Octonionic maxwells equations for biisotropic media. In this video we will discuss about maxwells equations for harmonically varying field on this channel you can get education and knowledge for general issu. Biquaternions algebra and its applications by solving of. On the changing form of maxwells equations during the last. A new model for the reformulation of compressible fluid equations. Pierce famously wrote to anyone who is motivated by anything beyond the most narrowly practical, it is worthwhile to understand maxwells equations simply for the good of his soul. A new wave quantum relativistic equation from quaternionic. The decoupling of the equations into the wave equation is very simple and natural. Biquaternionic model of electrogravimagnetic field, charges. Ecuatiile lui maxwell in teoria unitaria neolonoma. Looking for this representation was driven by the question of whether or not the compact geometric algebra representations of maxwell s equations \\ \\grad f.
Then maxwells equations are derived in terms of the biquaternionic repre sentations of differantial vector operator, electromagnetic bivector. Biquaternionic form of laws of electrogravimagnetic. P fe and m fb, which are denoted constitutive relations. With the publication of a dynamical theory of the electromagnetic field in 1865, maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light.
It is shown that maxwell s equation cannot be put into a spinor form that is equivalent to diracs equation. There also are perfect analogies between linear gravity and electromagnetism. Feb 03, 20 biquaternionic form of generalized maxwell dirac equation is constructed and its solutions are researched on base of the differential biquaternions algebra. We compare this approach to related descriptions of maxwell s equations by biquaternions and clifford algebras. Maxwell s equations are a set of coupled partial differential equations that, together with the lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Generalized dirac equation with vector structural coefficient. In the previous lecture, we discussed maxwells law i.
Struppa b dipartimento di matematica, universitd di milano, via saldini 50, milano, italy b department of mathematical sciences, george mason university, fairfax va 22030, usa received. The new term added by maxwell is called the displacement current and is responsible, with the. The term maxwell s equations is often also used for equivalent alternative formulations. Among all the subjects, we mention the treatment of fractional differential equations regarding the mathematical methods of. Mar 05, 2018 click here for a pdf of this post with nicer formatting motivation. Second, diracs equation implies a complex structure specific to. Aug 31, 2004 we express maxwell s equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the electromagnetic field. Lanczoss idea of classical electrodynamics as a biquaternionic field the. A new model for the reformulation of compressible fluid. It is also important to note, however, that for static fields, i. The quaternion form of maxwell s equations as stated in 2 is nearly indecipherable. A complex quaternion a biquaternion z is defined as.
Biquaternionic formulation of maxwells equations and their. As far as i am aware, this technique is not in the literature, up to an isomorphism meaning actually it is there but under a different name, math in disguise. The vector forms, written below, consist of 4 equations. After defining biquaternions with complex numbers, the algebra of biquaternions and some properties are introduced. Then maxwell s equations are derived in terms of the biquaternionic representations of differantial vector operator, electromagnetic bivector. Analogies between lines of force and streamlines in an incompressible. Welcome to the website for a students guide to maxwells equations, and thanks for visiting. Maxwells equations result in electromagnetic waves that are transverse disturbances int he electric and magnetic. May 23, 2011 maxwell s equations are taken into consideration when it comes to biisotropic media in which the electric and magnetic fields are coupled by means of biisotropic constitutive relations. The modern quaternionic form of these equations can be found in 1. The theory of functions of a real biquaternion variable and the solutions of maxwells equations are recapitulated. Biquaternions, biqs, maxwells equations and regular conditions for a function of a biq variable.
Biquaternionic reformulation of a fractional monochromatic maxwell system. Generalized hypercomplex analysis and its integral formulas. The stressenergymomentum 2tensor of the electromagnetic field. Maxwells equations in vacuum t e b t b e b e o o w w u w w u x x ph 0 0 the vacuum is a linear, homogeneous, isotropic and dispersion less medium since there is no current or electric charge is present in the vacuum, hence maxwells equations reads as these equations have a simple solution interms of traveling sinusoidal waves. A derivation of maxwell equations in quaternion space core. Maxwells theory combined in one stroke electricity, magnetism, and light into an eternal braid1. Spinor representation of maxwells equations iopscience. A study of the application to diffraction of light. We compare this approach to the related descriptions of maxwells equations by biquaternions and clifford algebras. Deriving the maxwell source equations using quaternions 2. Maxwell s equations for electromagnetism have been called the second great unification in physics6 after the first one realised by isaac newton. Split biquaternions when the coefficients are splitcomplex numbers. In the last four lectures, we have been investigating the behavior of dynamic i.
There are three types of biquaternions corresponding to complex numbers and the variations thereof. In the first half century, an elegant presentation of maxwells equations and special relativity was achieved with the help of biquaternions, that is, quaternions with. By using generalized functions theory the fundamental and regular solutions of this. Philosophical transactions of the royal society of london, 1865 155, 459512, published 1 january 1865.
Gauge transformation and electromagnetism with biquaternions. The particular types of biwave equations were considered, which are equivalent to the systems of maxwell and dirac. Maxwell s equations without sources in the dimensionless form are given. Regular functions of biquaternionic variables and maxwells equations. Regular functions of biquaternionic variables and maxwell. Biquaternions when the coefficients are complex numbers. In this discussion, vectors are denoted by boldfaced underscored lowercase letters, e. In the modern approach initiated by gursey, these difficulties are solved by recognizing that most generalizations. On the equivalence of the maxwell and dirac equations. Quaternions, maxwell equations and lorentz transformations. Similarly, majernik 16 has used the biquaternion theory to generalize the maxwelllike gravitational. Feb 01, 2017 moreover, biquaternions can also be employed to generalize the maxwell equations in presence of electric, magnetic sources and massive photons. Chapter maxwells equations and electromagnetic waves. Moreover, close connections exist between the solutions to eqs.
It is known that biquaternions could describe maxwell equations in its original form, and some of the use of biquaternions was discussed in 234. The above equations may also be written in differential forms as 0 000 0 t t. The more familiar di erential form of maxwells equations can be derived very easily from the integral relations as we will see below. Three maxwell equations can be found quickly in the original set, together with o hm s law 1. Computations become more reliable, e cient, and transparent if speci c plasma models are used to obtain only the information that would otherwise be missing. A derivation of maxwell equations in quaternion space. Maxwell s original equations before heaviside, et al. Biquaternionic reformulation of a fractional monochromatic. In the absence of sources where, the above equations become j g q0, i0 00 0. They can be uncoupled by taking another derivative. Download the ios download the android app company about us. Comment on formulating and generalizing diracs, procas, and maxwell s equations with biquaternions or clifford numbers.
Many difficulties of interpretation met by contemporary researchers attempting to recast or generalize diracs, procas, or maxwell s theories using biquaternions or clifford numbers have been encountered long ago by a number of physicists including lanczos, proca, and einstein. Biquaternionic model of electrogravimagnetic field. The maxwell source equations will be derived using quaternions an approach james clerk maxwell himself tried and yet failed to do. Citizenship british alma mater university of edinburgh university of contents cambridge life known for maxwell s equations early life, 18311839 maxwell relations education, 18391847 maxwell distribution university of edinburgh, 18471850 maxwell s demon university of cambridge, 18501856 maxwell s discs marischal college, aberdeen. The drudebornfedorov constitutive relations defined with the complex representations of electric and magnetic fields are used for all calculations. Its generalized decisions are built with use of scalar potential. One the base of maxwell and dirac equations the one biquaternionic model of electrogravimagnetic egm fields is considered. Comment on formulating and generalizing diracs, procas. Complex symmetric formulation of maxwell equations for. The reason for this is that with the correct set of equations, maxwell was able to obtain solutions of the four equations, predictions of these equations that could not be obtained until. This assumption is false, because mathematics and maxwells equations have a universality not shared by speci c plasma models. A study of the application to diffraction of light by a slit or a hole in a screen is described. A quaternion formulation of the dirac equation metrics. The purpose of this site is to supplement the material in the book by providing resources that will help you understand maxwells equations.
Travis nasa goddard institute for space studies, 2880 broadway, new york, ny 10025, usa 1. Such a formulation has the advantage of being closely connected to the physical situation. Also, the connection with lorentz transformations and the very elegant description of maxwells equations using biquaternions has been studied. Pdf deductions from the quaternion form of maxwells. Deriving the maxwell source equations using quaternions 25. A derivation of the quaternion maxwells equations using.
Biquaternionic form of laws of electrogravimagnetic charges. Biquaternions algebra and its applications by solving of some. Comment on formulating and generalizing diracs, procas, and. The novelty of this work is the construction of laws of electrogravymagnetic interactions on the base of biquaternions.
In praise of quaternions mathematics and statistics mcgill. Versions of maxwell s equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. Maxwells equations for harmonically varying field youtube. Solving sourcefree maxwells equations september 4, 2018 abstract it is shown that the solutions of the sourcefree maxwells equations are the same as those obtained from maxwells equations which contain a source, and that the choice of the lorentz gauge is compulsory rather arbitrary.
It could be argued that these equations got their start 150 years ago. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Regular functions of biquaternionic variables and maxwells. Maxwells death by heavis ide to those very few and more manageable equa tions we now refer to as m axwell s equations, wh ich are written in modern vecto r notation foll owing the w. In the theory, the nature of minkowski space, lorentz transformations and maxwell s equations is reduced to those of biquaternions and their functions, the theory of functions of a biquaternion. In this work, we obtain one maxwell equation, 10, representing the electromagnetic field as a matrix and the divergence as a vector multiplying. James clerk maxwell, a dynamical theory of the electromagnetic field royal society transactions, vol. For the value of electric charge from gcre only the. A solution of maxwells equations require the compatibility condition. Biquaternionic formulation of maxwells equations and.
Jul 01, 1998 journal of geom physics elsevier journal of geometry and physics 26 1998 183201 regular functions of biquaternionic variables and maxwell s equations f. Jan 01, 2017 an exact matrix representation of maxwell s equations. The role of the paulilubanski vector for the dirac, weyl, proca, maxwell and fierzpauli equations. Newtons laws for a biquaternionic model of the electrogravimagnetic field. The closed system of biquaternionic wave equations is constructed for determination of free system of electric and gravimagnetic charges and currents and generated by them egmfield. The original quaternion form of maxwells equations. Intuitiveguidetomaxwellsequationsan intuitive guide to.
Jan 02, 2007 after defining biquaternions with complex numbers, the algebra of biquaternions and some properties are introduced. Maxwell equations have two types of asymmetries between the electric and magnetic fields. Maxwells equations and light san jose state university. Maxwells equations a dynamical theory of the electromagnetic field james clerk maxwell, f. Biquaternionic form of generalized maxwell dirac equation is constructed and its solutions are researched on base of the differential biquaternions algebra. The novelty of this work is the construction of laws of electrogravymagnetic interactions on the base of biquaternions algebra. A study of the application to diffraction of light by.
The divergence of the stress energy tensor gives the lorentz law in a very natural way. Maxwell was the first person to calculate the speed of propagation of electromagnetic waves which was same as the speed of light and came to the conclusion that em waves and visible light are similar these are the set of partial differential equations that form the foundation of classical electrodynamics, electric circuits and classical optics along with lorentz force law. An intuitive and visual guide to understanding maxwell s equations. Chapter 6 maxwells equations for electromagnetic waves. In chapter 4, lanczos rewrites maxwells equations in vacuum as 4. Differential algebra of biquaternions gives possibility to simplify mathematical record of systems of maxwell and dirac equations to construct their solutions and to study their properties. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. Introduction the theoretical basis for describing elastic scattering of light by particles and surfaces is formed by classical electromagnetics. In maxwell equations the density of the magnetic charges is. Generalisation of classical electrodynamics to admit a scalar field. Spacetime algebra for the generalization of gravitational field. A 2d meshless timedomain algorithm for solving maxwell s equations based on stegerwarming flux vector splitting approach. The maxwell equations predict the existence of light 1j.
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