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Electromagnetic Fields
Published in Christos Christopoulos, Principles and Techniques of Electromagnetic Compatibility, 2022
In the previous section electric and magnetic fields were treated as if they are completely independent of each other. This is acceptable at low frequencies or low rates of change of field quantities. When this assumption becomes invalid, it is found that a changing electric (magnetic) field induces a magnetic (electric) field. The ideas and equations presented earlier must therefore be generalized to include the coupling between these two manifestations of the electromagnetic field. Treatment in this section will not, however, be of the most general kind. The assumption will be made that any interactions between different parts of a system or circuit are practically instantaneous (quasistatic approximation). If the speed of propagation for electromagnetic disturbances is u (the speed of light for propagation in air) and the largest dimension of the system studied is D, then the time it takes for different parts of the system to interact electromagnetically with each other is of the order of D/u. If the period T of the electrical disturbance is much larger than the propagation time D/u, then, in electrical terms, interaction across the system is instantaneous, and the phenomena observed can be classed as quasistatic. Another way of expressing the same thing is to compare the wavelength λ of the disturbance with the largest dimension D. If λ ≫ D, quasistatic conditions apply. Otherwise, the phenomena observed belong to the truly high-frequency regime and are described in Section 2.3.
Fundamentals of electrodynamics of gyrotropic media
Published in A.G. Gurevich, G.A. Melkov, and Waves, 2020
A necessary condition for the applicability of the quasistatic approximation is that at least one of the sample dimensions (d) must be small as compared with the length of electromagnetic wave in the sample. If we take into account that the transverse diagonal components μ and ε are of the same order of magnitude as other components, respectively, of μ↔ and ε↔, then we may write this condition as () ωc|μ||ε|d≲π6.
A Model to Calculate Force Characteristics of a Magnetic Suspension of a Superconducting Sphere
Published in Kirill Poletkin, Laurent A. Francis, Magnetic Sensors and Devices, 2017
Sergey I. Kuznetsov, Yury M. Urman
In this chapter, we present a model based on a calculation of the interaction energy between a spherical superconducting body of finite size and a supporting field created by a system of permanent magnets or electromagnets and represented in terms of expansions in spherical functions. The expressions for energy are obtained first in terms of expansions in spherical functions, in which case a finite number of terms in the sum may be calculated numerically. For some important cases (the case of N point magnetic charges and the case of N coaxial current loops), these expansions are converted into simple close-form analytical expressions. By placing N point charges or N coaxial current loops, one may construct rather complex source configurations, obtain various field distributions, and look for those configurations that provide the best characteristics for a particular device. Simple analytical expressions for energy allow the calculation of force characteristics, determine stability, study qualitatively the dynamics of the levitated body, see their dependence on the distributions of the field sources, and pose an inverse problem: find the source distribution providing the desired characteristics of the suspension. Even though the derivations are done for a spherical body, it is possible to consider small deviations from the spherical shape of the levitated body [30] and investigate the effects of non-sphericity on the dynamics of the levitated body [32-34]. The model may be useful to find distributions of sources providing a stable and robust equilibrium while satisfying mechanical and potentially thermal demands on the entire system. We use a quasi-static approximation, excluding the electromagnetic waves from the consideration [35].
An asymptotic approximation of Love wave frequency in a piezo-composite structure: WKB approach
Published in Waves in Random and Complex Media, 2021
The equations of motion, electrostatics and magneto statics in the usual quasistatic approximation is given by [24] where is the displacement component, is the mass density.