Maxwell's equations in the field of electricity and magnetism
Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang–Mills This text is an introduction to some ...
Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang–Mills
This text is an introduction to some of the mathematical wonders of Maxwell's equations. These equations led to the prediction of radio waves, the realization that light is a type of electromagnetic wave, and the discovery of the special theory of relativity. In fact, almost all current descriptions of the fundamental laws of the universe can be viewed as deep generalizations of Maxwell's equations. Even more surprising is that these equations and their generalizations have led to some of the most important mathematical discoveries of the past thirty years. It seems that the mathematics behind Maxwell's equations is endless. The goal of this book is to explain to mathematicians the underlying physics behind electricity and magnetism and to show their connections to mathematics. Starting with Maxwell's equations, the reader is led to such topics as the special theory of relativity, differential forms, quantum mechanics, manifolds, tangent bundles, connections, and curvature.
More details: Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang–Mills
A Different Approach to the Theory of Electromagnetic and Gravitational Fields
This book is a strikingly new exploration of the fundamentals of Maxwell's electromagnetic theory and of Newton's theory of gravitation. Starting with an analysis of causality in the phenomenon of electromagnetic induction, the author discovers a series of heretofore unknown or overlooked electromagnetic interdependencies and equations. One of the most notable new results is the discovery that Maxwell's equations do not depict cause and effect relations between electromagnetic phenomena: causal dependencies in electromagnetic phenomena are found to be described by solutions of Maxwell's equations in the form of retarded electric and magnetic field integrals. A consequence of this discovery is that, contrary to the generally accepted view, time-variable electric and magnetic fields cannot cause each other and that both fields are simultaneously created by their true causative sources -- time-dependent electric charges and currents. Another similarly important discovery is that Lenz's law of electromagnetic induction is a manifestation of the previously ignored electric force produced by the time-dependent electric currents. These discoveries lead to important new methods of calculations of various electromagnetic effects in time- depended electromagnetic systems. The new methods are demonstrated by a variety of illustrative examples. Continuing his analysis of causal electromagnetic relations, the author finds that these relations are closely associated with the law of momentum conservation, and that with the help of the law of momentum conservation one can analyze causal relations not only in electromagnetic but also in gravitational systems. This leads to the discovery that in the time-dependent gravitational systems the momentum cannot be conserved without a second gravitational force field, which the author calls the "cogravitational, or Heaviside's, field." This second field, first predicted by Heaviside, relates to the gravitational field proper just as the magnetic field relates to the electric field. The author then generalizes Newton's gravitational theory to time-dependent systems and derives causal gravitational equations in the form of two retarded integrals similar to the retarded integrals for the electric and magnetic fields introduced previously. One of the most important consequences of the causal gravitational equations is that a gravitational interaction between two bodies involves not one force (as in Newton's theory) but as many as five different forces corresponding to the five terms in the two retarded gravitational and cogravitational field integrals. These forces depend not only on the masses and separation of the interacting bodies, but also on their velocity and acceleration and even on the rate of change of their masses. A series of illustrative examples on the calculation of these new forces is provided and a graphical representation of these forces is given. The book concludes with a discussion of the possibility of antigravitation as a consequence of the negative equivalent mass of the gravitational field energy.
Review: Causality, Electromagnetic Induction, and Gravitation
The book is written in the style and format of a textbook. The clear presentation, the detailed derivations of all the basic formulas and equations, and the many illustrative examples make this book well suitable not only for independent studies but also as a supplementary textbook in courses on electromagnetic theory and gravitation. The second edition of the book refines and improves the first edition, especially in the presentation and development of Newton's gravitational theory generalized to time-dependent gravitational systems. The book has been augmented by several new Appendixes. Particularly notable are Appendixes 5, 6, and 8. Appendixes 5 and 6 present novel "dynamic" electric and gravitational field maps of rapidly moving charges and masses. Appendix 8 contains the little-known but extremely important Heaviside's 1893 article on the generalization of Newton's gravitational theory.
James Clerk Maxwell, the discreet one
In the field of physics, he is placed in the same category as Einstein or Newton. However, he is usually a name unknown to the outside world: James Clerk Maxwell. He took important steps in understanding electricity, magnetism, and light. It is possible that his brave personality is the reason for his relatively limited popularity. The excellent Scotsman turned out to be very enthusiastic when he spoke to Kennislink.
A flash of lightning flashes across the sky and you immediately realize that electricity is one of the most impressive forces in nature. For a long time, people did not know exactly what electricity was. Much thanks to the Scotsman James Clerk Maxwell (1831 - 1879) we already know. In mathematical equations, he established that electricity, magnetism, and light were all expressions of the same thing: the electromagnetic field. The fact that we no longer envision a world without electricity today is largely due to him.
Lesser known is that he also achieved success in other areas, such as color vision, gas activity and Saturn's rings. He also made his first color photograph and is a virtuoso poet. Maxwell is a beloved man, with a warm and engaging personality. Kennislink conducted a 'fictional interview' with the mathematician and physicist.
Important: James Clerk Maxwell is an acceptor of free Ether and energy. His letter to Nikola Tesla speaks for itself. See: Magnetic energy generator problem in theory of electricity and magnetism
Maxwell: a short biography
1831: Born June 13 in Edinburgh, the only child of lawyer John Clerk and his wife Frances Cay. 1841: Beginning of Edinburgh Academy. 1847: Beginning of scientific research at the University of Edinburgh. 1850 : Entered Cambridge University without a degree to study Mathematics. 1854: Graduated from Cambridge and continued to work there. 1856: Began as professor of natural philosophy at Marischal College, Aberdeen. 1858: Married Katherine Mary Dewar. 1860: Professor of physics and astronomy at King's College, London. 1861: Shows the first color photograph. 1871: Appointed Cavendish Professor at Cambridge University. 1873: His book Treatise on electricity and magnetism is published containing his famous equations of electromagnetism. 1879: Died of abdominal cancer on 5 November in Cambridge.
Maxwell's Equation
Initially Maxwell gave a series of twenty equations containing twenty variables. It was then reduced by Oliver Heaviside, the Englishman, to four equations with two variables, as we know them today. To really understand what the math formulas say, you need to know a lot about algebra. Here we will briefly explain what each equation entails.
- ∇⋅E = ρ / ε 0 (Gauss Law) Simply put, here it says that the electric field (E) arises from the charge (ρ). A charged particle (for example an electron) will feel a force in this electric field. The formula also contains a constant, ε 0 , called the allowable degree.
- ∇⋅B = 0 (Gauss Law for Magnetism) Unlike an electric field, a magnetic field has no source. That's what this recipe says. A magnetic field is always running around, from the north pole through the south pole back to the north pole. In other words, magnetic monopoles (such as a positive charge) do not exist.
- ∇ x E = -∂B / ∂t (Faraday's Law) This is what Faraday discovered in cryptic form: a changing magnetic field induces an electric current. In physics, this is called magnetic induction.
- ∇ x B = μ 0 J + μ 0 ε 0 ∂E / ∂t (Ampère's Law, adding Maxwell) It is said here that a circular magnetic field is created around a wire when current flows through it. . The last part of the formula - containing the electric field E - was added by Maxwell and states that the magnetic field is produced by a changing electric current (see main story).
That Ampere's Law is incomplete. According to this law, a circular magnetic field is created around a conductor through which current flows. When I compare the equations of this law with others, I notice that they do not quite match. To make the whole thing symmetric, I added an additional term to the equation, displacement current. Adding this term, my equation says that a changing electric field produces a magnetic field. Yes, that has yet to be proven in experiments, but it must be a matter of time.
Related: Rotating Magnetic Field
