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.

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