Braking radiation and electromagnetic self action

[Ruslan_Sharipov]

It is known that a charged particle moving with some acceleration emits electromagnetic waves. For example, this may be a particle moving at a constant speed on the circle. In this case the radiation of a particle is called the braking radiation, the cyclotron radiation, or the synchrotron radiation.

Now imagine a uniformly charged dielectric ring rotating in its own plane with a constant angular velocity about its center. In this case the spatial distribution of charges and currents is stationary. Therefore the ring must create stationary electric and stationary magnetic fields without radiation. Assume that particles being accelerated in a circular accelerator (e. g. in LHC) are homogeneously distributed along the accelerating ring. This would eliminate energy losses due to braking radiation. This would also eliminate the currents induced by moving particles in metallic structures surrounding the accelerating channel. My first question is whether this method is applied in real accelerators?

My second question is more theoretical. The electrostatic field created by a point charge acts upon other charges determining the Coulomb interaction of the charges. There is no self action in electrostatics. The field produced by a point charge does not act upon this charge itself. This situation is a little bit strange from a conceptual point of view. It turns out that the electrostatic field (which is understood as a material substance separate from the charges created it) acts selectively, i. e. differently on different charges. That is in the formula for the Lorentz force $$\mathbf F=q\,\mathbf E+\dfrac{q}{c}\,[\mathbf v,\,\mathbf H]$$ we should write the electrostatic and magnetostatic fields produced by all charges except for the charge $q$.

Now let's return to the braking radiation. Consider a particle that enters the area of ​​a uniform external magnetic field with the velocity $\mathbf v$ perpendicular to the field lines. The particle begins to twist and brake at the same time due to losses on braking radiation. Twisting force perpendicular to the velocity is due to the influence of the external magnetic field. But the braking force is different - it is an example of the self-action of the charged particle. The particle slows due to the electromagnetic field created by itself. Here is my second question: is it possible to apply the above Lorentz formula for the braking force occurring due to the braking radiation? What is the correct way to exclude electrostatic and magnetostatic field components, which do not contribute to the self-action? How to do such an exclusion in the cases of more complicated geometry of charges and currents? Is there a universal algorithm for performing such an exclusion. Are there examples of such calculations?

The problem of self-action becomes even more interesting in quantum physics. For describing the hydrogen atom the Schrödinger equation is used with the Coulomb potential of the electron in the electrostatic field of the proton (or vice versa with the Coulomb potential of the proton in the electrostatic field of the electron). In order to introduce an external electromagnetic field to this Schrödinger equation the minimal coupling is used. Here is my third question. Is it possible to describe the external and the internal electromagnetic fields of a hydrogen atom in a uniform way?

In the hydrogen atom states that do not possesses the spherical symmetry the electron and proton create not only a stationary spatial distribution of the charges, but the stationary spatial current distribution. In this case they should interact magnetostatically as well. How to take into account this magnetostatic interaction in the Schrödinger equation? By means of adding one more term to the potential energy or through the minimum coupling mechanism?

 

[ZapperZ]

It is known that a charged particle moving with some acceleration emits electromagnetic waves. For example, this may be a particle moving at a constant speed on the circle. In this case the radiation of a particle is called the braking radiation, the cyclotron radiation, or the synchrotron radiation.

This is not "braking radiation". Braking radiation, or bremmshtralung, is when the charged particle is being decelerated. What you described is definitely synchroton, cyclotron radiation.

Now imagine a uniformly charged dielectric ring rotating in its own plane with a constant angular velocity about its center. In this case the spatial distribution of charges and currents is stationary. Therefore the ring must create stationary electric and stationary magnetic fields without radiation. Assume that particles being accelerated in a circular accelerator (e. g. in LHC) are homogeneously distributed along the accelerating ring. This would eliminate energy losses due to braking radiation. This would also eliminate the currents induced by moving particles in metallic structures surrounding the accelerating channel. My first question is whether this method is applied in real accelerators?

This is very difficult to understand. You have a (i) a dielectric ring, (ii) a CHARGED dielectric ring, (iii) the charged dielectric ring spinning in its own plane. So why would the ring create "stationary electric field? After all, the ring is rotating/spinning. dE/dt may be a constant, but it is not zero!

Secondly, there are now additional charge particles that are the ones being accelerated, but they are somehow "distributed homogeneously along the ring"?? Are they imbedded inside the ring? Or is the ring a hollow doughnut and these charges are inside the hollow ring? Very confusing!

Note that for any charge particle going through an accelerating structure, be it metallic or dielectric, there is an additional effect that must be considered, which is the electromagnetic wakefield that can seriously affect not only the beam energy, but also the beam profile/shape!

This question is actually not a QM topic, but rather a classical E&M topic.

Zz.

 

[Jano L.]

Assume that particles being accelerated in a circular accelerator (e. g. in LHC) are homogeneously distributed along the accelerating ring. This would eliminate energy losses due to braking radiation. This would also eliminate the currents induced by moving particles in metallic structures surrounding the accelerating channel. My first question is whether this method is applied in real accelerators?

It is said that physicists constructing the cyclotron were thinking in this way. They did not expect radiation, since an electric current due to uniformly charged rotating circle does not lead to net electromagnetic radiation (follows from Maxwell's equations). Nevertheless, the radiation was discovered. The explanation is that the charge is never distributed uniformly. Rather, in synchrotron the electrons move in separate bunches (cca 10 cm long, billions of electrons). The radiation we detect is due to correlated motion of the electrons in these bunches. From the microscopic point of view, it does not seem likely that we could distribute charge uniformly (particles are point-like), so there will be always some radiation.

is it possible to apply the above Lorentz formula for the braking force occurring due to the braking radiation?

Depends on how we model the charged particles. If we think of them as point-like, then it is not possible in Maxwellian theory, since the Lorentz force due to field of the particle is not defined. If the particle is extended (charged sphere), it is possible, see the work of Lorentz, Abraham, Poincare, and recently Yaghijian, but only approximately. They consider charged ball or sphere that keeps together by some non-electromagnetic forces and calculate approximately the Lorentz force acting on one part of the sphere due to (retarded) fields created by the other parts of the sphere. This has great problems though, as we do not know how to model the non-electromagnetic forces, how large is the sphere etc. My view on this is that radiation reaction force is logically unnecessary and the attempts to introduce such thing so far did not lead to any advance in understanding of the electron.

Similar procedure is done in the macroscopic theory of antenna, where it is much more reliable (antenna is rigid, we know size, density of charge, etc.) We can calculate approximately damping forces acting on the currents as Lorentz forces (see the book Fields by Landau & Lifgarbagez) and in principle derive approximately the radiation resistance of an antenna.

Is it possible to describe the external and the internal electromagnetic fields of a hydrogen atom in a uniform way?

As you say the description of internal interaction in the non-relativistic Schroedinger equation is based on potential energy, while the interaction with the distant things is described by "minimal coupling". These are very different mathematically and that is quite disturbing. I do not know whether it is possible to to modify the Schr. equation to remedy this. Perhaps in quantum field theory one can come closer to such a goal, but I do not know if/how.

How to take into account this magnetostatic interaction in the Schrödinger equation?

One way is to try to include the Darwin terms in the Hamiltonian or use the Breit equation:

http://en.wikipedia.org/wiki/Darwin_Lagrangian
http://en.wikipedia.org/wiki/Breit_equation

 

[Ruslan_Sharipov]

Is it possible to describe the external and the internal electromagnetic fields of a hydrogen atom in a uniform way?

As you say the description of internal interaction in the non-relativistic Schroedinger equation is based on potential energy, while the interaction with the distant things is described by "minimal coupling". These are very different mathematically and that is quite disturbing. I do not know whether it is possible to to modify the Schr. equation to remedy this. Perhaps in quantum field theory one can come closer to such a goal, but I do not know if/how.

I tried to write the Shrodinger equation for two charged particles with the "minimal coupling" only in this article: ArXiv:1308.0221. The result is different from the standard hydrogen atom. It needs to be explained why it is so.