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As far as modern poetry online, I can only assume they're referring to this.

(And don't get me started on the missing apostrophe.)

Prime Time with Magicicada!

That’s right folks. We have a very special guest with us this summer: the 17-year periodical cicadas!



All rights to Wikipedia something something.

Tell us a little about yourselves.

Zeeeeee Tick Zeeeeee Tick (Warning: offensive)

Fascinating. So, some interesting factoids about cicadas of which I was previously unaware: When the 13- and 17-year cycle cicadas are underground, they’re still alive, feeding off the root systems of trees. Also, they remain in an immature, but mostly anatomically complete state the entire time—just like humans.

There are plenty of other cicadas out there, but only eastern North America has the famed 13- and 17-year broods. The genus of which these several cicada species are a part is known as Magicicada, which near as I can tell means magic tree cricket.

Anywho, as far as I’m concerned the most interesting fact about cicadas is that their cycles correspond to prime numbers. I’m not the only person who finds this interesting, and there have been a number of studies investigating this rather curious phenomenon.

So what’s the verdict? The best current hypothesis is that it’s a confluence of three separate factors: predator satiation, the last ice age, and hybridization.

The first factor is a common adaptation not unique to the periodical cicadas. As the name implies, the trait involves satisfying the hunger of those that wish to eat you. The gist is that if every single cicada emerges and breeds at the same time, then there are only so many predators that can eat said cicadas, and the vast majority won’t be eaten. If they came out piecemeal throughout the year or over the years, then their predators would have time to get hungry again, and their survival rate would go down.

Okay, so that accounts for why they all come out at once, but it doesn’t account for why they have such long cycles. This is where the ice age factor comes in. Cicadas are used to the nice warm temperatures underground and they’re only able to fly around and mate when the surface temperature is similarly suitable. So they wait until the soil temperature has reached a comfortable 63 °F and then erupt from the ground en masse.

But during glacial periods there is no guarantee that the temperature will stay that high for the 3 or 4 weeks cicadas need to successfully reproduce. What this means is that when the cicadas decide to come out, they’re taking a gamble. If the temperature drops too low during their emergence, then the brood dies off without reproducing. So the best bet is to hold off and only emerge every decade or two, reducing the odds that they hit a cold spell.

(This might seem specious, because you’d think that (ideally) one summer’s temperatures are not correlated with another’s, so no matter when you pop your head above the soil, there’s an equal chance it ends up not being warm enough to breed. More on this later.)

Now we need the final piece of the puzzle: prime numbers. What could possibly account for the appearance of prime numbers in insect reproductive cycles? It turns out the answer is hybridization. Say you’ve got your cicadas emerging en masse every decade or so. There are separate broods of cicadas emerging at different intervals, which means that every once in a while broods will come out at the same time. When that happens, they will interbreed, and the resultant brood will be hybridized.

If this different mix of genes changes the length of the cicada life cycle, the effect is to smear out the arrival times of the cicadas. This smearing out, however, reduces the population density of cicadas on any given summer, which reduces the effectiveness of the predator satiation strategy, which ultimately reduces the number of cicadas that get to breed. So hybridization is a bad strategy. The best strategy is to make sure there is as little overlap as possible in the cycles of various cicada broods.

For example, if you have one brood on a 4-year cycle and another brood on a 16-year cycle, there’s going to be a lot of overlap because 4 is a factor of 16. So we want cycles with few or preferably no factors. We want prime numbers.

I think it’s important to point out here that a lot of talk about evolutionary strategies can be quite misleading. It sounds as if a species is getting together and deciding the best way to reproduce and then carrying out that plan. But that’s not what is happening. In fact, if that were happening, very different reproductive strategies would emerge.

If cicadas wanted to get together and come up with the best strategy, they might decide to evolve faster flight and the ability to survive warmer temperatures. But they’re not given that choice. Species don’t decide what mutations they get. They are stuck with what nature selects for them.

To see how this works out, imagine there are dozens of different cicada broods with a variety of different cycle lengths and patterns of reproduction. If some broods emerge piecemeal and others as one, we know that due to predator satiation, the broods emerging together are more likely to breed. Thus, over a long enough period of time, we’re not going to see the piecemeal broods anymore because they will have been selected against.

So now we have broods that emerge in unison but do so at varying intervals. The question we have to ask ourselves again is, after a very long time, which broods are we most likely to see? We could see the broods that emerge every year or those that emerge every 20 years. Both broods might have their luck run out after 50 cycles, but that corresponds to 50 years for the first batch and 1000 years for the second batch. So after a long enough period of time, we’re only going to see the 20-year broods. This isn’t a strategy; it’s a consequence.

Now we’re left with only long-period cicadas. Again, the cicadas don’t choose to make their cycles prime numbers. What we see isn’t a choice; it’s statistics. Because prime-numbered cycles correspond to higher reproductive fitness, after a long enough period of time non-prime-numbered cycles die off, and the end result is that we only see those cycles that survived for thousands of years. We see those cicadas that breed all at once on large prime number cycles.

Sorry, no physics in this post, just stamp collecting. I really am going to follow up on the lawnmower post at some point. Really.

TANSTAAFL

I’m still going to do a post about transferring energy from gasoline to lawnmower blades, but a thought occurred to me while I was obsessively rereading my last post that I think might make for a good post in and of itself.

I’ve got a copy of The Way Things Work on my bookcase, and sitting right beside it is a copy of The New Way Things Work. I gobbled up the former book as a kid; it was my first introduction to the idea that electrons release photons whenever they jump down shells, to the concept of computer logic gates, how fission and fusion work, and a whole host of other scientific ideas. I know the book is primarily about machines, but for some reason it was the microscopic stuff that stuck with me more than the mechanical bits.

Anywho, there’s a page about transformers in the book, and I still remember thinking when I read it that transformers were somehow cheating.

You're the man, David Macaulay. Seriously.

You wrap wire around a magnet, wrap more wire around the other end of the magnet, and voila, you’ve increased the electricity pumping through the wire. So why is energy in such short supply? Are there not enough magnets?

I didn’t have a clear understanding of what the conservation of energy meant as a kid (and it can certainly be argued that I don’t now, either, given that I have no idea how Noether’s theorem about symmetry leads to conservation laws), but it still struck me as somehow wrong. And it struck me as wrong for a very long time, right up until about this semester, when I learned what a transformer is really doing.

I think this speaks to the fact that we (the general public) have a very fuzzy idea of what energy is. We know that climbing an electric fence that says “DANGER: 10,000 VOLTS”...

Please don't sue me, Steven Spielberg.

...is somehow more dangerous than licking a 9-volt battery. We also know that “it’s not the voltage that kills you; it’s the amperage.” So both of these are somehow electrical quantities that correspond to bigger and more powerful, but are they related to energy? Related, yes, but they aren’t synonymous with energy.

It turns out that a volt is a measure of energy per charge. You can think of it as the amount of pressure an electron is under. An amp, on the other hand, is the amount of charge moving past some point per second. If you multiply these two quantities together, the charge cancels out and you get energy per second. This quantity is known as the watt, which most people recognize as being what the power company delivers.

But more importantly, this gives us a way to measure energy over time. Because energy is conserved, we know that the amount of energy pumped into a transformer must be equal to the amount of energy pumped out (assuming some ideal transformer with no real world problems). What a transformer does, however, is increase the voltage of the current flowing through it. That is to say, it increases the amount of energy packed into each charge. In order to ensure that the same amount of energy passes through a transformer, less charge must flow out.

(It’s also true, however, that charge is a conserved quantity, so you might think just as much charge has to come out as goes in. But that’s not what conservation really means. It just means that the amount of charge does not change with time; it says nothing of where that charge must go. What’s really happening is that electrons go into the transformer, dump their energy into the transformer’s magnetic field, and then circle back to wherever they came from. The energy they dumped into the magnetic field, however, is then transferred to new electrons at the other end of the transformer. The number of electrons never changes, and the amount of energy never changes, but there are fewer electrons coming out one end than going in the other.)

Say we have a current carrying 1 watt of power at 1 volt and 1 ampere through a transformer. After 1 second, 1 watt pumps 1 joule of energy into the transformer. Then we turn the current off. We expect that 1 joule is going to come out of the transformer a second later. If the transformer increases the voltage of the current to 10 volts, then we have 10 joules per coulomb coming out of the transformer. But since we only have 1 joule to begin with, this means we can only send out 1/10 of a coulomb of charge. Our transformer, then, has turned our 1 volt, 1 ampere current into a 10 volt, 0.1 ampere current. The power is still 1 watt, however, which means that the amount of energy delivered over time is constant.

As the title of the post says, there ain’t no such thing as a free lunch. Transformers don’t cheat; they just do some clever bookkeeping. (I’ve tried to imagine a mechanical analog to a transformer but haven’t been able to come up with anything that doesn’t seem tortured and contrived. Perhaps this is one of the places where the world of electricity and magnetism doesn’t mirror the real world, or perhaps I’m simply having a failure of imagination.)

Until next time, folks.

On the Physical Principles of Graminoid Elimination

On Facebook this past Saturday I commented that I had “mastered E&M and multivariable calculus, but can’t seem to figure out how to start a lawnmower.” Leaving aside the image of a pasty nerd being carried off by an out of control garden tool, I think it’s interesting that the relatively basic physics introduction I’ve been given over the past two semesters is sufficient to explain almost completely how a lawnmower works. So I’m going to share that neatness with my dear and devoted reader(s).

It starts with the starter cord (no surprises there). Pulling the cord almost literally spins the engine’s crankshaft into motion. The crankshaft pulls the piston, which lets fuel and air into the cylinder. The crank continues to turn, the piston compresses the mixture and then, well, then Faraday’s law happens.

There are some chemicals that ignite on contact with air. There are other chemicals that ignite when they are compressed to any degree. Gasoline is not one of those chemicals. Gas is volatile, but not too volatile, which makes it a good method of storing energy. In order to get energy out of gas, you have to mix it with air, compress it, and then raise its temperature all at once.

I don’t want to get too much into the process of combustion, partly because that’s more chemistry than physics, and partly because I don’t understand it very well. But the very basic principle is something Dr. Dave Goldberg explained to me a few years ago when I asked why matter and anti-matter annihilate. His answer: because they can. Essentially, because matter and anti-matter are oppositely charged, it’s very easy for them to disappear and be replaced by electrically neutral photons with the same amount of energy.

The same principle applies to combustion. Oxygen doesn’t have enough electrons, and gasoline has electrons to give. So when you mix the two together, gasoline gives up electrons, breaks down into simpler compounds, and releases energy, because it can. That energy heats and expands the mixture, pushing the piston out and turning the crankshaft, which subsequently turns the lawnmower’s blade.

Why this releases energy is complicated. The simple explanation is that more energy is stored in the system than it takes to release that energy. The usual analogy is to a ball sitting at the top of a hill. When you kick a ball, you transfer kinetic energy from your foot to the ball, and the ball moves away from you until that kinetic energy has been lost to friction and air resistance.

But if you kick a ball so that it rolls down a hill, the energy you provide is enough to “release” the gravitational potential energy stored in the ball, so that by the time it gets to the bottom of the hill it has way more kinetic energy than you provided from your foot alone. So gasoline, because of its structure (yes, that’s a copout), has a lot of potential energy just waiting to be released if you give it a big enough push.

The key is that gasoline is less likely to give up the electrons that form its chemical structure than it is to give up the electrons that bond one molecule of gas to another, so you need more energy to get at those deeper electrons. This energy is provided in the form of a spark that raises the temperature of the mixture.

Where does the spark come from? The spark plug, of course. A spark plug is essentially just a capacitor that is designed to fail. As I explained a couple posts back, charging a capacitor creates an electric field. If the electric field gets too strong, the insulating material between the capacitor plates ionizes. That is, its electrons are stripped away, which turns it into a conductor, causing all those stored electrons in the capacitor to flood across the gap at high speed, producing lightning. This lightning is a spark. But where does a tiny gas-powered lawnmower get the necessary voltage to produce lightning?

This guy:



Courtesy Wikipedia

Lawnmowers use a device known as a magneto. As the engine spins, it rotates a permanent magnet inside a coil of wire. This changing magnetic field induces a current in the coil. But with every turn of the engine, that current is interrupted by a contact breaker, so the current drops down to zero.

Or it would, if it weren’t running through a coil of wire. This coil acts like an inductor, and inductors create electromagnetic fields whose strength is determined by how quickly the local electromagnetic field is changing. So even if there is only a very small current through an inductor, if that current is immediately cut off then the change in current is very large, which in turn produces a very strong electromagnetic field.

But we're not done there. After the magneto comes this:



What would I do without you, Wikipedia?

There’s a second stage, a transformer, which increases the voltage even more, enough to induce a spark in the spark plug. The other pole of the permanent magnet has its own set of wires coiled about it, but it has far more windings than the first set. When the magnetic field produced by the inductor changes due to the broken contact, this induces an electric field in the secondary coil.

But the interesting thing about Faraday’s law is that it talks about changing magnetic flux rather than just a changing magnetic field. Flux is the flow of magnetic field lines through an area. The larger the area, the greater the flux. And the area we’re dealing with here is the circle formed by a loop of wire. But if you have a hundred loops of wire, you have a hundred times the area as far as Faraday’s law is concerned, which means the voltage induced is a hundred times larger.

So by exploiting Faraday’s law to create a rapidly changing magnetic field through a very large area, pulling on a starter cord can induce a large enough voltage to create lightning, which ignites the fuel in the lawnmower’s engine. Pretty cool.

I also wanted to look at converting the chemical energy of gasoline into the rotational energy of the lawnmower blade, but I think this post is long enough already. So tune in next time and there might be a discussion of energy density, thermodynamics, and angular momentum.

Out of Phase

With the semester winding down, I think it’s a good time to discuss a slightly annoying idiosyncrasy I’ve observed this spring. E&M has, as a co-requisite, vector calculus. My school expects these two classes to be taken together, and E&M involves a fair amount of vector calc. But despite this connection, or perhaps because of it, my math and physics have been, you might say, out of phase.

What do I mean by out of phase? I think a thousand words can sum it up best…

As you can see here, Lt. Cmdr. Geordi La Forge’s hand is out of phase with the engineering console. They’re in the same place at the same time, and yet somehow they’re not actually interacting. The solution, of course, is a concentrated burst of anyons. Yep.

Or maybe I mean something a little more like this:

  

We just got through a unit on AC circuits, which are significantly more complex (heh) than DC circuits. The main reason is that the three basic elements of a circuit all respond differently to alternating current. Resistors don’t much care. They reduce the voltage across them, but (ideally) their response isn’t dependent on the incoming voltage. Thus, the voltage across a resistor looks like whatever current is coming from the AC source.

A capacitor, on the other hand, has its maximum voltage when it’s fully charged. And when it’s fully charged, the current across it is zero. Thus, the voltage across a capacitor is at its peak when the blue line crosses zero.

And inductors resist changes in current, which means an inductor will always have a voltage that is positive when the source is decreasing, and negative when the source is increasing.

Why did I go through all of that? Well, for two reasons. First, the differences between those circuits give rise to the different voltage graphs above, and those differences are known as phase shifts. Resistors are “in phase” with their voltage source, whereas inductors and capacitors are “out of phase.” That is, they don’t line up.

When two waves that are in phase with each other combine, they produce one wave that is bigger than either of the two waves. Conversely, out of phase waves that hit cancel each other out. This is known as interference, and maximizing the acoustics of a room is all about making sure sound waves are in phase when they get to the audience.

Secondly, a complete description of the wave-like behavior of AC circuits involves complex numbers—the square roots of negative numbers—a topic which has only been given the briefest of overviews in any math class I’ve ever taken. Consequently, the entirety of the complex aspect of AC circuits was covered in a single line from my professor: “And if we increase this factor over here, you see that we get imaginary numbers.” So, you see, my math and physics classes are out of phase.

But it doesn’t stop there. Near the beginning of the semester, we learned that the electric field can be thought of as the change in voltage across space. The text and our professor briefly mentioned that this is referred to as a gradient and involves partial derivatives. But at that moment in math, we were learning how to plot three-dimensional vectors.

Later on, we learned about Gauss’ law for electricity, which lets you figure out the amount of charge enclosed by a surface if you know what the electric fields coming out of that surface look like (and vice versa). When you first learn how to use Gauss’ law, you’re only asked to apply it to surfaces that are highly symmetric, which makes the math a lot easier. But it turns out that Gauss’ law is a very general principle in vector calculus involving surface integrals. And what, pray tell, were we learning about in math at that moment? Partial derivatives, of course.

Now, as the semester draws to a close, our professor has shown us Maxwell’s equations in all their glory. All their integral glory, anyhow. We’ve seen them in one form or another already, and we’ve used three of them. The one we haven’t used is Gauss’ law for magnetism, which is like the electricity law except it says there’s no such thing as magnetic charge and there’s never any net flow of magnetic field lines through a surface. Why is there a law saying the answer to your question is no? Because, our professor explained, the law is more useful in its differential form, where it uses a vector calculus principle known as divergence. We’ll be getting to divergence next week in math. Right now we’re studying surface integrals.

Maybe I'm misinterpreting all this. Maybe learning about the idea in physics first primes us for the material in math later. But I'm not sure I buy that argument. I feel as if we'd be better served by having to take vector calculus first, or by having E&M and calculus classes that were more, well, in phase with each other.

Anywho, so when are we going to learn the math behind complex numbers and phases? Apparently it involves phasor arithmetic and phasor diagrams. I don't know about you, but I encountered phasor diagrams waaaay before I started taking math classes...

Subtle is the Blurb…

This post might be a little disjointed. I’m weaving together several different thoughts I’ve been having lately, but I think it’ll come together.

So I’ve been reading an Einstein biography, the classic ‘Subtle is the Lord…’ by Abraham Pais. While a little dated, it’s considered one of the more definitive accounts of Einstein’s scientific development.

Anywho, as accords such a renowned book, there are blurbs from other Nobel laureates on the back flap. When I picked the book up in the library, one in particular popped out at me:

I found it fascinating to read about the development of Einstein's ideas, particularly those connected with relativity. The various steps are clearly presented and the influence of other physicists and their reactions are described and coordinated, to provide a very readable narrative. – P.A.M. Dirac

I couldn’t help but laugh after reading this blurb, despite being in a library. The praise here is not exactly effusive. In fact, the whole thing is rather stilted. But more than that, Dirac mainly compliments the book’s clear steps and coordinated narrative, rather than any insight it provides into Einstein the man.

The reason I found the blurb so amusing is that Dirac is often mentioned by those who like to diagnose dead people with psychiatric disorders. Specifically, there are many who believe Dirac was autistic to some degree or another. Anecdotes to that effect abound.

I don’t really go in for that sort of post-mortem diagnosis, but surely Dirac could have sounded a little less like an emotionless robot in that blurb. (That said, I have an autistic friend who is one of the finest writers I know. It’s not the quality of the text I’m harping on but the focus on logic.)

Where this is leading is the notion that geniuses (of which Dirac most certainly was one) and scientists generally are viewed as being out-of-touch geeks who pay more attention to beakers than breasts. We only have to look at the stereotypes presented in the Big Bang Theory to see how common such a view is. But of course, there are equally extreme counterexamples. Feynman certainly got around, and Heisenberg was having an affair with a colleague’s wife when he formulated the first good model of QM.

The truth is, as it always is, somewhere in the middle. Scientists are just people, as Chad Orzel frequently points out. The most significant difference between scientists and non-scientists is that scientists do science for a living. That’s really all there is to it. There are certainly correlations that come with choosing science for a career, but no hard and fast rules. The closest I’ve found to a universal trait amongst scientists is that they (or at least the Scottish ones) realize that humans are prone to error, and that the only way to account for that is rigorous application of the scientific method. But even those scientists who think this way screw up, too, because scientists are humans.

Which brings us back to Einstein. The basic outline you see in every account of Einstein’s life is that in 1905 he began two revolutions: one quantum and one relativistic. Eleven years later he completed relativity by introducing the general theory of relativity. The rest of his life, however, was spent on quantum theory, a theory with which he was never satisfied. The quote everyone brings up here is, “I, at any rate, am convinced that He does not throw dice.”

Einstein was dissatisfied with the intrinsically probabilistic nature of quantum mechanics and argued throughout his life that quantum theory had to be incomplete. His most cogent critique of the theory came in the form of the EPR paper. In it, Einstein, Podolsky, and Rosen showed that two particles could be become entangled such that knowing the state of one particle instantly told you the state of the second particle, no matter how far away it was, without having to measure it. The two solutions to this apparent paradox were faster-than-light communication or hidden variables. Einstein and others believed the latter, that there was information about a system that we just didn’t know how to measure yet, and that this meant quantum theory was incomplete.

But Einstein was wrong. Decades later, John Stewart Bell showed that reality had to behave in a particular way if there were local hidden variables, and in a different way if quantum systems were non-local. Numerous experiments were carried out, and they confirmed that local hidden variable theories were incompatible with reality. Quantum mechanics, to the extent that it described the probabilistic behavior of quantum systems, was a complete theory.

(I should add that I have only a very surface level understanding of this stuff. I know that quantum entanglement has been carried out in labs. I also know that there are theoretical loopholes to Bell’s theorem that might vindicate Einstein but that most scientists think will not. I don’t really know the math and science behind this debate. Ask me again in a few years.)

Einstein was dead by the time all this happened, so we cannot know how he would have reacted to being proven wrong, but for as long as he lived he stubbornly refused to accept the reality of quantum mechanics. Despite it being an extraordinarily precise and well-tested theory, Einstein objected to it on philosophical grounds.

How could Einstein, a brilliant and revolutionary thinker who proposed that light waves were particles, that time could be stretched and space curved, fail to imagine that reality might be a bit random? Because he was human. And humans make mistakes.

Yet despite Einstein’s brilliance, despite his celebrity and his authority, most scientists eventually rejected Einstein’s notions about quantum mechanics and came to accept the non-local ramifications of the theory. They didn’t do so because they were smarter than Einstein or better scientists; they did so because no one individual is responsible for this thing we call science. Science is a process, a methodical and ruthless tool for separating fact from fiction, and it has proven enormously successful.

Individual scientists can screw up, entire generations can dogmatically subscribe to an incorrect theory, and charlatans can purposefully promote bad science, but science carries on. Eventually, the scientific method works itself out. This process we have discovered, of rigorously testing theory against observations, is perhaps the closest we humans have come to transcending our biological limits.

For reasons I don’t really want to get into right now, I can think of nothing more important for us to do. But the essence of it is that I believe becoming something greater than ourselves, greater than the sum of our parts, is the only way we can truly understand the universe. And that’s why I’ve decided to become a scientist.

I have seen the ∂2E(x,t)/∂x2 = µ0ε0∂2E(x,t)/∂t2!

(There's some vector notation that I wasn't able to figure out how to get into a title. And that's an exclamation point, not a factorial.)

So, I'm reading ahead in my physics textbook, and I've reached the culmination of all this electricity and magnetism stuff. That equation up there, a nice little second-order partial differential equation, is ostensibly the pinnacle of 19th century physics and Maxwell's greatest contribution. (It's my understanding, however, that Maxwell didn't actually use that notation, and that there are plenty of other possibly more useful ways to write the equation. Nevertheless, that's my textbook's presentation.)

So what does that equation say? Well, a more or less literal translation says that the way in which an electric field is distributed through space is related to the way in which an electric field is distributed through time by the constants µ₀ and ε₀. This relationship arises from Maxwell's equations. The ones that are most relevant here are Faraday's Law and Ampere's Law.

Faraday's Law tells us that a changing magnetic field induces an electric field. I mentioned that in this post. The most frequent application of this law is in, well, almost every method of power generation we have. Some process (burning coal, burning gas, burning uranium) causes water to boil, and that water spins a turbine connected to a magnet, and that magnet's magnetic field moves through space, which sets up an electric field (and a corresponding current) in some conveniently placed wires. The faster the magnetic field changes, the stronger the current. But as soon as the magnetic field stops changing, the current dissipates. It is only while the field is changing that an electric field is generated.

Ampere's Law, in this context, tells us something similar. It says that a changing electric field, multiplied by µ₀ and ε₀, produces a magnetic field.

I suspect you can see the symmetry here. A changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. If the rate at which an electric field changes is increasing, then the magnetic field it produces will also be increasing. And a magnetic field that is increasing in strength will produce an electric field that increases in strength in a direction opposite to the first electric field, which will tend to diminish the strength of the first electric field. This symmetry creates a sort of back and forth seesaw effect between electric and magnetic fields.

The way this connects back to the equation of the title is that the rate at which a rate is changing is known as a second derivative. The most common example is acceleration. Velocity is the rate at which position changes. Acceleration is the rate at which velocity changes, or the rate at which the rate at which position changes. And as you saw, an "accelerating" electric field produces a changing magnetic field, and vice versa. The 2 in ∂²E(x,t)/∂t² means we're talking about second derivatives.

Okay, what's the point of all that? The point is that, traditionally, an electric field is set up by charged particles. An electron sitting by itself creates an electric field that extends radially beyond it. And magnetic fields are usually caused by moving charges. An electron flying off by itself at a constant speed will have a magnetic field encircling it. But Maxwell's equations say that you can get an electric field just by shaking a magnetic field around, and you can get a magnetic field just by shaking an electric field around. You don't need any charges at all (beyond an initial one to set up whichever field comes first). Electric and magnetic fields sustain each other, giving rise to electromagnetism.

On a basic level, I knew this beforehand. I didn't know the details or the math, however. But what really made the concept click for me was a discussion in an earlier chapter about LC circuits--that is, circuits composed of inductors and capacitors. I already discussed what a capacitor does in my last post, but an inductor is something new. An inductor is a circuit element that takes advantage of Faraday's Law in order to modulate the current in a circuit.

So, a current is just a bunch of moving charges, which means that all currents create magnetic fields. But when the current changes, as it does in AC circuits, you set up a changing magnetic field, which in turn creates an electric field that opposes the current change. The faster the current changes, the stronger the resultant magnetic and electric fields are. In essence, energy is being taken out of the current and put into the magnetic field. An inductor is an element designed to maximize the energy pumped into the magnetic field. This sounds a lot like a capacitor, where energy is being taken out of a current and stored in an electric field.

What happens when you put these two together? Well, as we saw, when you discharge a capacitor, it expels its electric energy very quickly at first and then slows down. But an inductor opposes current change, so the sharp increase from zero current is curbed by the creation of a strong magnetic field in the inductor. The effect is to take the energy stored in the electric field and place it into the magnetic field.

Once the capacitor is fully discharged, the current should stop, but again, inductors oppose current change. So instead, the inductor dissipates the energy from its magnetic field to increase the flow of current. The capacitor, down to zero charge, now begins to charge negatively. That is, it builds up electrons on the opposite plate. The energy from the magnetic field is transferred back to the electric field. With no resistance, this oscillation continues indefinitely, trading energy between the electric and magnetic fields of the circuit. The rate at which this happens depends on the properties of the circuit, but the general shape of the interaction is going to be a sine wave.

Yes, that's right, when you move energy between electric and magnetic fields, you get a wave. One might even be tempted to call it an electromagnetic wave. In fact, the equation in the title takes the form of the general "wave equation" that can also apply to the wave-like motion of a spring or a sound wave or a whole host of other physical phenomena. The general equation looks like this: ∂²u/∂x² = 1/v² * ∂²u/∂t², where u is some wave-like phenomenon, and v is the speed at which that wave propagates.

If that's the case, then µ₀ε₀ takes the place of 1/v² in an electromagnetic wave. Carrying out the algebra means that the speed of an electromagnetic wave should be 1 divided by the square root of µ₀ε₀. µ₀ = 4πx10^-7 and ε₀ = 8.85x10^-12. Multiply these and your product is 1.11x10^-17.Take the square root of that and you get 3.33x10^-9. Find the reciprocal of that and you arrive at 2.99x10⁸. This, of course, is the speed of light in vacuum. Light is a self-propagating wave of electromagnetic energy.

I have seen the light.

(There's a bit of cheating here. Those constants are now defined in relation to the speed of light, so of course the algebra works. But way back in the 19th century, ε₀ was a proportionality constant that described the ability of the vacuum to act as a capacitor, and µ₀ was a proportionality constant related to the magnetic force between two lengths of current-carrying wires a meter apart. And it was thought of as a rather interesting coincidence that putting those two constants together got you the speed of light. Maxwell set 'em all straight.)