Friday, April 12, 2013

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.

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