In the classical electrodynamics, there is one problem - to find the EM fields created by the charge moving with the constant velocity in the circular orbit or in the helical path (free electrons in the magnetic field; radiation of helical antennas in the near zone). The exact solution of this problem is unknown. Schott gives the expressions for this field as a series of the harmonics [Sec. 78 of his book].
Meanwhile if one goes to the rotating frame (this frame rotate with respect to the inertial frame with the constant frequency), the charge in this system is at rest. It seem the EM fields of this charge can be found much easier than to solve the transcedental equation for the retarded time and then to compute very complex integral.
Then one can apply the Lorentz transformations to the calculated E field and obtain the expressions for the EM field in the lab frame.
The system of the Maxwell equations in the rotating frame are known:
- L.I. Schiff, A Question in General Relativity, Proc. Nat. Acad. Sci. V. 25, 391 (1939);
- W.H. Irvine, Electrodynamics in a Rotating Frame of Reference, Physica 30, 1160 (1964).
The LT from the rotating to the lab frames is known too. The problem is to find this E field.
My questions:
- if someone knows where such an expression for the electric field is given (in the cloased form but not as equation)?
- If someone knows how to solve Schiff's or Irvine's systems of the equation for this case?
Unfortunately, the Maxwell equations in the rotating frame should have unphysical solutions.
Actually, these equations are given in the only paper, of Schiff. The other authors use (maybe in some diffrent form) the equations of Schiff.
Schiff reasonably suggested that the fields in the rotating frame must be connected with the fields in the lab frame by the Lorentz transformations ad he derived the Maxwell equations by means of these transformations. But it seems nobody even tried to solve these equations. I made simple analysis which results are given in the atttached file.
Instead of original equations of Schiff, I use the system of equations given in very detailed review on EM fields in the rotating frame made by Kirk McDonald from Princeton Univ: https://www.hep.princeton.edu//~mcdonald/examples/rotatingEM.pdf
'Unphysical' property of the solutions for the EM fields are due to the fact that if one makes inverse Lorentz transofrmation to the lab frame, the transofmation of the Coulomb field cannot give the radiated component. It is known that the charge, circulating in the closed orbit, radiates.
Formally all math steps are correct. So it is unclearly why the unphysical solution appears.
And how to find the physical solution?
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