# Dynamics of Electron in TEM Wave Field

Keywords: wave equation, generalized Fourier series, TEM wave

### Abstract

Big amount of works deals with solution of differential equations, associated with electron motion in electromagnetic field, using methods of classical electrodynamics. Solution of equation of an electron motion in TEM wave field is interesting task because this equation is mathematical model of big number of wave processes, which are used for researches of different physical processes. The proposed work dedicated to finding the solution for the equation of an electron motion in TEM wave field in laboratory system of coordinates using the theory of almost periodic functions. The work demonstrates that the projections of electron velocity on coordinate axis conform to the wave equation, and, consequently, could be expanded into generalized Fourier series at any value of the wave and electron parameters. In the present work, the formulas received before for electron velocity projection on coordinate axis, are transformed to a well-behaved form, and are broken down into non-perfect generalized Fourier series. Non-perfect Fourier series for projections of electron velocity on coordinate axis are found by means of plotting of complex series, which are called in the theory of almost periodic functions as ”closure of set”. For approximate computation of electron velocity it is possible to restrict oneself to finite number of series harmonics. Application of method of electron velocity components transformation into generalized Fourier series made it possible to find in electron velocity components series terms, which do not depend on time and are equal to average magnitudes of the respective values. Electron velocity components present functions of initial magnitudes of electron velocity components, of generalized phase magnitude and of the wave parameters. Initial magnitudes are not preset at random, but calculated from the equations, the type of which is specified in the work. Electron trajectory in coordinate space is calculated by integrating of the respective expressions for velocity projections on coordinate axis. For demonstration purpose the work deals with the example of electron dynamics in wave polarization plane with consideration of only permanent addends and first harmonics of Fourier series for electron velocity projections on coordinate axis. An approximate solution of the equations of electron dynamics in the plane of polarization of the wave is given. Solution for the equation of electron motion in TEM wave field in the laboratory coordinate system using the theory of almost periodic functions made it possible to solve the problem of dynamics of relativistic electron in the field of progressing TEM wave. It made it possible to demonstrate the availability of time-independent summands in the value of the speed of the electron, which moves in TEM wave. A very important circumstance is also the fact, that the theory makes it possible to investigate electron dynamics depending on the original wave intensity.

### References

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G. Bor, Almost periodic functions, (Editorial URSS, Moscow, 2009), 128 p. (in Russian)

H.A. Avetissian, Relativistic Nonlinear Electrodynamics, (2nd ed.), (Springer International Publishing, Switzerland, 2016) Vol 88, pp. 463-499, https://doi.org/10.1007/978-3-319-26384-7_1.