Conventional wisdom, even to this day, dictates that accelerating charges necessarily give off electromagnetic radiation. This is seen, for instance, in large-scale particle accelerators (synchrotrons), such as the Tevatron at Fermilab and the soon-to-be-operational LHC at CERN: the charged particles moving around the ring are constantly shedding radiation over a range of frequencies, including X-rays.

In the first post in my series on the physics of invisibility, however, we discussed a little-known 1910 paper by Paul Ehrenfest, in which he demonstrates theoretically that one can have accelerating *extended* distributions of charge which produce no radiation fields. Ehrenfest was attempting to explain one of the most vexing problems of physics at the time: the presence of electrons in the atom. The atom was known to have electrons moving about within it, and these electrons should have been radiating constantly, according to the known physics of the time, but were not seen to do so.

Soon after Ehrenfest's paper, Bohr produced his model of the atom, which eliminated the need for radiationless orbits and ended most speculation on atomic structure. Ehrenfest's work was mostly forgotten, but other researchers independently discovered other radiationless motions of charges, and this research would lead eventually to more detailed studies of invisibility. One of the most important researchers on radiationless motions was G.A. Schott, who in 1933 produced a beautiful and amazing theoretical result* which we discuss in this post.

We've mentioned Schott before, in our discussion of failed atomic models in the period 1903-1913. Schott had volunteered his own speculations on the origins of atomic structure, but they were discarded in the wake of Bohr's successful explanation. Schott, however, was evidently not convinced. He remained what we might call one of the last respectable 'anti-quantum' scientists, and he continued to seek a stable model for the atom which could be explained purely through classical mechanics and Maxwell's electromagnetic theory.

I've quoted Schott's obituary before, but it's worth doing so again, as it is directly relevant to his invisibility work (Obituary Notices of Fellows of the Royal Society, Vol. 2, No. 7, (Jan., 1939), pp. 451-454):

The first serious attack on the classical theory was made by Bohr in 1913. Although Planck’s constant appeared in the theory of light in 1900, Bohr was the first to use it to give an electrical explanation of line spectra. There were two violent departures from the classical theory. Firstly, the existence of radiationless electronic orbits, secondly the expression for the change in energy. Schott devoted the most of the remainder of his life attempting to fit both those in the classical theory… The mathematical difficulties are enormous, and the skill showed in getting numerical results shows Schott’s mastery at its highest. It might be called the supreme attack of a heroic defender before his death. Defeated? Who shall say? I like to think that in future years the work of Schott will always be consulted for inspiration to tackle the difficulties which come across the path of all theories.

So what did Schott achieve? He demonstrated mathematically that *a uniformly charged spherical shell of radius $latex a$, undergoing periodic motion of period $latex T$, will not produce radiation if the following condition is satisfied:*

$latex a = m c T/2$,

*where $latex m$ is an integer (1,2,3,...), and $latex c$ is the speed of light*. For those unfamiliar, the period $latex T$ refers to the amount of time it takes for the sphere to finish one full period and return to its starting position. A slowed-down movie of Schott's sphere in motion might look something like this:

This result is astounding for a number of reasons. First, the result is *completely independent* of the specific path of motion: the sphere can be moved in a circular path, an elliptical path, a square path, or any other shape, as long as the sphere repeats the pattern in the time $latex T$. Second, unlike the simple examples Ehrenfest provided in his 1910 paper, Schott's example has no inherent symmetry which prevents radiation. So why doesn't the sphere radiate?

The answer is, in short, complete destructive interference. The time-dependent electric field produced by a charge on one point of the sphere is canceled out by contributions to the field from all other charges. The net result is that the accelerating sphere produces only a *static* electric field, and if we were to put an appropriate negative point charge in the center of the sphere's orbit, we could negate even that.

Before we get too excited, however, and start planning 'radiationless synchrotrons' and other unusual applications of this effect, we should make a number of observations. First, we consider Schott's condition for radiationless motion. Since $latex cT$ is the distance light travels during the course of one 'orbit' of our electrified sphere, this suggests that

$latex cT= 2a/m$.

In other words, light cannot travel a distance further than the sphere's diameter. This in turn suggests that the sphere cannot move in a broad orbit; rather, its motion is constrained to be a small 'wobble' or 'vibration'. In simple illustration,

Schott's theorem allows for WOBBLE:

but *not* ORBITS:

Schott himself was aware of this when he wrote his paper, as he wrote, "...if one of our spheres were used as a model of the electron, the radiationless orbits of its centre would be far too small, since they would lie entirely inside the electron..."

Another concern with such a model is *radiation reaction*, the interaction of a charged object with its own electric field. Schott's model neglects the reality that his sphere must necessarily pass through its own radiation field, and its is unclear whether such a reaction might wreck the radiationless motion. We will see in future posts that Schott later looked at the radiation reaction of his sphere, and found even more unusual results!

Another concern is the inherently relativistic nature of Schott's model. For a sphere of any practical size, the period of vibration must be exceedingly fast, and this translates into a high velocity or acceleration of the sphere. It is unclear whether relativistic concerns invalidate Schott's theorem, though one later paper suggests that they do.

In spite of such concerns, Schott tried to ground his thought experiment in reality with the following 'concrete example' of his sphere:

imagine a metal sphere suspended by a fine metal wire in such a manner that it can be earthed or insulated at will. Surround it by a closely fitting insulating coating, e.g. two thin hollow hemispheres of ebonite fitted together, and then place around the whole and concentric with it a larger insulated metal sphere made of two hemispheres with a very small hole through which the suspending wire passes without touching the outer sphere. Now connect the outer metal sphere to one pole of a battery and the inner one momentarily to the other pole, and again insulate the latter sphere. This receives a charge which resides on the ebonite in contact with it according to the theory of Maxwell and a well known experiment of Faraday. Remove the outer metal hemispheres and also the ebonite hemispheres. Joining the latter together we obtain a very nearly uniformly charged insulating sphere, and if the ebonite insulated perfectly, the charge would remain uniform however the sphere moved about as a whole. This charged ebonite sphere is a concrete example, realized approximately, of what is meant in the present paper.

By the time of Schott's paper in 1933, the quantum theory was firmly established and Schott himself seems to have realized that it was futile to attempt to use his sphere to try and construct a classical model of the atom. However, another unexplained phenomenon had just recently been discovered, in 1930: a new type of radiation which was uncharged and penetrated deeply into matter. In 1932, James Chadwick performed a series of experiments elucidating the nature of this new particle, the neutron, and by 1933 physicists were trying to understand how this uncharged particle, with approximately the mass of the proton, fit into the picture.

Schott suggested that the neutron might be a collection of one or more of his radiationless spheres:

Having now established the principal result of our investigation, we may perhaps be permitted to indulge in a little speculation, and, though models of the atom and its constituents, especially classical ones, are out of fashion, enquire whether such models, constructed out of charged sphere, like the one considered above, may not, after all, be of use in the elucidation of atomic problems... Obviously it does not help to account for Bohr's radiationless electron orbits, for, if one of our spheres were used as a model of the electron, the radiationless orbits of its centre would be far too small, since they would lie entirely inside the electron, as we saw in the last section.

But this very fact suggests that, if two of our spheres were taken as models of the electron and proton, it might prove possible to use them to construct a permanent model of the neutron, possibly also permanent models of atomic nuclei; for we require radiationless orbits of nuclear dimensions, which can be provided by our spheres, if they are of such dimensions, and have no spin.

Schott's model of the neutron, like his earlier atomic model, was overwhelmed by experimental reality, and the undeniable existence of additional fundamental forces of nature, namely the strong nuclear force and the weak nuclear force. Schott's beautiful calculation would continue to inspire a number of physicists through the decades to come to perform their own investigations of radiationless motion, and as we will see in future posts these investigations would eventually lead to investigations of the possibility of truly invisible objects.

* G.A. Schott, "The electromagnetic field of a moving uniformly and rigidly electrified sphere and its radiationless orbits," Phil. Mag. Vol. 15, Ser. 7 (1933), 752-761.

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A Mathematical Appendix: For those of a mathematical inclination who might be interested in trying to reproduce Schott's results, I give you the following outline:

1. The sphere is moving periodically, and is of finite size; therefore its charge and current densities can be expanded in terms of a mixed Fourier series in time/Fourier transform in space:

$latex {bf j}({bf r},t)=sum_{n=-infty}^infty int {bf j}_n({bf K})e^{i({bf K}cdot{bf r}-omega_n t)}d^3K$.

2. In the Lorentz gauge, one can express the scalar potential $latex phi$ and vector potential $latex {bf A}$ in terms of an integral over the retarded forms of the charge and current densities.

3. We may also write the charge and current densities as follows:

$latex rho({bf r},t)=e f({bf r}-{bf xi}(t))$,

$latex {bf j} ({bf r},t)=e {bf dot{xi}}f({bf r}-{bf xi}(t))$,

where $latex {bf xi}$ is the vector position of the center of the sphere as a function of time.

4. Determine the radiation fields far away from the region of the sphere, i.e. in the 'far-zone'. If you do everything correctly, you get a frightfully complicated series form which includes, in each term,

$latex tilde{f}(k_n{bf hat{r}})$,

the three-dimensional Fourier transform of $latex f$, evaluated at spatial frequencies $latex k_n=n 2pi/T$, and $latex {bf hat{r}}$ is the direction of observation.

5. Since we are assuming that our sphere is an infinitely thin shell of charge, its Fourier transform is a Bessel function with evenly-spaced zeros. Requiring the function to be zero for those special spatial frequencies listed above results in Schott's condition.

6. No, it isn't easy!

I very much enjoy these posts. You may be interested in the work of Randell Mills, who uses the nonradiation condition as the basis for the stability of the bound and free electron, and as the cause of superconductivity. I also suggest you add to a wikipedia article on the nonradiation condition which I will be posting soon.

Yes, I may not understand it all, but at least I found it all. And now onwards

to part three 🙂 And may I congratulate you on your clarity of thought.