I love teaching undergraduate electromagnetism. It has such an elegance, logic, and completeness about it. It introduces such a host of powerful techniques and ideas to the student, taking them across the threshold into maturity in their physics studies: Once you’ve done electromagnetism, you don’t usually think about large chunks of Physics in quite the same way ever again.
Today saw me give one of my favourite lectures, in any subject. It’s always a thrill. Summarize all that has gone before in their studies of electromagnetism – Gauss’ Law for the electric field produced by charges, the statement of the absence of magnetic monopoles (the Law with No Name), Faraday’s Law for the electric field produced by changing magnetic fields (induction:- another really fun set of lectures there), and Ampère’s Law for the magnetic fields produced by a current density. Write them all down next to each other and stare at them. Realize that they are not internally consistent, in general, as Maxwell did (he was motivated slightly differently, but in an essentially equivalent way). The culprit is Ampère, and the problem is fixed by Maxwell’s realization of the existence of the displacement current term. Ah… symmetry. Changing electric fields produce magnetic fields. All is well. Do some nice examples to show how it all works in concrete terms.
The resulting beautiful and consistent set of equations sent a shiver down my spine when I first saw and appreciated them as an undergraduate. They still send a shiver down my spine, and I hope your spine shivers too. Maxwell’s equations:
[tex]
\begin{eqnarray}
\nabla\cdot \mathbf{E}\, &=&\,\frac{\rho}{\epsilon_0} \ ;\nonumber \\
\nabla\cdot \mathbf{B}\, &=&\, 0 \ ;\nonumber \\
\nabla\times\mathbf{E}\, &=&\, -\frac{\partial \mathbf{B}}{\partial t}\ ;\nonumber \\
\nabla\times\mathbf{B}\, &=&\, \mu_0\mathbf{J} +\mu_0\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\ .\nonumber
\end{eqnarray}
[/tex]
After the shiver, a lovely warm feeling. From here to light, radiation, Relativity, and beyond…
Every time, I especially love giving this lecture. It never gets old.
-cvj
From the old days: Bleaney and Bleaney – physically motivated, and the book Hawking learnt his E&M from (it features in a BBC documentary on the lad, in which two of his contemporaries recount how he banged off all the exercises in one chapter in less than half an hour, while they slaved away for an entire week -as I recall it was chapter 8 if you think you’re hard enough). Panofsky and Phillips – just as good, but with a trans-Atlantic vibe. Stratton: just excellent. And the Master himself: ‘Classical Electrodynamics’ by Schwinger, DeRaad, Milton and Tsai. What is there not to like? I would also second pgm’s censure of CB’s ill-considered remark – typical of those that got hep-th a bad name. (Gell-Mann’s gibes about squalid atate set the tone, and things have not really got any better since.)
Carl Brannen wrote:
E&M felt more like … [a] condensed matter theory than something that said much about nature.
*uncomfortable cough* … egad … oh dear…that’s not very nice, now is it? If you’re looking for theories that “say much about nature”, I would recommend you give condensed matter physics a closer look.
Thomas, I left them out for a reason or two: (1) That’s another lecture, not the one I was talking about. (2) Nobody’s found any monopoles yet, as far as I know. This is not really a class intended to spend too much time on the more speculative aspects of physics. It will get a mention on the grounds of symmetry, and I’ve already mentioned some amusing things about charge quantization, but that’s not really for this class.
Cheers,
-cvj
and I meant “E-field curl”, not divergence, of course. Okay, NOW I’m definitely going to go to sleep.
Oops and I guess you’re “Professor Johnson”, not “Professor Clifford”. Apologies, I blame the jet lag (I moved forward +15 hours today!).
“left out”
Professor Clifford – a minor quibble, you left the contribution to B-field divergence from monopoles, and the E-field divergence term for monopole currents.
🙂
Enjoy Griffiths’ super-happy-fun textbook! I loved it back in undergrad EM, and I love it more than ever in grad EM.
Candace:- Enjoy!
-cvj
Griffith has been purchased — thanks, Clifford and Amara.
“… the problem is fixed by Maxwell’s realization of the existence of the displacement current term.”
Maxwell’s original papers on this from 1861 are online at http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf
It’s all about “theory of molecular vortices” and pressures in the aether; see page 38 (on your PDF reader) for Maxwell’s graphical illustration of the “displacement current” mechanism. Page 49 on the PDF reader (labelled page 22 on the document) gives Maxwell’s claim to have predicted the velocity of light; using the formula for transverse waves in an elastic solid he gets the right answer and immediately declares in his own italics:
“… we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”
That 1861 quotation is quote common, but Maxwell rebuilt the mathematics of the theory in 1965 in a more abstract way and that averts some of the physically suspect theorizing (aetherial gear cogs and idler wheels in space) that the 1861 papers contains. Nowadays, the mathematics is all that is important.
It is curious that a lot of Maxwell’s problems seem related to the Dirac sea issues of quantum field theory, for example if the vacuum is polarizable without any limits, it would be able to polarize around an charge just enough to completely cancel it’s electric field completely.
This is clearly one reason for assuming lower limit electric field strength for polarization in quantum electrodynamics. What amazes me from studying lecture notes like http://arxiv.org/abs/hep-th/0510040 is that there are no vacuum loop effects beyond 1 fm from a fermion. The field below the IR cutoff is qualitatively different and more classical in nature than the close-in field which is affected by loops of charges appearing and annihilating. This isn’t made clear in popular accounts, which always seem to say that the vacuum is filled with pairs appearing and annihilating everywhere. Clearly the IR cutoff, requiring an intense field strength or frequency to create loops, prohibits any polarization of the vacuum and thus “displacement current” in radio or light waves, which don’t contain strong enough fields to polarize the vacuum.
Instead, a time-varying electric field is a curling magnetic field (4th Maxwell equation) just as a moving electric field (relative to observer!) is a straight magnetic field; the Lorentz force law, which assumes E = v x B, is literally to be interpreted as saying that “an magnetic field is a moving electric field”. (Maxwell’s detailed mechanism by which the electric field polarizes vacuum charge inducing “displacement current”, to create a magnetic field, is just complex and superfluous. You simply don’t need moving charge to create a curling magnetic field, because a varying electric field will do the job directly.)
This is a reversion from Maxwell’s aether to Faraday’s 1846 paper Thoughts on Ray Vibrations which argued how that oscillations of electric and magnetic fields could constitute light.
Few realize how fundamentally important this is to sound reinforcement/sound reproduction. Hell, i worked in the business for decades before i even knew it. Then, a dozen years ago, i had the chance to spend a couple of months with one of the finest audiophile, analog-system engineers in the world, rebuilding a complete sound system for live performance. He would spend hours with these equations recalibrating cross-over frequencies and signal voltages etc., to bring out the very best, and most accurate, signal from the speakers’ electro-magnets, without the dreaded noise. Next time you go to a concert and flinch at hiss and noise and overburdened equalization, consider that the folks who brought you that sound system, didn’t know their physics.
I second the book by Griffith..!
Candace:- A very popular and quite good book over here is the one by Griffith: “Electricity and Magnetism”. In my Imperial College days, we used Duffin. Also good. Griffith might be a bit more stripped down and to the point. Better for a first pass, perhaps…
-cvj
I am going to try and take it take it to a “whole…nother…..level.” 🙂
Maxwell’s equations are just so elegant. I had a prof who taught Electrodyn the deductive way, writing down the equations and then examining them. It maybe wasn’t so didactically but surely impressive. I had a T-shirt with Maxwell’s equations on it. A friend of mine as well. I recall his mother asked him whether that’s Russian on his shirt 😉 (no kidding). I was even more impressed when I learned what to do with differential forms! Suddenly all the in dices were gone, and there we were ddA=0!
So what E&M books do y’all recommend? I’m taking 2nd year E&M this term and the lecturer is pretty good but we don’t really have a text at our level, ie more mathy than a general physics book (serway for us) but not rigorous enough. Grant and Phillips has also been mentioned…but not recommended highly. Any suggestions?
Since I was a math major, I never took undergraduate E&M, just the graduate class using JD Jackson. So I never got the E&M epiphany. As a graduate student interested in particle theory guy, E&M felt more like engineering or condensed matter theory than something that said much about nature.
The story of physics is that one can go very far with linear or first order approximations, but nature herself is highly nonlinear. If QM were linear, twice a state vector would correspond to a state with twice the number of particles or twice of something. E&M is truly linear (in vacuum), and that shows that it is that much farther away from nature. It is a simplification, a fable, a story for undergraduates, an approximation that works only for very tenuous situations.
And I should mention that in the density matrix / operator formalism of quantum mechanics, the fundamental equation is the simplest non linear equation, [tex]\rho^2 = \rho[/tex]. This expresses perfectly the difference between the easy linearity of E&M and the difficult non linearity of the real world.
That’s a variation on the one that was around when I was a young ‘un: The T-shirt would have “And God said…” then there’d be Maxwell’s equations, followed by “…and then there was light”. I wonder if people still make/print those?
Good Luck with the E&M fun-ness!
-cvj
My dad’s an EE, and his company makes these shirts- http://www.coolmagnetman.com/images/maxwell1.jpg I grew up with that being my dad’s favorite, and have one of my own but it’s too big. 🙁
And the first half of a year of E&M fun-ness begins for me in a few weeks. I guess I have a few EE genes in me, as I can’t wait!
Funnily, Maxwell’s equations were one of the great sticking points about electromagnetism for me. As far as I was concerned, the action was all in the 4-potential, so why did I have to keep taking these derivatives? And as Feynman pointed out, E and B aren’t the real fields anyway.
Then recently I found Carver Mead’s ‘Collective Electrodynamics,’ which works through this point of view. It’s really a big improvement.
Thanks!
-cvj
I love your enthusiasm! (Not all professors are as earnest when it comes to teaching…)
Yes…. that was the origin of my parenthetical remark about induction and Faraday….. wonderful isn’t it?
Cheers,
-cvj
There is also the bit before that, arguing that changing magnetic field has to induce electric field. Moving a conducting loop through a magnetic field generates a current, but moving the magnet instead would not, unless the equations are modified. Ties nicely to relativity, and the fact that Lorenz force and Maxwell equations are not independent.