One of the blogs I hyped at the science blogging panel at Worldcon was Built on Facts, Matt Springer's blog explaining introductory physics concepts. You might not think that you want to read a blog that goes through freshman physics problems in detail-- I would've been dubious on the concept, had you explained it to me that way-- but it's really excellent stuff.
He's recently completed a series of posts on Maxwell's Equations, with one post for each of the four equations, plus one bringing them all together: Gauss's Law for Magnetism, Gauss's Law for Electricity, Faraday's Law, and the Ampere-Maxwell Law (the only one Maxwell added anything to), and Maxwell's Equations & Light. These posts are excellent, concise, readable introductions to these equations and why they matter. Somebody get this kid a textbook-writing contract.
With those posts as a resource, this is an excellent time for a Dorky Poll:
I know they're all lovely equations, but you really have to choose only one of them to be your favorite. Unless you have multiple computers, and vote multiple times.
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Lots of people I know work with magnetohydrodynamics (MHD), which is a sort of a combination of fluid mechanics and electrodynamics. Usually they neglect the displacement current, and most of the time this is a reasonable approximation. But the displacement current does have an identifiable physical effect: it constrains the group velocity of hydromagnetic (Alfvén) waves to be less than the speed of light. There are some astrophysical applications (the aurora borealis among them) where that term becomes important.
Displacement current also has a dark side. Some modelers include a displacement current term with an artificially low speed of light as a cheap way of making sure their code meets the Courant stability criterion (waves cannot propagate more than one spatial grid point in one time step). This practice goes by the name "Boris correction". See J. P. Boris (1970), "A physically motivated solution to the Alfven problem", Technical Report 2167, Naval Research Laboratory, Washington, DC. [cited in J. G. Lyon et al. (2004), J. Atmos. Solar Terr. Phys. 66, 1333].
Would have been easier to respond if you'd written out which one is which. I was never good at remembering which one was Faraday's Law and which was Gauss's Law, etc. I just remember that I like the one that equals zero!
How can you have a favourite? All four are necessary to understand electromagnetics.
Dave
Maxwell's equations were probably the first thing that I learned in physics that I thought were really cool so they have a special place in my heart. I have to say I like the differential form versions (without sources) the best:
dF=0
d*F=0
That would be Guass's Law for Magnetic Fields.
If I have to choose one, it has to be the Ampere-Maxwell law: first, as a theorist, because it was found (Maxwell's term) by a theoretical argument, just what we're trying to do with quantum gravity today. Second, because its discovery revealed the nature of light, one of the great moments in scientific history.
Hahaha! Really, though, what are the odds of that being true for any of your audience!
âF = J, as per http://scienceblogs.com/catdynamics/2009/08/maxwells_equations.php#comm…
Differential Geometry FTW.
Thank you for the kind words!
For the poll though I'm going to cheat and bit and say my favorite is the one where you can compact all four into one equation, assuming you're willing to allow tensors and restrict yourself to the Lorenz gauge.
If cheating isn't allowed, Gauss' Law.
Most people here are thinking about Heaviside's Equations, if they are considering 4 equations with differential operators. Maxwell (whom I tremendously admire) had a bigger set of equations, in coordinates. In any case, there are higher order terms in electrodynamics, so what what most people here are thinking about is truncated at 2nd derivatives. The differences matter if one tries to provide a complete model of the moving electron (which was tried and failed for a century), for high velocity (i.e. Alven waves in colliding interstellar plasma clouds having near-C velocity and thus a new type of shock waves emergent), in extreme conditions such as near supernova, charged black hole, Big Bang. Unified Field Theory is, in a sense, an attempt to modify Maxwell's Equations for curved space.
Ampere-maxwell, because that is were life got complicated enough for me to begin thinking very hard. Sigh, no more electrodynamics.