Having covered most of what you need to know about classical physics, the traditional next step is to talk about electricity and magnetism, colloquially known as "E&M," though really, "E and B" would be more appropriate, as the fundamental quantities discussed are the electric field (symbol: E) and the magnetic field (symbol: B), whose effect is given by today's equation:
This is the "Lorentz force law," giving the force experienced by a particle with charge q moving at a velocity v through a region with both electric and magnetic fields. This is, in some sense, what defines those fields, at least from an operational point of view.
So, why is this important, and what does it tell us?
The first thing to notice is that the two fields have very different effects. The force due to the electric field is just a simple multiplication of E by the charge, while the magnetic field includes yet another cross product. That means the force on a moving charge due to a magnetic field is at right angles to both the field and the direction of the velocity.
That's a pretty strange result, if you think about it, and doesn't necessarily seem to fit with what we know about magnets and other objects. That's mostly because we don't often encounter charges moving in large enough magnetic fields for this to be obvious. It's absolutely true, though, and is about to be a big part of the background to a news story.
Tomorrow, when all the blogs are in a tizzy over the announcement that the LHC may or may not have found the Higgs boson, a big part of the way they'll know whether they found it or not will come from this equation. The force on a charged particle moving in a magnetic field acts at right angles to the field and the velocity, which means it tends to push the particle to the side, bending it into a curve. The radius of the curve depends on the charge, the mass, and the strength of the magnetic field.
This result makes particle physics possible. If you're going to be in the business of identifying exotic particles produced in an accelerator, you need a way to sort out their charges and masses so you can identify them, and the Lorentz force gives you a way of doing that. You apply a magnetic field in a known direction with a known strength, and let your unknown particles pass through it. Those with positive charge will bend one way, while those with negative charge will bend the other. Those with large masses will follow a track with a large radius, while those with small masses will follow a track with a smaller radius.
So, keeping track of what sort of paths the various particles follow is a crucial first step in determining what new particles have been created in a collision. So, when you wake up tomorrow and read a bunch of stories about the maybe-sorta-kinda detection of the Higgs (or not), take a moment to acknowledge the importance of the Lorentz force law.
And come back tomorrow for another equation of the season.
A magnetic field's ability to change the direction of charged particles is also an important reason why we have an atmosphere! http://en.wikipedia.org/wiki/Solar_wind#Magnetospheres