Magnetic Vector Research Articles

Understanding Cyclotron Motion: A Visual Approach to Electron Dynamics in Electric and Magnetic Fields

Physics material is often material that is difficult for students to understand because of its abstract concepts, one of which is in the material of electric and magnetic fields that discuss the motion of cyclotrons in connecting theoretical concepts and practical applications. Cyclotron motion is accelerated by an electric field that is influenced by the Lorentz force on the combination of electric and magnetic fields. This paper is made with the aim to analyze and visualize the electric field and magnetic field around the cyclotron motion particles using Python animation. In this study, a helix-shaped trajectory is used to represent the movement of electron particles. The animation developed depicts the electric and magnetic field vectors moving around the helix trajectory. In running the particle trajectory animation, numerical equations are involved. The involved numerical equations are second-order ordinary differential equations. This research not only provides theoretical insights into the electrostatic and magnetostatic properties of electron particles but also provides effective visualizations for further education and research. The resulting animation facilitates to enhance the understanding of the concept of how electric and magnetic fields interact and move in the context of electron particles moving in a helix trajectory, opening up opportunities for further exploration in the development of knowledge related to cyclotron motion that is often difficult to understand only through theoretical approaches.

Fiber optic probe integrated with colloidal nanoparticles with directional diffraction selectivity for magnetic field vector detection

A fiber optic probe integrated with colloidal nanoparticles with directional diffraction selectivity is proposed for wide-bandwidth magnetic field vector detection. The probe is constructed with the multimode fiber in which the end-surface is integrated with the Fe3O4@C colloidal nanoparticles by a silicone tube. The colloidal nanoparticles form a three-dimensional photonic crystal structure by magnetic field for diffraction selectivity. The lattice constant and diffraction angle are adjusted by the intensity and direction of the magnetic field, respectively. Obtaining the directional diffraction light through the magnetic field-induced photonic band gap shift with the wavelength blue shift and reflectivity change is confirmed by theory and experiment. The results show that the maximum sensitivity reaches up to 19.7 nm/mT in response range from 13 mT to 200 mT. For vector detection, the peak wavelength shift from 740 nm to 485 nm and reflectance shift from 71% to 7% covering the 0–45° region is verified. In addition, the proposed method could decouple intensity and direction of the magnetic field completely. The fiber optic probe integrated with colloidal nanoparticles has wide detection range and high sensitivity with rapid response. It will open up new horizons for inspiring design and application of magnetic field vector detection in robot posture control and motion perception.

Modeling the Magnetic Vectors of Interplanetary Coronal Mass Ejections at Different Heliocentric Distances with INFROS

The INterplanetary Flux ROpe Simulator (INFROS) is an observationally constrained analytical model dedicated to forecasting the strength of the southward component (Bz) of the magnetic field embedded in interplanetary coronal mass ejections (ICMEs). In this work, we validate the model for six ICMEs sequentially observed by two radially aligned spacecraft positioned at different heliocentric distances. The six selected ICMEs in this study comprise cases associated with isolated coronal mass ejection (CME) evolution as well as those interacting with high-speed streams (HSSs) and high-density streams (HDSs). For the isolated CMEs, our results show that the model outputs at both spacecraft are in good agreement with in situ observations. However, for most of the interacting events, the model correctly captures the CME evolution only at the inner spacecraft. Due to the interaction with HSSs and HDSs, which in most cases occurred at heliocentric distances beyond the inner spacecraft, the ICME evolution no longer remains self-similar. Consequently, the model underestimates the field strength at the outer spacecraft. Our findings indicate that constraining the INFROS model with inner-spacecraft observations significantly enhances the prediction accuracy at the outer spacecraft for the three events undergoing self-similar expansion, achieving a 90% correlation between observed and predicted Bz profiles. This work also presents a quantitative estimation of the ICME magnetic field enhancement due to interaction which may lead to severe space weather. We conclude that the assumption of self-similar expansion provides a lower limit to the magnetic field strength estimated at any heliocentric distance, based on the remote-sensing observations.

A Method for Calculating Critical Current of High Temperature Superconducting Machine Based on Magnetic Vector Potential

A Method for Calculating Critical Current of High Temperature Superconducting Machine Based on Magnetic Vector Potential

Structure-preserving and efficient numerical simulation for diffuse interface model of two-phase magnetohydrodynamics

In this paper, we propose a novel diffuse interface model of two-phase magnetohydrodynamics (MHD) based on a magnetic vector potential formulation in the three-dimensional case. This model ensures an exact divergence-free approximation of the magnetic field by introducing a magnetic vector potential A and defining the magnetic field by B=curlA. The resulting framework constitutes a highly coupled, nonlinear saddle point system consisting of the Cahn–Hilliard system and MHD potential system. To solve the model efficiently, we present two fully decoupled, first-order, linear, and unconditionally energy-stable schemes and strictly prove their well-posedness and energy stability. Finally, we present several numerical examples that demonstrate the stability and effectiveness of our schemes.

Iron Loss and Temperature Rise Analysis of a Transformer Core Considering Vector Magnetic Hysteresis Characteristics under Direct Current Bias.

Direct current (DC) bias induced by the DC transmission and geomagnetically induced current is a critical factor in the abnormal operation of electrical equipment and is widely used in the field of power transmission and distribution system state evaluation. As the main affected component, the vector magnetization state of a transformer core under DC bias has rarely been studied, resulting in inaccurate transformer operation state estimations. In this paper, a dynamic vector hysteresis model that considers the impact of rotating and DC-biased fields is introduced into the numerical analysis to simulate the distribution of magnetic properties, iron loss and temperature of the transformer core model and a physical 110 kV single-phase autotransformer core. The maximum values of B, H and iron loss exist at the corners and T-joint of the core under rotating and DC-biased fields. The corresponding maximum value of the temperature increase is found in the main core limb area. The temperature rise of the 110 kV transformer core under various DC-biased conditions is measured and compared with the FEM (Finite Element Method) results of the proposed model and the model solely based on the magnetization curve B||H. The calculation error of the temperature rise obtained by the improved model is approximately 3.76-15.73% and is much less than the model solely based on magnetization curve B||H (approximately 50.71-66.92%).

Application of Geophysical Methods in the Identification of Mineralized Structures and Ranking of Areas for Drilling as Exemplified by Alto Guaporé Orogenic Gold Province

Mineral exploration works conducted in the Alto Guaporé Gold Province (AGGP), situated in the southwest region of the Amazon Craton in Brazil, faces the challenges of many gold provinces around the world, i.e., declines in the discoveries of new economic deposits and increases in exploration costs. Ground geophysical methods, combined with structural analyses and geological mapping, are valuable tools that have potential to improve accuracy in selecting exploration targets and in determining drilling locations. AGGP deposits are primarily associated with regional N20°–W50° inverse faulting and sheared geologic contacts between Meso-Neoproterozoic siliciclastic metasedimentary rocks and Mesoproterozoic basement (granite and volcano–sedimentary sequences). Mining currently occurring in the central portion of the province drives exploration works towards the many existing targets at the area. Among them, the ABP target is one of the most promising for being located few kilometers north of the Pau-a-Pique mine. At the ABP target, gold is associated with hydrothermal alteration located in the sheared contacts and in the hinge zone of folded metasedimentary sequence. Hydrothermal phases include Fe-oxides, sulfide (py), muscovite and quartz veins. In this study, we use magnetic and geoelectric (induced polarization) surveys coupled with structural and geological mapping to identify potential footprints within the ABP target. The results from induced polarization (IP) profiles successfully mapped the shape and orientation of the main structures down to approximately 350 m at the ABP target, indicating potential locations for hydrothermal alteration hosting gold. Additionally, 3D magnetic data inversions illustrated the distribution of magnetic susceptibilities and magnetization vectors associated with shear zone structures and isolated magnetic bodies. Magnetic data highlighted fault zones along the contacts between metamorphic rocks and granites, while IP data identified areas with high chargeability, correlating with sulfidation zones mineralized with gold. These findings suggest a metallogenic model where gold deposits are transported through deep structures connected to regional faults, implying significant tectonic and structural control over gold deposition. The results underscore the potential of multiparameter geophysics in identifying and characterizing deposits in both deep and strike, thereby advancing our understanding of mineral occurrences in the region and enhancing the search for new mineralized zones.

The Formation of Filaments and Dense Cores in the Cocoon Nebula (IC 5146)

We present 850 μm linear polarization and C18O (3 − 2) and 13CO (3 − 2) molecular line observations toward the filaments (F13 and F13S) in the Cocoon Nebula (IC 5146) using the JCMT POL-2 and Heterodyne Array Receiver Program instruments. F13 and F13S are found to be thermally supercritical with identified dense cores along their crests. Our findings include that the polarization fraction decreases in denser regions, indicating reduced dust grain alignment efficiency. The magnetic field vectors at core scales tend to be parallel to the filaments, but disturbed at the high density regions. Magnetic field strengths measured using the Davis–Chandrasekhar–Fermi method are 58 ± 31 and 40 ± 9 μG for F13 and F13S, respectively, and it reveals subcritical and sub-Alfvénic filaments, emphasizing the importance of magnetic fields in the Cocoon region. Sinusoidal C18O (3 − 2) velocity and density distributions are observed along the filaments’ skeletons, and their variations are mostly displaced by ∼1/4 × the wavelength of the sinusoid, indicating core formation occurred through the fragmentation of a gravitationally unstable filament, but with shorter core spacings than predicted. Large-scale velocity fields of F13 and F13S, studied using 13CO (3 − 2) data, present a V-shape transverse velocity structure. We propose a scenario for the formation and evolution of F13 and F13S, along with the dense cores within them. A radiation shock front generated by a B-type star collided with a sheet-like cloud about 1.4 Myr ago. The filaments became thermally critical due to mass infall through self-gravity ∼1 Myr ago, and subsequently, dense cores formed through gravitational fragmentation, accompanied by the disturbance of the magnetic field.

First-principles calculation of Hubbard U for Terbium metal under high pressure

Using density functional theory (DFT) and linear response approaches, we compute the on-site Hubbard interaction U of elemental Terbium (Tb) metal in the pressure range ∼ 0–65 GPa. The resulting first-principles U values with experimental crystal structures enable us to examine the magnetic properties of Tb using a DFT+U method. The lowest-energy magnetic states in our calculations for different high-pressure Tb phases—including hcp, α-Sm, and dhcp—are found to be compatible with the corresponding magnetic ordering vectors reported in experiments. The result shows that the inclusion of Hubbard U substantially improves the accuracy and efficiency in modeling correlated rare-earth materials. Our study also provides the necessary U information for other quantum many-body techniques to study Tb under extreme pressure conditions.

Multimodal Approach Reveals the Symmetry-Breaking Pathway to the Broken Helix in EuIn2As2

Understanding and manipulating emergent phases, which are themes at the forefront of quantum-materials research, rely on identifying their underlying symmetries. This general principle has been particularly prominent in materials with coupled electronic and magnetic degrees of freedom, in which magnetic order influences the electronic band structure and can lead to exotic topological effects. However, identifying symmetry of a magnetically ordered phase can pose a challenge, particularly in the presence of small domains. Here we introduce a multimodal approach for determining magnetic structures, which combines symmetry-sensitive optical probes, scattering, and group-theoretical analysis. We apply it to EuIn2As2, a material that has received attention as a candidate axion insulator. While first-principles calculations predict this state on the assumption of a simple collinear antiferromagnetic structure, subsequent neutron-scattering measurements reveal a much more intricate magnetic ground state characterized by two coexisting magnetic wave vectors reached by successive thermal phase transitions. The proposed high- and low-temperature phases are a spin helix and a state with interpenetrating helical and Néel antiferromagnetic order termed a “broken helix,” respectively. Employing a multimodal approach, we identify the magnetic structure associated with these two phases of EuIn2As2. We find that the higher-temperature phase is characterized by a variation of the magnetic moment amplitude from layer to layer, with the moment vanishing entirely in every third Eu layer. The lower-temperature structure is similar to the broken helix, with one important difference: Because of local strain, the relative orientation of the magnetic structure and the lattice is not fixed. Consequently, the symmetry required to protect the axion phase is not generically protected in EuIn2As2, but we show that it can be restored if the magnetic structure is tuned with uniaxial strain. Finally, we present a spin Hamiltonian that identifies the spin interactions that account for the complex magnetic order in EuIn2As2. Our work highlights the importance of a multimodal approach in determining the symmetry of complex order parameters. Published by the American Physical Society 2024

Feasibility of Induced Magnetic Gradient Surveying for Seepage Detection in Earth-filled Dams: Insights from Synthetic and Field Studies

Seepage is a common hydrogeological hazard in engineering. Determining the seepage paths is vital for de-risking the instability of embankment structures. With the improvement of the acquisition accuracy of magnetic sensors, the magnetometric resistivity method has become an emerging technology for detecting seepage paths through earth-filled dams. This technique is non-destructive and gives prominent signals. However, the resulting magnetic data have seen ambiguity in fully determining the targets. We propose an induced magnetic gradient surveying approach to monitor seepage paths in earth-filled dams. First, we briefly review the electromagnetic theory for the magnetic gradient tensor based on Maxwell’s equations. To match against the measurements, we present an accurate modeling framework using the third-order finite element method and a novel compact difference scheme. We verify our approach on both semi-analytical 1D and 3D models. Systematic modeling studies are then carried out to investigate the spatial distribution characteristics and sensitivities of the induced magnetic gradient to the seepage in typical dam scenarios. In addition, we conducted two field experiments in the Zhongmou experimental base and Xixiayuan Reservoir in Henan Province, China,respectively. The induced magnetic field vector and its gradient components were both acquired. Cross-validation with a-priori geological information shows that the seepage path can be spatially identified by the induced magnetic gradient components Byy, Byz, Bzy, and Bzz while the field components failed to locate the seepage pathways. This successful application indicates that the proposed approach could be a promising solution for seepage path discrimination in earth-filled dams with high resolution.

Features of magnetic structures in perforated films due to the finite thickness of the sample

The paper examines ferromagnetic films with strong uniaxial anisotropy of the ‘easy plane’ type, in which vortex-like inhomogeneities can arise in the presence of artificially created perforations. A universal approach has been developed that makes it possible to reduce the problem of calculating demagnetizing fields in a film of arbitrary thickness to the simplest case, when the film thickness is large. It has been shown that the influence of demagnetizing fields causes the magnetization vector to necessarily move out of the sample plane, and the spatial distribution of magnetization corresponding to this effect has been studied. It has been revealed that the contribution of demagnetizing fields to the total energy of the magnet changes slightly during the transition from a homogeneous structure to an inhomogeneous one.

Mathematical Assessment of Pollution Factors from Eddy Currents in the Durrës Seaport

Eddy currents, a significant phenomenon in electromagnetic fields, were first distinguished from general electric currents by Foucault. These currents, influenced by the magnetic field vector (H), magnetic flux density (B), and electric field (E), are present in various environments, including homogeneous and non-homogeneous areas, linear and non-linear environments, and open boundary problems. In marine environments, these currents, referred to as planetary currents, contribute to electrical pollution, impacting maritime infrastructure, machinery, and ecosystems. This study aims to assess the impact of eddy currents in the Durrës Seaport, Albania, focusing on how meteorological factors such as temperature, atmospheric pressure, and humidity influence pollution levels. Using the TES-92 Electro-smog meter and the PCE-THB 40 Thermo-hygrometer and barometer, we measured electromagnetic field characteristics and meteorological parameters. Our results highlight the significant impact of stray currents on marine environments, emphasizing the need for effective measures to minimize their effects and improve the quality and selection of materials and equipment in port facilities. This study provides valuable insights into the environmental and infrastructural challenges posed by electromagnetic pollution and proposes recommendations for mitigating these impacts.

Electrically engineering synthetic magnetic fields for polarized photons

Polarized photons are, in essence, neutral particles and therefore do not couple directly to external fields, thus hampering the effective interaction of photons with external fields. Here, we theoretically identify an equivalent spin-1/2 model for polarized photons and synthesize a magnetization vector for coupling differently polarized photons in an engineered anisotropic medium. The synthetic magnetic field can be electrically engineered to manipulate the magnetic moments of the pseudo-spin-1/2 photons, leading to observation of the Lorentz force and analogous Stern–Gerlach effect. We experimentally demonstrate these fundamental effects by using different spins, including purely single-polarization spins and mutually two-polarization mixing spins. We also demonstrate the higher-order Stern–Gerlach effect by using spins having nontrivial topological structures. Our findings could enable polarization-based elements with potential applications in polarization selection and conversion, benefiting classical and quantum information processing.

A decomposition of light’s spin angular momentum density

Light carries intrinsic spin angular momentum (SAM) when the electric or magnetic field vector rotates over time. A familiar vector equation calculates the direction of light’s SAM density using the right-hand rule with reference to the electric and magnetic polarisation ellipses. Using Maxwell’s equations, this vector equation can be decomposed into a sum of two distinct terms, akin to the well-known Poynting vector decomposition into orbital and spin currents. We present the first general study of this spin decomposition, showing that the two terms, which we call canonical and Poynting spin, are chiral analogies to the canonical and spin momenta of light in its interaction with matter. Like canonical momentum, canonical spin is directly measurable. Both canonical and Poynting spin incorporate spatial variation of the electric and magnetic fields and are influenced by optical vortices. The decomposition allows us to show that a linearly polarised vortex beam, which has no total SAM, can nevertheless exert longitudinal chiral pressure due to equal and opposite canonical and Poynting spins.

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Application of the G’/G Expansion Method for Solving New Form of Nonlinear Schrödinger Equation in Bi-Isotropic Fiber

Bi-isotropic media (chiral and non-reciprocal) present an outstanding challenge for the scientific community. Their characteristics have facilitated the emergence of new and remarkable applications. In this paper, we focus on the novel effect of chirality, characterized through a newly proposed formalism, to highlight the nonlinear effect induced by the magnetization vector under the influence of a strong electric field. This research work is concerned with a new formulation of constitutive relations. We delve into the analysis and discussion of the family of solutions of the nonlinear Schrödinger equation, describing the pulse propagation in nonlinear bi-isotropic media, with a novel approach to constitutive equations. We apply the extended -expansion method with varying dispersion and nonlinearity to define certain families of solutions of the nonlinear Schrödinger equation in bi-isotropic (chiral and non-reciprocal) optical fibers. This clarification aids in understanding the propagation of light with two modes of propagation: a right circular polarized wave (RCP) and a left circular polarized wave (LCP), each having two different wave vectors in nonlinear bi-isotropic media. Various novel exact solutions of bi-isotropic optical solitons are reported in this study. Introduction: The investigation of exact solutions for nonlinear partial differential equations (PDEs) holds significant importance in understanding nonlinear physical phenomena. Nonlinear waves manifest across various scientific domains, notably in optical fibers and solid-state physics. In recent years, several potent methodologies have emerged for identifying solitons and periodic wave solutions of nonlinear PDEs. These include the -expansion method [1-6], the new mapping method [9-10], the method of generalized projective Riccati equations [11-16], and the expansion method [17]. Consequently, an original mathematical approach is proposed to evaluate nonlinear effects in bi-isotropic optical fibers, stemming from magnetization under the influence of a strong electric field [19-20]. The extended -expansion method emerges as a potent technique for deriving solution families of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method employs a perturbation expansion in powers of the dimensionless parameter and is applicable for both weak and strong nonlinearities. It accommodates varying dispersion and nonlinearity, rendering it suitable for modeling a wide array of optical fibers. Results and Conclusion: This investigate is concerned with a new formulation of constitutive relation linking to the magnetic effect, to understand rigorously the physical nature of biisotropic effects and to generalize the main macroscopic models. We inferred the nonlinear Schrodinger equation for a bi-isotropic medium term with a nonlinear term of magnetizing. In this article, the extended -expansion method is a powerful technique for determining a family of solutions of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method is based on the use of a perturbation expansion in powers of the dimensionless parameter, and it is valid for both weak and strong nonlinearities. The method allows for the inclusion of varying dispersion and nonlinearity, making it well-suited for modeling a wide range of optical fibres. Overall, the extended -expansion method is a valuable tool for understanding the dynamics of nonlinear optical systems, and it is expected to have a wide range of applications in the field of nonlinear optics.

Suppressing Competing Solvent Reduction in CO2 Electroreduction with a Magnetic Field.

The presence of an external magnetic field is found to affect the competition between the H2O and CO2 reduction reactions by increasing mass transport via the Lorentz force. Increasing the magnetic field strength at the electrode surface from 0 to 325 mT increases the selectivity of CO over H2 by 3×, while an increase in current density from 0.5 to 5 mA/cm2 increases the selectivity of CO production by 5×. Cyclic voltammetry and finite-element simulations reveal that the origin of the enhanced CO selectivity is attributable to a magnetic field lowering the electrode-electrolyte interfacial pH. A drop in interfacial pH enables increased production of CO from CO2 reduction due to a decrease in the activity of H2O reduction and increase in CO2 solubility near the electrode surface. The insight provided in this study offers new opportunities to control reaction selectivity in electrocatalysis with magnetic field vectors.

Revisiting the dry friction-like magnetic vector hysteresis model

Physically correct simulations of magnetic devices require precise and energy-consistent hysteresis models. Electrical steel sheets of, e.g., power transformers or rotating machines represent one crucial element in such a simulation since the hysteresis is revealed uni- and multi-axial, and the steel sheets may have anisotropy. A dedicated vector hysteresis model based on dry friction-like pinning is revisited and derived in terms of thermodynamic state equations in a theoretical manner. The model is then linked to an incremental energy minimization procedure used in elasto-plasticity theory. A numerically efficient two-dimensional solution scheme of the hysteresis model is extended to the three-dimensional case. Rotational losses are also in the scope of this paper since the model cannot describe this loss type by nature. Therefore, numerical adaptations are discussed to account for vanishing rotational losses in saturation.

Influence of ionospheric resonators on daily dynamics of the first Schumann resonance spectral parameters according to data from a meridional chain of ULF magnetometers

We have experimentally studied the influence of the local ionosphere, namely the ionospheric Alfvén resonator (IAR) and the lower ionospheric resonator at altitudes 80–300 km (sub-IAR) on the amplitude and polarization of the first Schumann resonance. The study is based on spectral analysis of data from simultaneous monitoring of ULF magnetic noise components at a meridional chain of stations: high-latitude stations Barentsburg and Lovozero, mid-latitude observatory NNGU NIRFI (NL, Nizhny Novgorod Region), a low-latitude station in Israel. We have also used monitoring data from the Borok and Crete observatories. At the stations in dark conditions, significant variations were found in the spectrum of the polarization parameter ε at the frequency of the first Schumann resonance (SR). Moreover, these variations had different character at different observatories. Analysis of the daily dynamics of the parameter ε has shown that these variations are associated with the influence of local sub-IAR having different optical thickness and quality factor at these observatories. The influence of sub-IAR on polarization in the SR band was found to depend on the ratio of the boundary frequency fb (the frequency that separates the negative and positive polarization of ULF magnetic noise) to the frequency of the first SR. The IAR influence on the polarization and amplitude of magnetic fields in the frequency band of the first Schumann resonance was discovered only at the NL and Lovozero stations: a high-quality Alfvén resonator in the ionosphere above the NL station could cause the SR frequency band to change and its central frequency to shift. Analysis of low-frequency data from the observatories separated by distances of 400 km has revealed that the influence of local ionospheric resonators can lead to a difference in the amplitude characteristics of the first SR even at such distances. It has also been shown that the influence of the IAR and sub-IAR resonators on the azimuthal angle of the magnetic field vector in the frequency band of the first SR is less noticeable and can generate variations in this parameter by 10°–20°. Numerical calculations performed for the spherical waveguide model made it possible to adequately interpret the features of the daily dynamics of the parameter ε in the frequency range of the first SR.