Why is "C" the speed of light?
|Regardless of wavelength and energy, all electromagnetic waves move at the same speed.|
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.
|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 equation for the magnetically-created attractive force between the wires.|
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.
If you changed the permeability of an area of space to zero, then the velocity would necessarily approach infinity. Relativity is based on this fundamental result of maxwell's equations. Calling c the constant is bonkers when the product of mu and epsilon are what c always depends on.
Friday, November 25, 2016 at 1:13 AM
James Sutherland said...
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.
Tuesday, February 23, 2016 at 10:49 AM
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.
Monday, January 18, 2016 at 8:11 AM
Torbjörn Larsson said...
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 . 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 ."
[ 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.]
Monday, January 18, 2016 at 5:13 AM
Torbjörn Larsson said...
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.
- 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.]
Monday, January 18, 2016 at 4:44 AM
Chandrahas Halai said...
Excellent! Thoroughly enjoyed reading it.
Sunday, January 17, 2016 at 9:11 PM
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.
Thursday, January 14, 2016 at 5:46 AM
venkatraman sampath said...
one doubt sir,
for wireless comm we encode the info on a light photon and transmitt it right? how does this encoding take place?
Wednesday, January 13, 2016 at 10:25 PM
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.
Wednesday, January 13, 2016 at 8:23 PM
I thought the article was going to be about why the 'c' was chosen to represent the speed of light.
Wednesday, January 13, 2016 at 5:44 PM
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:
Thanks for reading!
Wednesday, January 13, 2016 at 5:30 PM
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.
Wednesday, January 13, 2016 at 5:08 PM
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:
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.
Wednesday, January 13, 2016 at 5:08 PM
Short answer: because light to subject to the maximum speed of causality. Your answer may be more rigorous, but it's also circular.
Wednesday, January 13, 2016 at 2:37 PM