scholarly journals Foundations of the Electromagnetic Theory

2018 ◽  
Vol 3 (1) ◽  
Author(s):  
Constantinos Krikos
Keyword(s):  
Experimental Data ◽  
Maxwell’S Equations ◽  
Vector Fields ◽  
Maxwell's Equations ◽  
Electromagnetic Theory ◽  
Theoretical Derivation ◽  
Systematic Classification

In this paper equations in R3 which are illustrations of “linear” ellipses, i.e. ellipses which tend to become segments of a geodesic of R2, because their eccentricities tend to unit () will be found. During a linearization process of ellipses, varying vectors will be mapped, from which ellipses and their relations in R2 , to varying vector fields and their relations in R3 are defined. These vector fields and their relations in R3 are called “holographic”. At the limit , the holographic relationships are formalistically similar to Maxwell's equations. This is a theoretical derivation of Maxwell’s equations and not a systematic classification of experimental data as Maxwell did.

Basic Electromagnetic Theory

2020 ◽  
pp. 68-77
Author(s):  
Magdalene Wan Ching Goh
Keyword(s):  
Electromagnetic Wave ◽  
Magnetic Fields ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Electromagnetic Theory ◽  
Electromagnetic Spectrum ◽  
Wave Polarization ◽  
Basic Principles ◽  
Electric And Magnetic Fields

Electromagnetic theory covers the basic principles of electromagnetism. This chapter explores relationships between electric and magnetic fields. The chapter describes the behaviour of electromagnetic wave. The four sets of Maxwell's equations which underpin the principles of electromagnetism are briefly explained. An illustration on wave polarization and propagation is presented. The author describes the classification of waves according to their wavelengths (i.e. the electromagnetic spectrum).

scholarly journals Abstract Nonconforming Error Estimates and Application to Boundary Penalty Methods for Diffusion Equations and Time-Harmonic Maxwell’s Equations

2018 ◽  
Vol 18 (3) ◽  
pp. 451-475 ◽  
Author(s):  
Alexandre Ern ◽  
Jean-Luc Guermond
Keyword(s):  
Error Analysis ◽  
Error Estimates ◽  
Material Properties ◽  
Maxwell’S Equations ◽  
Vector Fields ◽  
Maxwell's Equations ◽  
Heterogeneous Material ◽  
Test Functions ◽  
Time Harmonic ◽  
New Formulation

AbstractWe devise a novel framework for the error analysis of finite element approximations to low-regularity solutions in nonconforming settings where the discrete trial and test spaces are not subspaces of their exact counterparts. The key is to use face-to-cell extension operators so as to give a weak meaning to the normal or tangential trace on each mesh face individually for vector fields with minimal regularity and then to prove the consistency of this new formulation by means of some recently-derived mollification operators that commute with the usual derivative operators. We illustrate the technique on Nitsche’s boundary penalty method applied to a scalar diffusion equation and to the time-harmonic Maxwell’s equations. In both cases, the error estimates are robust in the case of heterogeneous material properties. We also revisit the error analysis framework proposed by Gudi where a trimming operator is introduced to map discrete test functions into conforming test functions. This technique also gives error estimates for minimal regularity solutions, but the constants depend on the material properties through contrast factors.

scholarly journals GLOBAL CLASSIFICATION OF A CLASS OF CUBIC VECTOR FIELDS WHOSE CANONICAL REGIONS ARE PERIOD ANNULI

2011 ◽  
Vol 21 (07) ◽  
pp. 1831-1867 ◽  
Author(s):  
M. CAUBERGH ◽  
J. LLIBRE ◽  
J. TORREGROSA
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Vector Fields ◽  
Topological Type ◽  
Radial Symmetry ◽  
Phase Portraits ◽  
Arbitrary Degree ◽  
Systematic Classification ◽  
Global Classification

We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincaré disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

scholarly journals The Noether current in Maxwell's equations and radiation under symmetry breaking

2018 ◽  
Vol 376 (2134) ◽  
pp. 20170452
Author(s):  
Dhiraj Sinha ◽  
Gehan Amaratunga
Keyword(s):  
Symmetry Breaking ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Dielectric Materials ◽  
Electromagnetic Theory ◽  
Global Symmetry ◽  
Electrodynamic System ◽  
Theme Issue ◽  
Physical Systems ◽  
Noether Current

Symmetries in physical systems are defined in terms of conserved Noether Currents of the associated Lagrangian. In electrodynamic systems, global symmetry is defined through conservation of charges, which is reflected in gauge symmetry; however, loss of charges from a radiating system can be interpreted as localized loss of the Noether current which implies that electrodynamic symmetry has been locally broken. Thus, we propose that global symmetries and localized symmetry breaking are interwoven into the framework of Maxwell's equations which appear as globally conserved and locally non-conserved charges in an electrodynamic system and define the geometric topology of the electromagnetic field. We apply the ideas in the context of explaining radiation from dielectric materials with low physical dimensions. We also briefly look at the nature of reversibility in electromagnetic wave generation which was initially proposed by Planck, but opposed by Einstein and in recent years by Zoh. This article is part of the theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.

III. On the connexion between electric current and the electric and magnetic inductions in the surrounding field

1885 ◽  
Vol 38 (235-238) ◽  
pp. 168-172 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Electric Current ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Electromagnetic Theory ◽  
Magnetic Intensity ◽  
The Wire

This paper describes a hypothesis as to the connexion between current in conductors and the transfer of electric and magnetic inductions in the surrounding field. The hypothesis is suggested by the mode of transfer of energy in the electromagnetic field, resulting from Maxwell’s equations investigated in a former paper (“Phil. Trans.,” vol. 175, pp. 343—361, 1884). It was there shown that according to Maxwell’s electromagnetic theory the energy which is dissipated in the circuit is transferred through the medium, always moving perpendicularly to the plane containing the lines of electric and magnetic intensity, and that it comes into the conductor from the surrounding insulator, not flowing along the wire.

scholarly journals The first and second order equations of the quantum theory

1929 ◽  
Vol 124 (793) ◽  
pp. 143-150 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Electromagnetic Theory ◽  
Second Order ◽  
Electromagnetic Mass ◽  
First Order ◽  
Second Order Equations

The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator ( h /2 πi ∂/∂ x α - eϕ α ) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc . Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.

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