I.V. Demichev1

1 Military University of Radio Electronics (Cherepovets, Russia)

The mathematical apparatus of vector algebra has been widely used to describe the electromagnetic field. However, its use in solving a number of electrodynamic problems imposes certain limitations. At the same time, the electromagnetic field is a set of vector quantities that functionally depend on both spatial location and time. In this connection, it is advisable to use hypercomplex numbers – quaternions to describe spatio-temporal changes in the electromagnetic field. The article presents the possibility of extending the well-known Euler formula for ordinary complex numbers to a set of quaternions, which makes it possible to represent the quaternion in both trigonometric and exponential forms of notation. It is shown that the application of the Hamilton spatial differentiation operator over the quaternion combines the rotor, divergence and gradient operations, which generalizes the consideration of vector and scalar fields of one physical process as a single entity and simplifies the analytical record. Based on the development of the elements of the quaternion algebra, a generalization of the laws of electrodynamics is carried out. It is shown that the presented form of writing based on the quaternion algebra is adequate to the existing theory and fully reflects the known properties of the electromagnetic field. The relationship between the electromagnetic field and its manifestation through charges and conduction currents is reflected. Based on the expressions obtained for the electromagnetic field, equations satisfying the Helmholtz wave equations are derived. It is shown that an electromagnetic wave interacts with both a scalar field of electric charges and a vector field of conduction currents induced in a real medium. An analytical record of the wave equations for the electric and magnetic field strength based on the D'alembert operator and the extension of the Hamilton operator into the space-time domain – the four-dimensional differentiation operator is given. The expressions obtained indicate the relationship between the scalar field of charges and the vector field of conduction currents induced in a real environment, both with the electric and magnetic fields of oxen. The use of the mathematical apparatus of the quaternion algebra for the basic equations of the electromagnetic field simplifies the mathematical notation and fully reflects its properties. Generalization of electromagnetic fields and waves based on the quaternion algebra not only expands the parametric description of spatiotemporal changes in the electromagnetic field, but also allows them to be applied in various physical models of electrodynamic systems, which opens up new opportunities for fundamental science. At the same time, the extension of the results obtained in the work to inhomogeneous, nonlinear and anisotropic media is of particular relevance for practical application.

Demichev I.V. Wave equations for electromagnetic field based on quaternion algebra. Electromagnetic waves and electronic systems. 2023. V. 28. № 3. P. 5−10. DOI: https://doi.org/10.18127/j15604128-202303-01 (in Russian)

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7. Demichev I.V., Ognev V.A. A method for measuring spatially polarizing parameters based on the quaternion algebra when registering ionospheric RF radio waves. Electromagnetic waves and electronic systems. 2019. V. 24. № 8. P. 15–22. DOI 10.18127/j15604128-201908-02. (in Russian)
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9. Patent for the invention RUS2720588 dated 12.05.2020. Method and device for spatial selection of electromagnetic waves with subsequent polarization signal processing. Demichev I.V., Kolesnikov R.V., Ivanov A.V., Laptev I.V. (in Russian)