Fact which aren’t.

I’ve learned a long time ago that in an electromagnetic wave electric and magnetic fields are perpendicular to each other (and to the direction of propagation):


But recently someone pointed out to me that this is not necessarily the case, and I did not believe them at first.

Here is this example:

E=E_0\cos(kz)\left(cos(\omega t)\hat{x}-\sin(\omega t)\hat{y}\right)

B=\frac{E_0}{c}\sin(kz)\left(cos(\omega t)\hat{x}-\sin(\omega t)\hat{y}\right)

(I may have screwed up the overall sign of the magnetic field. If so, pretend that this is a left-handed basis. 🙂 This is a circularly polarized electromagnetic wave “propagating” along z direction. The trick which leads to the electric and magnetic fields being parallel is that they are shifted one quarter wavelength spatially.

It took me some time to believe this, after confirming that the expressions above satisfy the vacuum Maxwell equations. My mind was blown. Something I took as a true fact inescapably following from the classical E&M turned out to be false. Even now I am not 100% sure, maybe I screwed up some place. I even googled it, just to feel more comfortable, and found the following paper:


Gray, J.E. Electromagnetic waves with E parallel to B. US Naval Surface Warfare Center, Dahlgren, VA, USA. 

I don’t have access to the full text of the paper (paywalls must die!), but it’s clear from the abstract that I am not the first person to be surprised by this.

I haven’t look into this in any depth, but shifting E and B fields relative to each other ought to require some rather special boundary conditions. I am not sure if it’s even possible to obtain by reflecting the standard electromagnetic wave from some special boundary.

But the point of this post is not about some unusual boundary conditions, but about how what I thought was true and would probably bet money on ended up being false. And given that one example is probably a good indication that there are others. Let me emphasize how unexpected this was. Now if someone tells me an example of an object accelerating away from another body due to their mutual gravitational attraction, I will no longer dismiss it as obvious nonsense.

There are still some “facts” which appear inviolate, however. Like, if someone claims to have constructed a perpetual motion engine out of cogs and chains, I will probably still not take them seriously.

But, to quote Tom De Marco’s The Deadline, “It’s not what you don’t know that kills you, it’s what you do know that isn’t s0.” And so I remain worried and curious, as to what other false facts I am absolutely sure of, to the degree where I don’t even see them as facts, but just the way things are. I ought to attempt to notice these transparent facts more often, and critically examine them. Maybe I’ll find something interesting.


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