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.|
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.