it. You can forget about reeling in the wires and just imagine instead that charge is flowing through the wires, and then the thought experiment works. As some others have observed, it still doesn't explain WHY c is the speed of light (no one can explain that), but it does show you a neat way to extract the speed of light from the experiments of Ampere and Coulomb rather than from Maxwell's equations, which unified the results of Ampere and Coulomb and others. Maxwell's wave math is a bit difficult, whereas this thought experiment shows you how to bypass Maxwell to find the speed of the wave with the earlier, simpler equations. That's pretty neat.

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    Replies
    1. To clarify: reeling in a wire is only equivalent to putting a current through it when there's a net charge in the wire—if you've got a positively charged wire with more protons than electrons per meter, pulling it along its length will create a net flow of charge.

      Relativity does make things interesting here—you're correct that, from each of the wires' perspective, the other is stationary and there ought to be no magnetic field. In the "rest" frame, however, the magnetic field does arise! Understanding this apparent contradiction requires delving into special relativity, and the length contraction involved becomes kind of difficult to talk about in an example with infinitely long wires, but here's a great resource on it:
      http://physics.weber.edu/schroeder/mrr/MRRtalk.html
      Thanks for reading!

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    2. I understand that the magnetism is a function of length contraction. The problem is, if the wires are moving together, neither of them "sees" any length contraction in the other. For an electron in one of the wires to feel a magnetic force coming from the other, it has to "see" that the protons in the other wire are contracted. That happens if you have an electron moving parallel to a wire in which the other electrons are also moving but the protons are not. It doesn't happen if you have an electron moving parallel to a wire in which both the electrons and the protons are moving. If you take a relativistic perspective, there is no way to distinguish the situation you are describing from a situation in which the wires are not being reeled in at all.
      I don't see anything in the link you provided that suggests otherwise.

      Also, your units don't work in your equations for Fe and Fb. What you really have there is expressions for force per unit length, not force.

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  3. thank you, I fucking love science, my problem is that I am slow learned and the internet let me see it and review as many times until I think that I got it. so I got to do it my own way.

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  4. I thought the article was going to be about why the 'c' was chosen to represent the speed of light.

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  5. one doubt sir,
    for wireless comm we encode the info on a light photon and transmitt it right? how does this encoding take place?

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  6. I really enjoyed reading it but you didn't explain as to why c is the maximum limit for anything to move.Please anyone help.

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  7. Excellent! Thoroughly enjoyed reading it.

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  8. Interesting, I don't think I've seen that derivation before or (more likely) I have forgotten it. It should work if we inject currents and look at the current speed.

    Two notes:

    - The speed of light in vacuum becomes elevated to the universal speed limit by relativity theory.

    [Basically because other interactions should play nice with EM forces under causality.

    By this derivation the reason seems to pivot on having EM forces playing nice in the first place, which is why the first comment seems insightful - it removes the peculiar forcing by EM unto other interactions.]

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  9. I am not sure if the last part of my previous comment was cut, so I repeat the last point:

    - The 2*pi factor doesn't cancel because the distance does. It cancels because a degree of freedom of the geometry also cancels. I.e. keeping the symmetry but changing the wires to parallel plates would produce the same result. (But with more awkward math.)

    @Lloyd: Terminology choices is most often just a question of history of science.

    In this case, it is:

    "Although c is now the universal symbol for the speed of light, the most common symbol in the nineteenth century was an upper-case V which Maxwell had started using in 1865. That was the notation adopted by Einstein for his first few papers on relativity from 1905. The origins of the letter c being used for the speed of light can be traced back to a paper of 1856 by Weber and Kohlrausch [2]. They defined and measured a quantity denoted by c that they used in an electrodynamics force law equation. It became known as Weber's constant and was later shown to have a theoretical value equal to the speed of light times the square root of two. In 1894 Paul Drude modified the usage of Weber's constant so that the letter c became the symbol for the speed of electrodynamic waves [3]."

    [ http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/c.html ]

    @Anonymous of January 14: It is a deep result in relativity theory. Einstein derived it by looking at how to preserve the [some specific] laws of physics despite having observers and interaction sources move with speeds relative to each other. [ https://en.wikipedia.org/wiki/Speed_of_light ]

    The pity abstraction is that a universal speed limit is necessary in order to have causality for local observers. (That is, relativity allows that time order is reversed when an observer looks at non-local processes.)

    Why the specific value of c and why causality are more or less open questions, as far as I know. Baby steps.

    @venkatraman: That depends if you do analog or digital encoding, see radio techniques. Typically you don't study encoding in individual photons but in signals (photon flows of radiation fields).

    The digital encoding used today is encoding bit states on phase shifts in the signal. [ https://en.wikipedia.org/wiki/Quadrature_amplitude_modulation ; that isn't the latest modulation scheme by all means but it is a fair introduction.]

    You _can_ study topological encoding of the wave fronts in EM signaling, and if you look at astronomical objects where few photons reach us, the individual photons wave packets become interesting. But eventually you run up against quantum effects of wave packets, and that is above my pay grade. ;-) [ http://arxiv.org/ftp/arxiv/papers/1104/1104.3099.pdf ; FWIW I have meet Tamburini and Thidé and helped them in the lab there they did some experiments related to their work on this - just with the lab setup, mind you, not the experiment. Fun times.]

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  10. A very interesting description. Although to me it's more of a proof of a balance, in where 'c' becomes a limit. It do makes you wonder how 'big' our infinite universe may be though :) As seen from 'somewhere else' naturally. Whatever that 'somewhere else' might be, or probably 'won't be' :)

    Thanks for the example.
    Yoron.

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  11. Thank you for providing a engaging answer to this question to the "why" question for c. On similar sites I've seen answers which try to sound clever to avoid answering the question (e.g. "c isn't a speed, it's a vector") or are just straight up unhelpful (e.g. "because the maths says it is"). Personally I find the "deep result from relativity" that Torbjorn mentions the most satisfying explanation - relativity suggests that the exact value of c is a maximum speed in space time because c is the point at which dilation of space time makes the idea of having relative 4 dimensional coordinates meaningless. As speed is a relationship between distance and time this not surprisingly makes relative speed meaningless at c. This is, in my opinion, quite a satisfying "why" explanation for c but also hints at something even more interesting - the nature of light's existence outside of our 4 dimensional space time. The article mentions how the most interesting questions give us an insight into the working of our universe - this question goes one better by giving us an insight into the workings of other universes too.

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