Nonradiating orbital motions!

You know what I haven’t talked about much lately? My own research! Well, today is a great day for it, because a paper I wrote with my student Ray Abney just came out in Physical Review A, titled “Nonradiating orbital motions.” It’s actually invisibility-related, and I thought I would say just a few words (and pictures) of what it’s all about!

So, one of the earliest physical phenomena studied that can be connected to invisibility is known today as a nonradiating source. The oxymoronic name refers to a source of electromagnetic radiation (or more generally other types of waves) that, in fact, does not produce any radiation at all.

This is counterintuitive because the mathematical formulas that describe electricity, magnetism and light, called Maxwell’s equations after their discoverer, predict that an oscillating electrical current will produce electromagnetic waves. All of our wireless communications technology is based on that principle; when you see a radio antenna, such as the mast radiator pictured below, you are looking at a metal structure that has an oscillating electrical current driven through it to produce radio waves.

Image of an AM radio tower in Chapel Hill, North Carolina, via Wikipedia.

Your cell phones produce a signal in a similar manner; they have an antenna for broadcasting and receiving. Apple infamously ran into trouble with the iPhone 4 when they changed the antenna design and put it around the edge of the case, causing dropped calls when people held the phone “wrong!”

If charges are accelerated more strongly, they can produce higher-energy electromagnetic waves such as X-rays. The Advanced Photon Source at Argonne National Laboratory takes advantage of this and sends electrons around an 1,100 meter ring at nearly the speed of light; the circular path of the electrons causes them to constantly shed X-rays that can be used for basic and applied research.

Night view of the Advanced Photon Source, via Wikipedia.

So it is widely known and assumed that accelerated electric charges produce electromagnetic radiation of some form. But this is not always the case!

Physicists have known about the relationship between accelerating charges and electromagnetic radiation since Maxwell first introduced his equations in the 1860s. But in the early 1900s, this became a problem: atomic researchers knew that atoms possessed electrons, and that these electrons must be moving around in some way in the atom, but those moving electrons didn’t produce radiation!

One remarkable explanation was proposed by the Austrian theoretical physicist Paul Ehrenfest. In 1910, he published a paper where he demonstrated theoretically that it is possible to have an extended distribution of electric charges that oscillate in such a way that they produce no radiation at all! (I blogged about this remarkable paper waaaay back when my blog first began.)

When I say an “extended” distribution of charges, I mean a collection of charges that is stretched over a finite surface, or distributed through a volume of space. The radio antenna I showed earlier essentially has charges moving along a line, and the electrons moving in a circular path at the APS are essentially point-like. Ehrenfest considered examples like a vibrating plane of electric charge and a pulsating sphere, and showed that these examples produced no radiation. All the electric and magnetic fields produced by the accelerating charges are trapped in the region of the charges themselves. The entire structure later became known as a nonradiating source.

Much later, this phenomenon was explained as a very unusual and previously unrecognized form of destructive interference. When two or more waves combine, they can enhance each other or cancel each other out, depending on whether they are “waving” in the same direction or in opposite directions at the point they meet. Ehrenfest has discovered a new type of destructive interference, where the waves from all the charges in the oscillating distribution combine to cancel each other out everywhere outside the distribution.

It should be noted that this is not a common situation. One has to design a source structure very carefully to make it nonradiating. I still recall my PhD advisor wondering if there was any sort of physics that causes nonradiating sources to spontaneously appear in nature, and we never found any.

Ehrenfest’s radiationless motions didn’t draw much attention, because the same year he introduced them, Rutherford and his assistants discovered the atomic nucleus, thus providing a new direction for atomic research. Then, only a few years after that, it became recognized that new physics was needed to describe the structure of the atom, what of course became known as quantum physics, and Ehrenfest’s radiationless motions became irrelevant.

Since then, nonradiating sources have popped up again and again in physics, with various authors rediscovering them and wondering about their significance. I talk about this a lot in my book on invisibility, so I won’t dwell on the history too much, but one classic and noteworthy paper was published by G.A. Schott in 1933 (I have also blogged about this one). In a paper titled, “The electromagnetic field of a moving uniformly and rigidly electrified sphere and its radiationless orbits,” Schott noted that a spherical shell of electric charge of radius a, set into periodic motion with a time period T, would be nonradiating as long as cT = 2a/m, where c is the speed of light and m is a positive integer. The incredible thing about Schott’s result is that the specific shape of the path the sphere takes doesn’t matter — it could be a circle, an ellipse, a rounded rectangle, as long as it has a period T.

I read this paper back as a PhD student and was amazed by the beauty of Schott’s result. However, I noticed that the title is a bit misleading: Schott refers to the motion of the sphere as an “orbit,” but the nonradiating condition I gave above indicates that the length of the path must be equal to or smaller than the diameter of the sphere. When we think of an “orbit,” we really think of something like the Earth orbiting the Sun, or the Moon orbiting the Earth, where the orbiting object traces a path much larger than the object itself.

The way I’ve described Schott’s result in talks ever since is that the sphere is really doing a “wobble”:

more than an “orbit:”

In fact, this is true of basically every nonradiating source result that has been published over the years, from Ehrenfest’s paper to the present. Every nonradiating source demonstrated theoretically to date has either been a bunch of electric charges oscillating in place, or a “wobbly” set of electric charges that are moving but moving over such a small distance that they look effectively like they are oscillating in place.

This led me to ask the question: is it possible to make a nonradiating source where a rigid distribution of electric charges orbit around a central point? I pondered this question for years, and then a couple of years ago I tasked my student Ray to do some math to test it.

Without going into all the mathematical detail, we showed, theoretically, that it is possible to create a “nonradiating orbital motion.” An illustration of one of the results from the paper is shown below.

Figure 1 from “Nonradiating orbital motions.”

These figures show the waves produced by two different nonradiating sources with different structures at two different times. If we were to “start the clock,” we would see these waves circle around the center of the figures without changing their shape. The phenomenon is quite general, and we can create nonradiating sources that orbit with an arbitrary radius, and we also have great freedom in designing the structure of the sources.

Our nonradiating orbital motions were created by first designing the orbiting field we wanted to see, and then determining the source that would produce this field through use of the wave equation. We also confirmed that the result worked by going in reverse: starting with the orbiting source, we numerically calculated the field produced and showed that it was nonradiating.

Our results show that the nonradiating phenomenon is much more general than is typically appreciated; this is quite timely, as nonradiating sources have recently had a bit of a revival of interest in the optics community. Of particular interest are structures called anapoles, which have both an electric and magnetic response that can cancel each other out; anapoles are even being considered as candidates for dark matter (though I have no expertise to judge how seriously they are being considered).

It is also fascinating to think that this result suggests that, in principle, it is possible to make a synchrotron source (like the Advanced Photon Source) where charged particles orbit but don’t produce radiation! This seems unlikely to be feasible in practice, since it would require near perfect control of the structure of the packets of orbiting particles, but shows that classical physics still can produce some unexpected surprises.

Our nonradiating orbital motions are also a good illustration of how, if one studies a problem in depth, one can find questions to ask that nobody has ever thought of!

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2 Responses to Nonradiating orbital motions!

  1. Very interesting! Thinking this has more in common with some ultra-low temperature physics, so I know I’m way off-target here. But this reminds me of research into self-confining aluminum-plasma toroids in the 90s (SDI)… essentially stable (milliseconds?), MJ-energy, magneto-hydrodynamic “smoke-rings” accelerated to a few percent of c, the idea being that they could be used as (relatively) non-radiating, electromagnetic projectiles.

  2. EB says:

    I was recently listening to a podcast about charged black holes. When black holes accelerate, they generate gravitational waves. But would accelerating charged black holes also radiate electromagnetic waves? Or would they be a different kind of nonradiating source? Is there a nonradiating analog for the gravitational waves produced by accelerating masses?

    BTW, great picture of the APS! I still fondly remember biking around the ring to go and pick up dinner from the food cart before a long night of data collection.

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