scholarly journals Maxwell’s Equations on Cantor Sets: A Local Fractional Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yang Zhao ◽  
Dumitru Baleanu ◽  
Carlo Cattani ◽  
De-Fu Cheng ◽  
Xiao-Jun Yang
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Cold Dark Matter ◽  
Differential Forms ◽  
Cantor Sets ◽  
Unified Theory ◽  
Fractional Operators ◽  
Vector Calculus ◽  
Electric And Magnetic Fields ◽  
Fractional Differential

Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell's equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.

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).

WAVE AND MAXWELL'S EQUATIONS IN CARNOT GROUPS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250032 ◽  
Author(s):  
BRUNO FRANCHI ◽  
MARIA CARLA TESI
Keyword(s):  
Vector Potential ◽  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Higher Order ◽  
Carnot Groups ◽  
New Class ◽  
Euclidean Setting

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

Fields—Maxwell’s Equations

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Electromagnetic Field ◽  
Charged Particle ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Dipole Moments ◽  
Scalar Fields ◽  
Vector Calculus ◽  
Klein Gordon ◽  
Energy And Momentum ◽  
Scalar Potentials

Chapter 3 explores the concept of the field, which is necessary to describe forces without resorting to action at a distance, and uses it to describe electromagnetism, as encapsulated by the Maxwell equations. First, scalar fields and the Klein–Gordon equation are discussed. Vector calculus is introduced. The physical meaning of Maxwell’s equations is explained. The equations are then solved for electrostatic fields. Non-uniform charge distributions and dipole moments are discussed. The vector and scalar potentials are introduced. Electromagnetic wave solutions of Maxwell’s equations are found and the Hertz experiment is described. Magnetostatics is discussed briefly. The Lorentz force is described and used to determine the motion of a charged particle in a cyclotron or synchrotron. The action principle for electromagnetism is described. The energy and momentum carried by the electromagnetic field are calculated. The reaction of a charged particle to its own electromagnetic field is considered.

About Maxwell’s equations on fractal subsets of ℝ3

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Alireza Golmankhaneh ◽  
Ali Golmankhaneh ◽  
Dumitru Baleanu
Keyword(s):  
Differential Form ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Measure Theory ◽  
Computer Programs ◽  
Physical Equation ◽  
Fractional Differential ◽  
Local Derivative ◽  
Fractional Differential Form

AbstractIn this paper we have generalized $$F^{\bar \xi }$$-calculus for fractals embedding in ℝ3. $$F^{\bar \xi }$$-calculus is a fractional local derivative on fractals. It is an algorithm which may be used for computer programs and is more applicable than using measure theory. In this Calculus staircase functions for fractals has important role. $$F^{\bar \xi }$$-fractional differential form is introduced such that it can help us to derive the physical equation. Furthermore, using the $$F^{\bar \xi }$$-fractional differential form of Maxwell’s equations on fractals has been suggested.

Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell’s Equations

2013 ◽  
Vol 13 (4) ◽  
pp. 1107-1133 ◽  
Author(s):  
Tony W. H. Sheu ◽  
L. Y. Liang ◽  
J. H. Li
Keyword(s):  
Dispersion Relation ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Angular Frequency ◽  
Numerical Dispersion ◽  
Local Conservation ◽  
Symplectic Scheme ◽  
Electric And Magnetic Fields ◽  
The Difference ◽  
Explicit Finite Difference

AbstractIn this paper an explicit finite-difference time-domain scheme for solving the Maxwell’s equations in non-staggered grids is presented. The proposed scheme for solving the Faraday’s and Ampere’s equations in a theoretical manner is aimed to preserve discrete zero-divergence for the electric and magnetic fields. The inherent local conservation laws in Maxwell’s equations are also preserved discretely all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme. The remaining spatial derivative terms in the semi-discretized Faraday’s and Ampere’s equations are then discretized to provide an accurate mathematical dispersion relation equation that governs the numerical angular frequency and the wavenumbers in two space dimensions. To achieve the goal of getting the best dispersive characteristics, we propose a fourth-order accurate space centered scheme which minimizes the difference between the exact and numerical dispersion relation equations. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions.

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