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.(adsbygoogle = window.adsbygoogle || []).push({});

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 [tex]\mathbf F=q\,\mathbf E+\dfrac{q}{c}\,[\mathbf v,\,\mathbf H][/tex] we should write the electrostatic and magnetostatic fields produced by all charges except for the charge [itex]q[/itex].

Now let's return to the braking radiation. Consider a particle that enters the area of a uniform external magnetic field with the velocity [itex]\mathbf v[/itex] 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 occuring 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 possess 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?

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# Braking radiation and electromagnetic self action

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