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# Electromagnetism ### INTRODUCTION

A good knowledge of electromagnetism, part of the physics which deals with the relations between electrical and magnetic phenomena, is one of the necessary bases for the electrical engineer; we have therefore endeavored to present a logical, precise, useful presentation that can even increase the general culture of the electrical engineer.

Before detailing each point, let us first note that we have adopted the notations and the system of units defined by French and international standards [International Bureau of Weights and Measures (1985), Union Technique de l’Électricité (1981), International Union of Pure and Applied Physics (1965)] (cf. articles Legal units and conversion factors [A 24] in the treatise on Plastics and Composites and MKSA system of units by Giorgi [D 50] in the treatise on Electrical engineering). We therefore use the following vocabulary:

• for the basic sizes: (1)
• for sources: (2)
• in the case of linear relations:

These details are very important in the field of electromagnetism where it is difficult to say that unanimity is reached! All the considerations which lead us to show that such and such a system is better than all the others appear to us to be artificial: a system is good if it makes it possible to identify the different types of quantities by different types of symbols while remaining as close as possible to physical reality.

• Considerations of pure logic show that a relation whose domain of validity is ignored is useless and can even be dangerous. We therefore sought to show the origin of the various relations by distinguishing from the beginning of the article, on the one hand, the general laws of electromagnetism (Maxwell’s equations and energy relation) and, on the other hand, the special relationships corresponding to the different materials. It is thus necessary to distinguish a general law, for example) from a particular relation which is not a law but a valid relation in certain cases (frequently realized, it is true). Moreover, there are two types of presentation (microscopic and macroscopic) of Maxwell’s equations depending on whether, for example, in a solid, the volume charge density is defined either by sliding between atoms (clearly less than 10 –3 nm3, the volume of a atom being of the order of 30 · 10 –3 nm3), or, on the contrary, by considering ones comprising at least 10 4 atoms. It is this last presentation (macroscopic) that we have considered because it is the simplest and makes it possible to deal with almost all the problems (the field of superconductors being excluded, cf., in this treatise, article [D 2 700] Superconductors) which present themselves to the electrotechnicians. Finally, so that the reader can follow the logic of the exposition, we have always either given the details of the simple demonstrations, or only indicated the diagram of the unfolding of the ideas in the complicated cases; it is thus possible, for example, to see that the only general law on the volume density of magnetic energy is its variation and not its value which only has meaning for ideal bodies, that is to say governed by a law of strict proportionality between.
• For an account to be precise, it must include only well-defined quantities intervening only in intrinsic relations. The intensity of a current can only be well defined after having indicated the direction by relation to which this current is identified. The potential difference between two points must also be specified by U = VA – VB or U = VB – VA. The charge Q of a capacitor does not make sense: it is necessary to indicate the charges Q i and Qj of the electrodes i and j, etc. The surface charge density σ at the limit of two media must be expressed in the form of the intrinsic relation is the unit normal directed from the middle to the middle. This relation is quite intrinsic since the permutation of does not change the result; the same is not true for the widely used expression: σ = D 1n – D 2n. We can combine the two types of imprecision in a relation of the type Q = CU that is completely asexual (that is to say unsigned), while the intrinsic expression of the charge of an electrode of an ideal capacitor is Qi = C (Vi – Vj).
• Electrical technicians are increasingly using non-sinusoidal currents and base frequencies higher than 50 Hz. We have therefore, for certain problems (evaluation of losses for example), considered the evolution of phenomena as a function of frequency and shown that one could be satisfied, with a fairly good precision, to use two asymptotic laws respectively valid for. We have thus developed a method which makes it possible to calculate fairly simply the losses by the Joule effect in a conductor of any section through which any periodic current flows. In addition, the use of non-sinusoidal currents and high frequencies shows that it is increasingly necessary for electrical engineers to acquire a good basic knowledge of magnetic materials. The example shows that sometimes there are large gaps in this area and the definition of magnetization as the volume density of magnetic moment (which suggests that is a spatially continuous quantity) has done a lot of harm in this regard: we can only truly understand the behavior of ferromagnetics by considering the truth, i.e. the existence of Weiss domains and their separation by Bloch walls. We have tried to present these concepts in the simplest possible way in paragraph 2.2.4.
• To highlight the characteristics of the different physical quantities, we can first distinguish two types of vectors :
• polar vectors (such as force, electric field, electric displacement) which have the symmetry of an arrow;
• axial vectors (such as torque, magnetic field, magnetic induction) that have the symmetry of a spinning top; at this point it is necessary to have a corkscrew to define the three components to the right of this type of “vector” while a stuffing and grabbing corkscrew would provide the three components to the left. necessary to acquire this difference then makes it possible to foresee the framework of the possible relations and thus to better understand the phenomena: with regard to flow, for example, we show that a good physical quantity can only concern the flow of through a closed surface limiting a volume or the flow of through a surface resting on and limited by a closed contour.A deeper and more general analysis shows that each physical quantity can be characterized by means of two criteria:
• its dimensional nature (linked to the modifications of the measurement of this quantity when we change the base units);
• its tensorial nature (linked to the modifications of the components of this quantity when we change the base vectors which make it possible to locate the space); this second criterion [very often neglected and wrongly considered to be very difficult, which has led us, in order not to put off readers, to postpone all that concerns it in appendix A 4] allows us to acquire more synthetic notions on physics in general and electromagnetism in particular. A general relation of equality can therefore only unite two quantities of the same tensorial nature, that is to say quantities whose components react in the same way when we modify the basic vectors of space. Since it is possible to show that they do not have the same tensorial nature, there cannot therefore exist, even in the case of a vacuum (and whatever the system of units chosen), a general relation of pure proportionality between; nevertheless, in this case, if one obliges oneself to use only triorthogonal base vectors, the relation is usable as well as the relation “the length of the ship = the age of the captain” remains valid if one obliges oneself to keep the same units of length and time. It is also possible to show that they do not have the same tensor nature and it is therefore impossible to claim in general that, in the case of a vacuum, the difference between these quantities n ‘ is only a question of units.It is always the tensorial nature of the quantities which shows (contrary to what one can read in certain works) that the theorem of Gauss must involve in general the electric charges and the flux de whereas Ampère’s theorem must link the currents and the circulation of. The errors that we have just pointed out (example: in a vacuum) remain in the literature because their users, as long as they limit themselves to the use of triorthogonal basis vectors, obtained nnent correct results without being penalized; nevertheless, one should not confuse a convenient process and its conditions of use (which we will use) with the reality of the phenomena. It is in this sense that we can show that the so-called axial vectors are in reality quantities of a type which, in 3-dimensional space, have 3 2 = 9 components (the 3 ex-components on the right , the 3 ex-components on the left and 3 zero components), while a true vector (ex-polar vector) is defined by 3 components. If, in love with modern physics, we consider the relativistic doctrines (which date from 1905) in which four coordinates (x, y, z, t) of space-time intervene in an indissolubly linked way, we show that electromagnetism is essentially constituted from two quantities of the component type (four of which are zero): the first quantity is formed from the components of, the second from. The comparison thus made between, on the one hand, on the other hand, shows that the adopted vocabulary [relations ] is not rational. However, it seems difficult to change the designation of physical quantities with each new advance in knowledge. For example, the supporters of the change, since the discovery (1939) of the possibility of fission of certain atoms should speak of the fission of the breakable uranium 235 (and no longer of the uranium 235 atoms). complete tensorials of, the type being only a partial indication, then show that must necessarily satisfy a certain relation which leads, in the language of 3-dimensional space, to: (3) (4) while , for, the simplest authorized relation (and therefore the first to try) is translated under the same conditions by: (5) (6) We thus find the classical Maxwell equations 1.2.1 from pure tensorial considerations. : The article includes three parts respectively devoted to the basics of electromagnetism, its different aspects and its applications to electrical engineering. Appendix A 4 is devoted to the tensor nature of quantities while appendix B 5 concerns the various differential operators and their applications.