Why Special Relativity is Obvious

I’ve been meaning to write this post about why special relativity is obvious for a while, and since the new Microsoft Word comes with equation writers, I’ve got an excuse to finally do it.

Special Relativity is a favorite whipping boy among cranks. You’ll see attempts to disprove it all the time (I got one in my inbox a few weeks ago). It also has a reputation as being difficult to understand. I hope to address both of those with this post.

I was in my electrodynamics lecture, and the professor was going through the derivation of wave solutions to Maxwell’s equations in a vacuum (if you’re interested, the derivation is here, with H in place of B, for some weird reason). The final product is below (which is where Word’s new equations came in handy).



Which is immediately recognizable as a wave traveling with speed c. This is pretty easily derived for waves in matter too (with D and H replacing E and B respectively. D and H are simply the electric and magnetic fields in matter, and for most materials end up being just a factor different, which leads to a different propagation velocity).

So what? Everyone knows that electromagnetic waves exist. And what does this have to do with relativity?

Well, for those not familiar with it, special relativity has two postulates, the rest of the theory rest on their back. The first is that there exist inertial reference frames, and the laws of physics are the same in all inertial frames (a frame is more or less just a coordinate system with a clock (so you can measure position and time), and an inertial frame is defined as one where the laws of physics hold, so the second part is something of a tautology). This postulate is hard to argue with, you’d essentially be saying that physics is invalid.

So if a frame is inertial (like say, the one you’re in while you’re sitting at your computer reading this), then any frame moving with a constant velocity relative to it is also an inertial frame. How do we know that? Because if you’re moving with constant velocity in a closed train (meaning you can’t see the outside), there’s no way to tell that you’re moving. You might not believe this at first, but most of your cues that you’re moving come from tiny accelerations, you really can’t tell when you’re moving with constant velocity; it’s impossible, which means that it’s an inertial frame.

The second postulate is more interesting, and that is that the speed of light is the same in all inertial frames. But this is obvious: I just showed above that Maxwell’s equations have waves that propagate with speed c, and since Maxwell’s equations are a law of physics they’re true in all frames. That means that an observer in any inertial frame must see light moving at speed c, no matter how they’re moving relative to anything else.

I think we should really speak of the postulate of relativity, because the second comes naturally from the first, and it might clear up why this wacky theory has to be correct (and is totally obvious).

Of course, the consequences of this aren’t obvious at all: Time Dilation, Lorentz Contraction, the relativity of simultaneity, the weird addition of velocities, and everyone’s favorite: the equivalence of mass and energy. Those are all truly bizarre consequences, but they have to be true, and they’re Einstein’s brilliant insight. I’m not going in to them now, the Wikipedia articles should satisfy the curious , but they have to be true given the two postulates.

Or, at least, they have to be true if Maxwell’s equations and the rest of physics is correct. If Maxwell were wrong pretty much no modern technology would work, rendering that highly implausible. So unless you think that inertial frames don’t exist, you must think that special relativity is correct. I hope I’ve made it clear exactly why this is true.

What's the deal with magnetic monopoles?

So what are magnetic monopoles, and why are they important?

A monopole is just something with only one pole (enlightening, I know). Normal permanent magnets (or the Earth) are dipoles, they have a North and a South pole. If you cut a magnet in half you don’t get one North and one South pole, you get two smaller and weaker dipole magnets.

Imagine an electron. You probably picture it as a point in space, and you probably picture its electric field as lines pointing radially outward. That’s a monopole, an electric monopole. Now if you switch from it having electric charge to having magnetic change, you’ll have a magnetic monopole.

You might be saying right now, “Well that’s all well and good, but I’ve never seen one of the magnetic monopoles, why should I think they exist?” If you are thinking that, then you’re on to something. No one ever has seen a magnetic monopole. If you look at Maxwell’s Equations, you won’t see any magnetic charge anywhere. But Maxwell noticed that it could easily be added, but the lack of evidence for them dissuaded him from including them.

I just said that no one has seen a magnetic monopole, and this is true, in a sense. Valentine’s Day 1984, Blas Cabrera, my advisor, saw a signature on a detector that is exactly what you’d expect if a monopole had passed through. However, further experiments made it vanishingly unlikely that this was an actual monopole, and Blas will tell you it almost certainly was not. (Personally I’d like to believe it was, and I’m impressed Blas can admit that it wasn’t.)

Now you’re probably thinking, “If we’ve never seen one, why do we care about them?” That’s also a good question. It turns out that if you take quantum mechanical principles and mix them with electrodynamics, you can prove that if there exists one magnetic monopole anywhere in the universe, then both electric and magnetic charge will be quantized. This is called Dirac’s quantization, and is a pretty stunning result. I’ve seen the calculation done and it’s quite beautiful (but too complicated and lengthy for a blog entry). There’s no real reason for the quantization of charge without this (at least that I know of), so the fact that charge is indeed quantized is a good indicator that there is a monopole somewhere out there (maybe it did go through Palo Alto in 1984). Unfortunately for any monopole lovers, there’s probably less than one per cosmic horizon, which means your odds of finding one are just about nil.

Besides that, Grand Unification Theories and other high-level theories, such as String Theory, demand their existence. For a while, theories demanded too damn many of them, and people were concerned about why there were so few. Alan Guth’s Inflationary Cosmology did a fantastic job of explaining the small level of monopoles, which is one of the many reasons it’s so widely accepted.

I hope that now you have a decent grasp of magnetic monopoles, what they are and why they matter.

Magnetic Fields Do No Work

Every time I head about a perpetual motion machine (or some other kind of free energy), I know immediately how it works. It’s one of a short list of well-known but poorly-understood (at least among the general population) physical principles. The two big ones are magnetic fields and the Casimir Effect. I’m going to be talking about magnetic fields today, and why they can never be used to make any kind of perpetual motion machine.

Everyone loves magnets, and if they don’t they should. I know I do, I have about 40 neodymium super-magnets on my desk. They’re fascinating, and with their invisible and seemingly magical attraction, they have mystified child and adult alike for generations. Surely there must be some way to harness this power and get free energy!

There’s just one problem with this: magnetic fields do no work.

Some physics background for those who don’t have it, work is basically force times distance. If you apply a force over a given distance you do work, any time you move an object you’re doing work (however, sitting at your computer and playing solitaire is not work). Work has the units of energy, and (ignoring friction) the work done moving an object is exactly equal to the change in its energy.

So if magnetic fields don’t do work, then we can’t get any energy out of them without dissipating the field itself.

But, you ask, how do I know that magnetic fields don’t do work? The answer to that requires some vector calculus (unfortunately, no one likes vector calculus), but it’s not too bad. Skip it if you don’t care, but I promise I’m not going to kill you with Math.

The magnetic field (B) is defined as:



That X in the middle does not mean “times”, it’s the cross product, which basically means that the force from a moving charge in a magnetic field is perpendicular to both the field and to the velocity of the charge.

Work, in the true mathematic form, is:



F is the force and ds is a bit of the path, “dotted” into the force. But we know that ds is equal to the velocity time a small bit of time, dt (because that’s the part of the path that the object moves in time dt).

Now recall that the force is equal to the field crossed with the velocity, and to get work we have to dot it with the velocity:



The cross and the dot products have a peculiar property that if this happens, the result is always zero. Geometrically this happens because the cross product creates a vector that is perpendicular to both the initial vectors, but the dot product evaluates the length that two vectors have in common. If they’re perpendicular, then the answer is always zero.

So magnetic fields do no work, and hence you can’t get any energy from them (without dissipating the field).

If you don’t like that argument (although it’s perfectly solid), I’ve got another one for you. The energy density of the magnetic field is:



If you integrate that over all space, you get the entire amount of energy contained in the field. Because every magnetic field falls to zero as the distance away increases, that integral is finite, and hence the total amount of energy one can extract from a field is finite. This is why I repeatedly added “without dissipating the field” to the end of “magnetic fields do no work.” This is also why objects (such as paper clips) will go flying towards magnets: they modify the field, changing how much energy is stored in it. You could conceivably extract energy from this, but only as much as you put into it (and actually less because of losses due to friction), just like every other physical system.

So, while magnets are fun and fascinating, anyone who claims that they’ve harnessed free energy from them is either mistaken or a liar, and they have an incomplete grasp of electromagnetism. Remember, magnetic fields do no work!

The bizarre and intriguing story of Oleg Jefimenko and the solutions to Maxwell's Equations

I recently heard the story of Oleg Jefimenko during a lecture on Electrodynamics, specifically the general solution to Maxwell’s Equations.

Jefimenko’s tiny bit of fame comes from Jefimenko’s Equations, which are the general solution to Maxwell’s equations expressed solely in terms of sources, that is charge and current distributions. The equations are messy and difficult to work with, and aren’t used much in practice. But they do reveal certain bits of physics (such as the applicability of the quasistatic approximation (the link goes to a thermodynamics page, but the idea is the same) and that fields must be created by sources), and it’s always nice to have the general solution to a problem available.

These equations weren’t written down until 1966, about a century after Maxwell’s Equations were known. Some people will claim (as the Wikipedia article cited does) that Jefimenko’s Equations were written down earlier, but those earlier versions are always slightly different and not quite complete. What’s really funny is that Jefimenko wrote them down in an attempt to formulate an alternative to Maxwell’s equations.

When my current Professor, David Griffiths, was in the process of writing a paper on the subject, he independently derived Jefimenko’s equations, and tried to figure out if anyone had done it before. Other than some slightly tricky and annoying math, they’re not hard to derive, so someone must have done it. He found that Jefimenko had written them in a book that was published by a company that had only published one other work, also by Jefimenko (apparently regular publishers wouldn’t take his books, so he went to a prestige press). He contacted Jefimenko, and Jefimenko didn’t believe that he had solved Maxwell’s equations, but that he had created an electromagnetic theory separate from (and doubtless better than) Maxwell’s. Of course he had done no such thing, his formulation is exactly equivalent to Maxwell’s, but he wasn’t buying it.

According to Griffiths, Jefimenko currently submits one or two papers a week to American journals, gets denied, then publishes them in Europe (where review is apparently not as stringent). I don’t know what they’re about, the Wikipedia article says he focuses on overthrowing Einstein’s General Relativity and Maxwell.

I found this story behind some esoteric equations to be pretty amusing, and thought others might agree. I hope you’ve enjoyed the convoluted and intriguing story behind Jefimenko’s equations.

[Most of my information comes from a lecture with Griffiths, and as such could not be found online. Anything that is available online has been referenced.]

Why do electrons stay in wires?

What keeps electrons in wires?

That’s a question I had never asked myself, and it’s one few people have considered. The answer isn’t as easy or obvious as you’d expect, but if you’re curious about it, read on.

The short answer is that the electron clouds and the nuclei of the atoms at the edge of the metal form a dipole. A dipole is essentially a set of two charges that form a mathematically unique field (see the Wikipedia article for a picture). The field from this dipole keeps the electrons in, as they’d need a kinetic energy of roughly 4 electron volts (eV) to get through the dipole field.

If you’re having trouble imagining it, an analogy is a water channel, where the dipole field is the walls and the electrons are the water. The water doesn’t have enough energy to get over the boundary (this case the potential barrier is gravitational rather than electrical), and so it flows along the path of the channel. The analogy is good for another reason too, water has to flow downhill (following the gravitational field), and so do the electrons (following an electrical field). (One difference is that the gravitational field is created by the Earth, whereas the electric field is created by the current.)

This effect has to do with the work function, the energy needed to pull an electron off of a metal into free space. Experiments around the turn of the century had discovered that when a metal was bombarded with light of the right frequency, electrons would fly off, creating a current. This is called the photoelectric effect. Physicists couldn’t quite explain it until Einstein realized that it meant that light is both a wave, with a frequency and wavelength, and a particle with quantized energy and momentum. One of his 1905 trio of amazing papers put forth this hypothesis, and it’s what won him his Nobel Prize (nope, Relativity never won him one, and the other two were on Brownian Motion and Special Relativity).

So that’s why electrons stay in wires (and metals in general).