[hamdani yusuf (OP)]

The laws of classical mechanics (i.e. the Larmor formula) predict that accelerating electric charges will release electromagnetic radiation, hence will lose energy.

https://en.wikipedia.org/wiki/Bohr_model#Origin

In the early 20th century, experiments by Ernest Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus.[2] Given this experimental data, Rutherford naturally considered a planetary model of the atom, the Rutherford model of 1911 – electrons orbiting a solar nucleus – however, the said planetary model of the atom has a technical difficulty: the laws of classical mechanics (i.e. the Larmor formula) predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would rapidly spiral inwards, collapsing into the nucleus on a timescale of around 16 picoseconds.[3] This atom model is disastrous, because it predicts that all atoms are unstable.[4]

But experiments using superconductor ring show that circulating electrical current can go on indefinitely, which means they don't lose energy through radiation.
https://arxiv.org/ftp/cond-mat/papers/0506/0506426.pdf

An extremely sensitive method for the purpose, pioneered by Onnes himself, is the
technique of estimating the upper limit of the resistivity by studying the decay rate of the
persistent current in a superconducting ring. Once established, the time dependence of the
current I(t) through the ring is given by I(t) = I0 e ^{– (R/L) t} where I₀ is the current at t = 0, R is the
resistance and L is the inductance of the ring. If the superconductor had zero resistance, the
current would not decay even for infinitely long times. However, an experiment can be
performed only over a limited amount of time. In a number of such experiments no detectable
decay of the current was found for periods of time extending to several years.

In a minor variation of the experiment, after the loop became superconducting, the
source current was switched off, the superconducting loop being driven into the persistent
current mode. It was observed that even now the field generated by coil B remained much
larger than the value in the normal state, indicating that the resistances in the two paths are
exactly zero. This provides additional evidence that no extraneous effects such as differential
terminal resistances have any role to play.
 In summary, we have demonstrated that the dc resistance of a superconducting wire
is indeed zero and not just unmeasurably small, thus resolving the uncertainty that had lingered
on for nearly a century after the discovery of the phenomenon of superconductivity.

[Bored chemist]

Good question.
I think the classical argument would be that the energy lost by electrons on one side of the "loop"  is gained by the electrons on the other side.

[hamdani yusuf (OP)]

The problem is, which side would be gaining energy at any given time, which time is losing, since they supposed to be symetrical?

[Bored chemist]

I presume that, on average, it all cancels out. The EM radiation emitted by one electron is absorbed by anothe rone.

But I'd be interested to know if anyone has a better/ more complete explanation.

Otherwise , as you suggest, the electrons going round should emit radiation
More or less like this
https://en.wikipedia.org/wiki/Cyclotron_radiation
or maybe this?
https://en.wikipedia.org/wiki/Synchrotron_radiation

[hamdani yusuf (OP)]

I guess the electrons lose energy not because they accelerate per se. In case of cyclotron and synchrotron, they can be interpreted that the energy is lost due to the media the electrons are moving in have non-zero resistance.

[hamdani yusuf (OP)]

But experiments using superconductor ring show that circulating electrical current can go on indefinitely, which means they don't lose energy through radiation.
https://arxiv.org/ftp/cond-mat/papers/0506/0506426.pdf

I wonder if the electrical current can still go on indefinitely if the ring is wiggled.

[Bored chemist]

Is the case of a simple superconducting loop equivalent to a simple magnet?

If so, would the wiggly wire loop be equivalent to an array of magnets?

[evan_au]

But experiments using superconductor ring show that circulating electrical current can go on indefinitely, which means they don't lose energy through radiation.

How much acceleration? This determines how much radiation.

Typical electron velocity when copper is about to melt is something like 1-2mm/s.
- Superconductors carry DC quite well, but not AC, so the acceleration is about forcing the electrons to travel in a circle, instead of a straight line.
- The electrons complete one cycle in a time taken to travel around the circuit once, at 1-2mm/second (assuming the carrier velocity is similar to copper?).
- For an MRI machine with a superconductor loop 2m in diameter, this is something like 0.0001 Hz.

In contrast, in a VHF antenna application, the electrons change direction and start going in the opposite direction n x 100 million times per second. So I expect  the acceleration is far greater, and the radiation so much more.

@alancalverd knows all about superconductors in an MRI machine - how often do you have to recharge them?
- And did you need to take into account radiation from the superconductor coils when doing EMI tests?
- Or was it impossible to measure against all of the intentional EMI required to align all our protons?

[hamdani yusuf (OP)]

Is the case of a simple superconducting loop equivalent to a simple magnet?

I guess so, if you are referring to a permanent magnet.

If so, would the wiggly wire loop be equivalent to an array of magnets?

I'm not sure how an array of magnets can produce similar movement of electric charges to the wiggle ring.
Also, what are the effects of changing the number of wiggles in the ring? What would happen if the number of the wiggles is not an integer, which will make the curve of the ring non-smooth at the joint?
I think those questions can be easier to answer using real life experiment, rather than analytical calculation.

[hamdani yusuf (OP)]

- For an MRI machine with a superconductor loop 2m in diameter, this is something like 0.0001 Hz.

For a wiggly ring, you can increase the frequency significantly, let's say to 1 Hz.

@alancalverd knows all about superconductors in an MRI machine - how often do you have to recharge them?
- And did you need to take into account radiation from the superconductor coils when doing EMI tests?
- Or was it impossible to measure against all of the intentional EMI required to align all our protons?

I hope he can share his knowledge and experience with us here.

[alancalverd]

The principal losses in superconducting magnets are resistive losses due to the imperfect superconductivity of a practical magnet, and in practice you are more likely to have to recharge an MRI system after quenching it to service or modify the machine. A bit of drift of the primary magnetic field isn't a  problem as you can retune the RF components to maximise the received signal from a water phantom (a posh name for a bucket of hydroxylated protons) from time to time. 

If you consider the supercon as nothing more or less than an ideal wire carrying a constant current, classical Maxwell electromagnetism says that it doesn't radiate because di/dt = 0. If you can persuade it to carry an alternating current, it behaves as a classical loop aerial.

The synchrotron wiggler, on the other hand, is accelerating individual electrons traveling at relativistic velocities, so the acceleration is considerable at each point on its path. 

[hamdani yusuf (OP)]

If you consider the supercon as nothing more or less than an ideal wire carrying a constant current, classical Maxwell electromagnetism says that it doesn't radiate because di/dt = 0.

But why early 20th century scientists objected to Rutherford's planetary atomic model arguing that circular motion of electrons around nucleus must radiate energy, hence make the orbit unstable?
What do you think if the ring is wiggled?
What if it's elliptical?

[Bored chemist]

But why early 20th century scientists objected to Rutherford's planetary atomic model arguing that circular motion of electrons around nucleus must radiate energy,

The original model was for hydrogen, with just one electron.

The solution for a current- consisting of many electrons is still this:

I think the classical argument would be that the energy lost by electrons on one side of the "loop"  is gained by the electrons on the other side.

[hamdani yusuf (OP)]

The original model was for hydrogen, with just one electron.

Perhaps that's why hydrogen atoms naturally form diatomic molecules.

I think the classical argument would be that the energy lost by electrons on one side of the "loop"  is gained by the electrons on the other side.

That's a possibility. Although it's not clear yet how to make it work consistently. In a balanced symmetrical structure, how to determine which electron gives up energy, which one receive it? What's the mechanism to prevent the energy from going to any other direction?