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.)

Capacitors fully charged, Captain.

(I'm going to try to post about once a week from now on. We'll see.)

On Wednesday we did a fun lab that involved charging capacitors in an RC circuit. I only expect to hear about charging the capacitor banks in Star Trek and other such shows, so this was a neat experience.

The point of the lab was to observe the exponential decay of a discharging capacitor. A capacitor is a circuit element that takes the energy of a voltage source, such as a battery, and stores it for later use. The energy comes from moving charges, which are clumped together on capacitor plates. But like charges repel, so something needs to keep the charges in place. That something is another nearby capacitor plate where opposite charges are also being clumped.

If the space between the two plates is non-conducting, then the charges will feel an attraction to the other side but won't be able to do anything about, and this attraction balances the repulsion on their side. A capacitor has reached its, um, capacity when the voltage source is unable to overcome the repulsive force of the charges already on the plate. Each time you add an electron to the capacitor, it gets harder to add another one, because there's even more negative charge repelling the next electron. So every capacitor has a limit to the amount of charge per volt it can hold that is measured in farads.

Once the capacitor is fully charged, the energy is said to be stored in the electric field between the two capacitor plates. An analogous situation is a crane suspending a large hunk of steel in the air. The crane's engine supplies the energy necessary to lift the mass up, and the energy is then stored in the earth's gravitational field as potential energy. If the crane lets go of the steel, that potential energy is converted into kinetic energy and the steel slams into the ground. Similarly, if the capacitor is discharged, the electric potential transforms into kinetic energy, shooting the electrons out and establishing a current.

(In a further extension of this analogy, if a crane tries to lift something too heavy for too long, the cable will snap, damaging the whole setup. If too much voltage is applied to a capacitor, then the insulating material between the plates momentarily transforms into a conductor and charge shoots across, ruining the capacitor. This is lightning. When charge builds up during a storm, the air acts as an insulator between the clouds and the ground. Once the voltage gets too high, the air ionizes and creates a path for the charged electrons.)

What differentiates this from gravity is that the force pushing the charges off the plate is proportional to the number of charges on the plate in that instant. For the same reason that it gets harder and harder to add more charges onto a plate, the charges initially leave the plate very quickly. But then, as they leave, there are fewer repulsive charges in place, and the remaining ones leave more slowly.

When the rate at which some amount of stuff changes is proportional to the amount of stuff present, you have an exponential process. Radioactive decay is exponential because it depends on the amount of an isotope present in a sample. Population growth is ideally (that is, under perfect conditions, not as in I want it to be that way) exponential because the more people there are, the more children will be born.

The most recognizable feature of an exponential process is that the amount of time it takes for the process to double (or halve, or reach any specified multiple) is constant. This is where we get the idea of half-life. If some radioactive material has a half-life of a thousand years, and you've got 50 grams of it, then after a thousand years you will have 25 grams, but it takes another thousand years to get down to 12.5 grams.

Which leads us to this pretty graph:

As you can see, our capacitor discharged itself in a very cooperative exponential fashion. But it didn't cooperate fully. Every exponential decay has an associated "time constant" which is the amount of time that has to pass before the sample has reached 1/e of its original size. In an RC circuit, the time constant turns out to be the product of the circuit's resistance and capacitance. We had a 22 megaohm resistor and a 1 microfarad capacitor, which gave us a time constant of 22 seconds.

If you analyze the data we collected, however, our half-life turns out to be about 24 seconds. Now, there's a relationship between half-life and time constant, but right away it should be obvious that something is wrong. It absolutely has to take longer for the voltage to decay to 1/e (.368) its starting value than 1/2 its starting value. So if we accept that our calculated half-life is correct (which I'm willing to do, because it's such a pretty graph), then the time constant must be higher by about 13 seconds. For the time constant to be higher, there must be more total capacitance or resistance in the circuit.

But we have a fairly simple circuit that looks like this:

(We are fortunate that our circuit diagrams are not graded on artistic merit.)

When we close the switch, the battery charges the capacitor but is hampered by the resistor. The DMM is a digital multimeter which measures the voltage across the circuit. When we open the switch, the battery is no longer a part of the circuit, the electromotive force keeping the capacitor charged goes away, and the voltage discharges across the circuit. We measured this with the DMM.

Now, the wires have resistance, but not on the order of 12 or 13 megaohms. And it would be nearly impossible to add any capacitance to the system because capacitors in series have a lower net capacitance than that of each individual capacitor. Capacitors in parallel can add up, but we don't have anything in parallel when the switch is open. The only explanation is that the DMM has some internal resistance. As it turns out, the DMM is supposed to have an internal resistance of between 10 and 20 megaohms. So our 12 missing megaohms fit perfectly into that range. w00t.

As a final note, there was another section of the lab dealing with fast decay on the order of milliseconds. Because we can't measure that with our eyes, we used the oscilloscope to plot voltage versus time. As I explained before, the scope's display is a grid of squares. The width of each square corresponds to a length of time, the height to a voltage. How much time and voltage are calibrated with knobs. Anywho, we were measuring half-life, and we saw that the voltage decayed to half its original value across 1.5 squares. Our time knob was set to .5 milliseconds.

One of my partners will only take raw data, so he wrote down 1.5 * .5 milliseconds and left it at that, because he didn't want to make any math errors during the lab. My other partner whipped out his calculator and began typing 1.5*5x10^-4 into it. I looked at them both rather strangely and said "Guys, you just divide by two. It's .75 milliseconds." They looked at me like I was crazy and kept writing/calculating. Sigh.

Stand Back! I'm Doing Fake Science.

(Sorry, Randall Munroe.)

Okay. So it turns out that simultaneously working full time and going to class nearly full time is hard. Who knew? Then there's that short story I was working on. And theoretically there are friends who need reminders of my existence as well. Anyway, that doesn't leave a lot of time for blogging. But here I am, doing a post about motherfucking magnets.

How do they work? I don't know. Something about spin and charge. My guess is there's no good explanation for where magnetism comes from (by which I mean the magnetic field, not why magnets attract) until I get to quantum mechanics.

Anywho, we did a lab on Wednesday to demonstrate some of the principles of magnetism. There was a neat section of the lab where we played with solenoids, magnets, and batteries to see how a changing magnetic field induces a current and all that jazz. But the main part of the lab was playing with an e/m apparatus, which looks something like this:

Yes, it was really that awesome. So, the two parallel coils of wire are known as a Helmholtz Coil and they produce a nearly uniform magnetic field between them. The bulb in the middle houses an electron gun and is filled with helium. The contraption to the right (we had a different one) is just a power supply that can vary its voltage and amperage.

This setup is apparently a pretty classic lab and is supposed to mirror a series of experiments done by J. J. Thomson when he discovered the electron and the ratio of its mass to its charge. The electrons in this experiment are visible as that blue-ish ring in the bulb. The blue itself is just the helium being excited by fast moving electrons. And the electrons are moving in a circle because, well, because magnetism is a weird force.

If you'll allow me to demonstrate with a crappy MS Paint illustration. The magnetic field only works on charges that are in motion relative to the field. And, unlike more traditional forces that just push or pull, the magnetic field creates a force that is perpendicular to both the velocity of the charge and the direction of the magnetic field (this is known as the cross product in vector notation).

If we imagine that this is a box, and you have a charged particle (blue) moving west, and a magnetic field (purple) directed north, then the resultant force on the charge (if it's negative) will be up. But our electron is moving in a circle. The reason why is that as soon as the electron changes direction due to the magnetic field, the force acting on it also points in a slightly new direction so that it's still perpendicular to the electron's motion. The end result is that the electron experiences a force that is everywhere at right angles to its motion, and this gives rise to the equations for circular motion.

So, in the experiment above, the magnetic field is directed in a horizontal line between the two coils, and the electron gun is aimed down. The cross product of that is a fuzzy blue circle.

How big the circle is depends on four quantities: the strength of the magnetic field, the voltage applied to the electrons, and the mass and charge of the electrons. So, if you know those first two, you can measure the ratio of charge to mass. You can't separate the two variables, however, because there's just the one equation. Robert Millikan eventually found a way to measure the charge of the electron alone.

But today, scientists know both values to incredible precision, and our job wasn't to figure out that ratio. Rather, we were trying to measure the strength of the magnetic field according to the radius of the electron circle and its speed. The relevant equation is r = mv/qB, which tell us that the faster the electron is moving (which depends on the voltage of the electron gun), the stronger the magnetic field (B) has to be to keep it at a constant radius.

We can also predict the value of the magnetic field based on the geometry of the Helmholtz Coil and the current it receives. Which leads to this graph:

We varied the strength of the current between 1.6 and 2.9 amps, which resulted in a magnetic field of between 1.30 and 2.34 milliteslas. But, as you can see, those values are all slightly higher than the predicted values for a given amperage. The average discrepancy is small, about 3.5%, which amounts to .06 mT. What could be causing this systematic error? A variety of things, really. It could be an inaccurate reporting of the current strength, or an imprecise model of the magnetic field, or really any number of things.

Or it could be the Earth's magnetic field, which a little googling tells me is between .03 and .06 mT. Science: It works, bitches.

Knot Another Punny Title

So, this is more of an update post than anything else. Work and school have been pretty busy. I'm also writing (well, editing at this point) a short story, which is where the title of my post comes from. How's that work?

For reasons that escape me, I'm writing a short story that heavily features a branch of mathematics known as knot theory. Wait, there's a branch of mathematics about knots? Why yes, yes there is.

Math is about numbers, duh, but it's also about geometry. Many of the ancient Greek mathematicians did math purely through geometry, in fact, because they didn't have algebra to represent general forms or calculus to deal with infinitesimals. Anywho, there was a point in the 19th century at which studying geometry morphed into studying surfaces, and this led to ideas such as differential geometry and topology. Differential geometry is the math behind Einstein's general theory of relativity, and topology tells us that doughnuts and coffee mugs are the same thing.

Doughnuts and coffee mugs are the same thing? Apparently, yes, because in topology, objects are homeomorphic if you can transform (stretch, squeeze, rotate, twist) one into the other without creating any new shapes or holes. There's a little gif on wikipedia showing that this isn't quite as crazy as it sounds.

Now, there's a sub-branch of topology known as knot theory, which studies circles embedded in space (or spheres embedded in 4-space, etc.). It turns out that some circles are just circles (called the unknot), whereas others are true knots that cannot be transformed back into circles without cutting the knot. Take a rubber band, for example. It's just a circle.

No matter how many times you twist it around and tie it into knots, it's still just a circle that can (theoretically) be undone.

But if you cut the rubber band...

...tie it into a knot, and then reconnect the severed ends (and pretend it's not being held together with tape), then you've created a true knot that cannot be transformed back into the unknot.

You can take that one new knot and twist it all around into different-looking knots, but knot theory says that, as long as you don't cut the rubber band again, it is still fundamentally the same knot, the same way a coffee mug and a doughnut can be fundamentally the same topological space.

So, how'd I write a short story about that? Well, it turns out knot theory has uses outside of rubber bands. In fact, Kelvin kind of got some of the credit for starting this whole knot theory business when he hypothesized that atoms were just knots (vortices) in the aether. But then Michelson and Morley kind of threw a wrench in that whole thing.

There are some modern applications of knots, however. DNA gets itself tied up into knots, and knot theory can help explain how it undoes those knots. Quantum field theory can also be described in a knot-like way, and there might also be quantum computers based on knots.

My story extrapolates this all well into the realm of science fiction, to the point that it would probably piss off the 3 mathematicians who study knots and read short science fiction. But I think it goes in interesting directions. I tie in (ha, not intentional) notions of Buddhist and Celtic endless knots, the Gordian knot legend, and the knot-based number system used by the Inca, known as Quipu. Fun stuff, I hope. I currently have two beta readers attempting to determine whether or not it's ridiculously boring. We'll see.

(Hm. I guess that was a little more than just an update.)

The Gravity Thief

(I'm reading The Quantum Thief right now, so I figure vague, sciencey-sounding titles are appropriate.)

My good friend (read: successful and well-liked blogger on whose posts I sometimes leave inane comments) Phil Plait recently blogged about the impending non-Armageddon that is 2012 DA14. The brief summary is that a giant hunk of rock half the size of a football field will just barely miss us this Saturday, and astronomers are so excited they're polishing their mirrors in anticipation. For a slightly more nuanced explanation, check out the Bad Astronomer's blog.

What intrigued me about Plait's post is the minor detail that, currently, 2012 DA14 has an orbital period of 366 days, but after its interaction with the Earth its orbital period will shrink to 317 days. This means, obviously, that the asteroid will go around the sun more quickly from now on. But why? Is it speeding up, or just getting closer to the sun? The answer is yes.

Four hundred years ago, after pouring through Tycho Brahe's enormously detailed astronomical records, Johannes Kepler devised three empirical laws of planetary motion. The relevant one here is the third law, which states that the square of a body's orbital period is proportional to the cube of its semi-major axis (radius). So the closer a planet is to the sun, the shorter its period.

Newton later confirmed Kepler's observation with his universal law of gravitation, which states that the force between two objects is proportional to the product of their masses divided by the square of the distance between them. So, then, if an orbiting object's period decreases, its radius does also, and if the radius decreases, its speed increases.

We know from the kinetic energy formula that a faster object has more energy, which means that 2012 DA14 will gain energy from its interaction with the Earth. How much energy? Well, using Kepler's third law, we can see that if the orbital period shrinks by 49 days, the semi-major axis will shrink by about 9%.

With a bit of calculus, we can figure out that Newton's law of universal gravitation predicts that the gravitational potential energy of an object is inversely proportional to the distance between it and another object. So, a 9% reduction in radius is a 10% increase in energy. (Because gravitational potential energy is defined to be negative, this can be thought of as gaining negative potential energy, or losing positive potential energy. The end result, however, is an increase in kinetic energy).

And energy, as we know, is conserved. So if the asteroid is gaining energy from its encounter with the Earth, the Earth must be losing energy. It looks as if the asteroid has a mass of about 130 million kg, which is less than the mass of the Earth by a factor of 46 quadrillion. Thus, a 10% increase in energy for the asteroid is a 2.4x10^-17 % decrease for the Earth. And if we go backward through the math, we see that the Earth's orbital period will increase by about 1 nanosecond. Put another way, after 28 years, an extra 1 second will have elapsed when compared to the previous 28 years (all else being equal).

The all else being equal part, however, is quite a stickler. The duration of the Earth's orbital period and day vary quite significantly due to effects from the moon, the sun, earthquakes, glaciers, and a whole range of other factors. It's likely that more than just the asteroid's speed changes during its near-Earth encounter, and the same goes for the Earth. Rather than our orbit changing, it might alter the length of the day by a tiny fraction.

But there is an underlying principle here: energy conservation. Energy never disappears completely; it just moves from one object to another, changing forms as it does so.

Don't turn that dial!

Okay, so this is a little late, but I want to talk about my physics lab from last Wednesday. I haven't had a lab course since I took Chemistry my junior year of high school, and that was eleven years ago. Working with a lab partner, writing down results, adjusting the apparatus -- these are things I (until recently) thought I was probably done with. Since I'm trying to become a scientist, however, I may have to get used to this routine.

Anyway, the lab was split into two parts. The first part had me and a partner measuring the spring constant of a spring by pulling a force gauge attached to said spring. After recording the force from the gauge and the distance we'd stretched the spring, we could find the slope of a Force vs. Distance graph, and that slope was the spring constant in newtons/meter -- the higher the value, the stronger the spring.

The exercise was decidedly unrelated to electricity and magnetism (unless you want to talk about the fact that chemical bonds are electric in nature). I think the point was just to acquaint us with doing a lab, which I suppose is necessary given that some of the students haven't taken a lab course in eleven years.

After that, however, the professor had each of us individually play with an analog oscilloscope. We weren't attempting to measure anything with the oscilloscope -- just push buttons and turn dials so that we could familiarize ourselves with its function.

For those unfamiliar with an oscilloscope (I had heard of but never seen one before last Wednesday), it's a device that measures an incoming electrical signal and displays a corresponding Volts vs. Seconds graph.

Here's a picture of the model we were using:

Manipulating and calibrating the o-scope (as the prof called it) is a somewhat tricky enterprise. The display is divided into a grid, and you can adjust the number of volts and seconds per length of grid. As you can see, however, there are a wide variety of other knobs and controls that are required to produce a clean image of the incoming signal. And throughout this whole process, I was wondering, is this really necessary?

I mean, perhaps I'm putting the cart before the horse, but this is 2013; shouldn't there be an iPhone app that can do this for me? (Yes.) In fact, isn't that exactly what any piece of audio equipment does when it translates electric signals into sound -- measure the frequency and amplitude of the wave?

Yeah, I'm in an introductory E&M course. Yeah, we're going over the basics. But is learning to fine-tune an o-scope like a WW2 radio operator really going to be useful in our later physics careers?

I can imagine two ways in which it might be. The first possibility is that while of course there will be software that can identify the sinusoidal wave (or whatever) of an electric signal when we're working scientists, someone has to write that software. In that case, having an intuitive feel for what we're attempting to measure might be useful.

The other possibility I can imagine is that a working scientist will have to play with significantly more complex pieces of machinery, and training on a somewhat antiquated oscilloscope is a necessary first step toward that goal. I mean, you can't start with the LHC, right?

I thought of both these counter-arguments to my original complaint, but I don't know how much stock I put in either of them. Unless I'm actually going to use an oscilloscope as a scientist, isn't there something that more closely resembles a piece of modern scientific equipment that we could be using?

According to wiki:

Oscilloscopes are used in the sciences, medicine, engineering, and telecommunications industry. General-purpose instruments are used for maintenance of electronic equipment and laboratory work. Special-purpose oscilloscopes may be used for such purposes as analyzing an automotive ignition system, or to display the waveform of the heartbeat as an electrocardiogram.

Ah, I did not know that EKGs were essentially specialized o-scopes. That's neat! But I'm a little hazy on what "maintenance of electronic equipment and laboratory work" means for general-purpose o-scopes. A little more reading tells me that they can be used to test the changing voltage of a device to make sure it's working within expected parameters. That's obviously useful, but again I feel as if this is something that could be done by a piece of software. Do we really need a human to turn a dial until the display stops spazzing, or can we just use a do...while loop?

Anywho, I think that's enough griping for now. Perhaps I should be a theorist.

The Community College Cafeteria

Where timeless wisdom and crass marketing come together.

(I'll probably post something substantive later today.)