Lecture notes Electromagnetic field theory

electromagnetic field and waves lecture notes and lecture notes in electromagnetic field theory. uniform electromagnetic field in the theory of general relativity, how quantum mechanics work
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ELECTROMAGNETICS GENERAL THEORY OF THE ELECTROMAGNETIC FIELD CLASSICAL AND RELATIVISTIC APPROACHES THIRD EDITION Revised and Augmented INTRODUCTION 1. CONTENT OF ELECTROMAGNETICS In this work, the foundations of Electromagnetics, including theory and applications are treated. It is useful to note that Electromagnetics and Electromagnetism can be considered synonyms. The theory of electromagnetism includes the introduction (i.e., the definition) of several fundamental concepts among which: Field and substance, electric charge, electric current, state quantities of electric and magnetic fields. Also, it contains the study of forces acting upon electric charge carriers in motion, laws and energy of electromagnetic field. The applications concern the corresponding topics. 2. THE THEORIES USED IN THE STUDY OF ELECTROMAGNETISM We recall that the Electromagnetism is a branch of Physics in which the electromagnetic phenomena are studied. It contains the study of physical bases and of the propagation of electromagnetic field. This work refers to physical bases only. The principal domains of electromagnetism are the following ones: Electrostatics, Electrokinetics, Magnetostatics and Electrodynamics. These domains are very useful for the study of macroscopic phenomena and in practical applications. The study of the domains above can be carried out by using the general laws of electromagnetism in these various cases. Certain important problems of the mentioned domains are analysed in the present work. A more detailed study of the mentioned domains can be found in several works devoted to these subjects, including the works of the author, mentioned in Bibliography. In the study of electromagnetism, the following theories are utilized: Theory of electromagnetic field (Theory of Maxwell), Theory of electrons (Theory of Lorentz), Theory of relativity and Quantum Mechanics. The theory of Maxwell is the macroscopic theory of electromagnetic phenomena. In the framework of this theory, relationships between the quantities that characterize the electric and magnetic state of the substance are given in the form of a set of differential equations. The theory refers to media at rest. An extension of this theory to moving media was made by Heinrich HERTZ. The Theory of electrons is the microscopic theory of electromagnetic phenomena, which admits the existence of certain elementary charged particles, called electrons. The electron is characterized by its electric charge, mass, and magnetic moment. In the framework of this theory, the ponderomotive forces of the electromagnetic field are exclusively determined from the forces exerted upon particles and expressed by the General Theory of the Electromagnetic Field 24 Lorentz formula. The electromagnetic field equations are obtained by applying the Maxwell equations for empty space (i.e., vacuum) at microscopic scale. The theory of electrons can be presented in either quantum or non-quantum form, respectively. The non-quantum form of the theory of electrons has also two forms, namely: non-relativistic and relativistic one. In the framework of the non-relativistic form, the existence of a privileged reference frame is assumed. This reference frame is at rest with respect to the group of fixed stars and is referred to as Lorentz inertial reference frame. The theory of electrons refers to media at rest as well as to moving media. The non-quantum theory of electrons cannot be put in accordance with some properties of elementary particles and the utilisation of Quantum Mechanics then becomes necessary. Finally we shall recall that the fundamental physical interactions or forces, in nature, are of the following four types: Electromagnetic, Weak, Heavy and Gravitational. 3. SHORT HISTORICAL SURVEY In this Section, certain data of the history of the development of the knowledge of electromagnetic phenomena will be presented. The first knowledge about electric and magnetic phenomena refers to natural magnetism and to electrification by friction. Magnet and magnetism are so termed because the loadstone (iron ore) µȐȖȞȘȢ (magnes) was originally found in the Thessalian Magnesia. Also, in Antiquity the electrification by friction of amber, called in Ancient Greek µ (kehrimpari, read kechrimpari) or țIJȡȠȞȒȜİ (elektron, read ilektron), was known. This manner of electrification was described by THALES of Millet (640 547 BC). The development of electromagnetism was related to a great extent to the discovery of the law of the force exerted between two point-like bodies, charged with electricity (i.e., having electric charge). The establishment of this law had several stages due to the research of Benjamin FRANKLIN (1706 – 1790), Joseph PRIESTLEY (1733 – 1804), John ROBISON (1739 – 1805), Henry CAVENDISH (1731 – 1810) and Charles-Augustin de COULOMB (1736 – 1806). Coulomb performed experiments by two different methods. In the first method, he used a torsion balance and measured the angle proportional to the force exerted between electrified bodies. In the second method, he used an apparatus with an oscillating device and determined the number of oscillations that depends on the force exerted between the electrified bodies; the results were published in 1785. With respect to the previous experiments, he established the results more directly and also mentioned that the force is directly proportional to the product of the quantities of electricity (electric charges) of the two electrified bodies. He established with a high precision that the force exerted between two electrified point-like bodies is inversely proportional to the square of the distance between them. Carl Friedrich GAUSS (1777 – 1855) established important formulae in Electrostatics and Magnetostatics. Hans Christian OERSTED (1777 – 1851) experimentally remarked the action exerted by an electrical conductor carrying an electric current, on a magnetic needle. This experiment was determined by the remark that the magnetic needle of a compass makes ȡȚʌȐțİȤȡȚIntroduction 25 oscillations during a storm. The result was published in 1820. This result has been of a great importance, because it has allowed the establishment of the relation between two classes of phenomena, previously independently treated. At the same time, in the year 1820, Jean-Baptiste BIOT (1774 – 1862), Félix SAVART (1791 – 1841) and Pierre-Simon de LAPLACE (1749 – 1827) established the relation expressing the interaction between an element of electric current and a magnetic pole. Continuing this research, André-Marie AMPÈRE (1775 – 1836) established the same year 1820, that forces are exerted between two conductors carrying electric currents. He also introduced the difference between electric potentials (potential difference, voltage, electric tension) and electric current. He showed that a permanent magnet in the form of a bar is equivalent to a coil carrying an electric current. It is worth noting that at present the Ampère conception lies at the base of the theory of magnetism. Georg Simon OHM (1789 – 1854) established in 1826 the relationship between electric tension (voltage) and the intensity of the electric current. Two very important discoveries lie on the ground of the theory of electromagnetic field. The first one is the fundamental discovery made by Michael FARADAY (1791 – 1867) and consists in the fact that a magnetic field varying with time induces (i.e., produces) an electric field, what he experimentally established. A historical survey of the research carried out by several scientists on this subject can be found in literature 13, 25. The second one belongs to James Clerk MAXWELL (1831 – 1879). Maxwell established in a theoretical way that, conversely, an electric field varying with time induces (i.e., produces) a magnetic field. Therefore, the electric fields varying with time have the same effect as the conduction currents concerning the production of the magnetic fields. Hence, the variation with time of the electric field may be considered as corresponding to an electric current, called by Maxwell displacement current. To the previous two components of the electric current (i.e., conduction current and displacement current) it is to be added a third component namely the convection current. It is produced by the motion of electrified bodies with respect to a reference system. This component was studied by several scientists, among which Henry ROWLAND (1848 – 1901) and N. VASILESCU KARPEN (1870 – 1964). At the same time, Faraday introduced the concept of line of force, in order to visualize the magnetic field and subsequently the electric field. Faraday thought the seat of electric phenomena as going on a medium, whereas previously, mathematicians thought the same phenomena as being produced by centres of forces acting at a distance. The conception of Faraday allowed him to replace the concept of action at distance by the concept of a local interaction between electrified bodies and a field of forces, what has had a great importance for the subsequent development of the theory of electromagnetic field. After six years of experimental researches, Faraday discovered in the year 1831 the phenomenon of electromagnetic induction mentioned above. In the first experiment, he utilized a soft iron ring having the cross section diameter of about 2.22 cm and the exterior diameter of about 15.24 cm. On this ring, there were wound two coils of insulated copper wires. The ends (terminals) of the first coil could be connected with an electric battery (of cells). The ends of the second coil were connected each other by a copper wire placed in the neighbourhood of a magnetic needle. When connecting or breaking the connection of the first coil with the battery, he remarked oscillations of the General Theory of the Electromagnetic Field 26 magnetic needle that then ceased. This experiment led Faraday to the conclusion that the second coil carried during this time interval “an electricity wave”. Hence, the phenomenon of electromagnetic induction by transformation was discovered. In another experiment, he utilised a coil of wound wire forming a helix cylinder. When displacing, inside the coil, a permanent magnet in the form of a bar of about 1.905 cm in diameter, and of about 21.590 cm in length, he remarked that the needle of a galvanometer, connected with the ends of the coil, moved in different directions depending on the direction in which the permanent magnet had been displaced. Hence, the phenomenon of electromagnetic induction by the relative motion of a conductor with respect to the field lines produced by the permanent magnet was discovered. Therefore, the law of electromagnetic induction is also called the Faraday law. The mathematical expression of the electromagnetic induction law was subsequently established by Maxwell. Also, in 1831, Faraday invented the first direct current generator composed of a copper plate that could rotate between magnetic poles, and the external electrical circuit was connected between the centre and the rim of the plate. In 1851, he described a machine consisting of a rotating wire rectangle with an attached commutator, this being the prototype from which derived the direct current machines with commutator. The self-induction phenomenon was discovered by Joseph HENRY (1797 – 1878) in the year 1832. The conversion into heat of the energy due to electric currents flowing through conducting wires is called electro-heating effect. James Prescott JOULE (1818 – 1889) carried out experimental research on the heat generated by electric currents. He established the relation expressing that the heat produced by electro-heating effect, in a given time, is proportional to the square of the current, and his results were published in 1840. Heinrich Friedrich Emil LENZ (1804 – 1865) made investigations on the variation of the resistance of a conducting wire carrying an electric current and showed that the resistance increases with temperature, these results were reported in 1833. Afterwards, he performed research on the electro-heating effect. He also established the statement that an electric current produced by the electromagnetic induction phenomenon, in any circuit, flows in a direction such that the effect of that current opposes the cause that produced the current. This statement is known as the Lenz rule. Emil WARBURG (1846 – 1931) and John Henry POYNTING (1852 – 1914) established useful relations referring to the transformation and propagation of electromagnetic energy. The study of electromagnetic field for the case of moving bodies was developed in the researches of Heinrich HERTZ (1857 – 1894), Hendrik Antoon LORENTZ (1853 – 1928), Hermann MINKOWSKI (1864 – 1909), Albert EINSTEIN (1879 – 1955). Lorentz developed the theory of electrons which allowed the explanation of many electromagnetic phenomena; he also established the relation for the transformation of co- ordinates and of time, when passing from a reference frame to another, from the condition that the form of Maxwell equations remain unchanged. Introduction 27 Einstein developed the Special Theory of Relativity published in 1905, and the General Theory of Relativity formulated in the year 1916, a presentation of which can be found in 27. The Special Theory of Relativity, is also referred to by one of the following denominations: Theory of Special Relativity, Restricted Theory of Relativity, Theory of Restricted Relativity. In the framework of the Special Theory of Relativity, Einstein obtained the Lorentz transformation relations, without utilizing the Maxwell equations. Utilizing the theory of relativity, it has been possible to express the equations of the electromagnetic field in a general form for the case of moving bodies. The Theory of Relativity implies to assume a constant velocity of light in empty-space with respect to any reference frame. This assumption leads to a local time at the points taken in various reference systems. An interesting interpretation of the Lorentz theory was given by Henri POINCARÉ (1854 – 1912), 39. In this interpretation, he stated that when considering a body in motion, any perturbation propagates more rapidly along the direction of motion than along the cross direction and the wave surfaces would be no more spheres but ellipsoids. These considerations have been analysed by Édouard GUILLAUME but their development has not been continued 39. It is interesting to be noted that Einstein and Poincaré obtained the same formula for the composition of velocities but with quite different derivations. The derivation of Einstein starts from relations of Mechanics and the postulates of the Special Theory of Relativity, whereas the derivation of Poincaré starts from the transformation relations of Lorentz. After the special theory of relativity became known, it has been possible to derive the Maxwell equations starting from the Coulomb formula and the transformation relation of forces when passing from an inertial reference frame to another one. Several mathematical explanations of the special theory of relativity can be found in literature, among which a derivation starting from the four-dimensional structure assumed for the universe 37. The theory of relativity has been based on the postulate mentioned above, according to which the velocity of light in empty-space is constant with respect to any reference frame. This postulate was based on the experiments first carried out by Michelson in 1881, and repeated with improved accuracy by Michelson and Morley in 1887. These experiments concern the propagation of a monochromatic light, emitted from a source on the Earth, taking into account the revolution motion of the Earth around the Sun. For this purpose, an apparatus containing an interferometer was used. From the mentioned experiments, it follows that the velocity of light on the Earth is not affected by the orbital motion of the Earth around the Sun. Later, the above postulate was checked by several direct experiments. An example is mentioned in 18, p. 4 and refers to the experiment performed in 1964 by Alväger, Farley, Kjellman and Wallin. They determined the velocity of photons arisen from the o decay of S - mesons. It is recalled that from the decay of each of these mesons, two photons arise. The velocity of the mesons above was found, using the equations of the Special Theory of Relativity, to be very close to the velocity of light. The velocity of the photons obtained as mentioned above was found to be very close to that of the mentioned mesons, except a very small deviation. Therefore, the velocity of photons was not added General Theory of the Electromagnetic Field 28 to the velocity of mesons and, hence, the velocity of light was not surpassed. Thus, the mentioned postulate was verified in this case. Despite the success of the theory of relativity it cannot be considered to be a complete one. Indeed, there are electrodynamic phenomena that cannot be satisfactorily explained by the known theories, the theory of relativity included. Further, an example of such a phenomenon will be given namely the experiment of G. Sagnac 28-31. For a long time this phenomenon has been mentioned in literature, e.g., by Lucien FABRE 38, although not enough analysed. The experiment carried out by Georges SAGNAC (1869 – 1928) in 1913, 31, is a very curious one. The experiment consists in achieving the interference of two light beams travelling in inverse directions along the same way. The light source, the interferometer and the reflecting mirrors which ensure the desired paths (ways) for the beams (namely approximately a circular trajectory), photographic plate, hence the set of apparatus is placed on a disc, outside which nothing related with the experiment occurs. The light beams travelling around the same way but in opposite directions are reflected from the interferometer to a photographic plate. The disc can rotate with any angular velocity Ȧ . We recall that the ether (aether), mentioned below, is the denomination of a certain substance assumed by certain scientists to fill all space (between particles of air and other substances) through which electromagnetic waves and light may be transmitted. However, according to several researches, among which the experiment of Albert A. MICHELSON and Edward W. MORLEY, the concept of ether appears as being non-consistent. Sagnac obtained that the time for a light beam to travel around a way parallel to the disc surface differed, according to whether the travelling direction was with or against the rotation sense of the disc. Hence, the light beams had different velocities with respect to a reference frame fixed to the disc. The result, referred to as Sagnac effect, seems to be not in concordance with the Theory of Relativity. Indeed, the phenomenon appears, as if ether would exist at rest, independently of the existing motion 38, p. 111, 248-251. This result determined many thorough analyses, one of the most recent and interesting being carried out in papers 34-36. In these papers, A.G. KELLY made a thorough analysis examining the arguments for and against the theory of relativity by considering the Sagnac effect. His analysis is based on the most important studies and reports concerning this effect. His main remarks are the following ones: 1º Many experiments performed with a high precision, including laser light, have confirmed with a good accuracy the results of the Sagnac experiment. It can be mentioned a very precise experiment carried out by investigators using laser-light in a piping system filled with a helium-neon gas 35, p. 7. In fact, the Sagnac effect proves that light does not travel with the same velocity in both directions relative to the interferometer on a spinning disc 35, p. 10. 2º According to the internationally agreed method of synchronizing clocks on Earth, using electromagnetic signals, the following three effects are considered 35, p. 10, 16: a. Correction calculated according to the special theory of relativity. b. Correction calculated for the difference of the gravitational potential, according to the general theory of relativity. Introduction 29 c. Correction for the rotation of the Earth about its axis. The last correction corresponds to the Sagnac effect (although it is not denominated as such). The last correction is necessary because light does not travel around the globe Eastward and Westward with the same velocity (i.e., in equal times). 3º The measurements of high precision made by several investigators showed that the velocity of light on the Earth is not influenced by the rotation of the Earth around the Sun but it is influenced by the rotation of the Earth about its axis. 4º Some authors, among which A.G. Kelly, have considered that the Sagnac effect could not be explained by the Theory of Relativity. This opinion has been justified, because the modification of the light velocity in the Sagnac effect is much greater than 7 any relativistic effect, by a factor of the order of magnitude 10 34, p. 8, 35, p. 14. However, as we have proved, and also described in Appendix 9 of this book, the relations obtained using the relationships of the General Theory of Relativity are in good agreement with the results shown by the Sagnac effect. 5º Tests were carried out in order to determine the effect corresponding to the General Theory of Relativity on the time indicated by airborne clocks relative to a standard clock system fixed on the Earth. The clocks had to be carried Eastward, and Westward, respectively by aeroplane in both cases approximately at the same latitude. Atomic clocks with caesium were used. The results have not been conclusive because the clocks had not sufficient stability required by the experiment 36, p. 5. According to paper 35, p. 22, the light moves on the Earth together with the gravitational field of the Earth. Kelly has shown the importance of modifying the Theory of Relativity in order to avoid the mentioned discrepancies with respect to experiments. Other remarks concerning the difficulties in using the Theory of Relativity can be found in paper 20. Despite the difficulties encountered in utilizing the Theory of Relativity for explaining certain phenomena it can be considered as a very convenient mathematical and physical procedure for calculating the electromagnetic field state quantities in the case of moving bodies charged with electricity. Also, the Theory of Relativity proved a higher accuracy than any other known theory. We should add that the derivation of the Maxwell equations starting from the Coulomb law, was performed by Leigh PAGE (1882 – 1952) in 1912, and developed subsequently by certain authors in several works, among which the following references (treatises and textbooks) of the Bibliography 6, 11, 13, 18, 23, 25. Classical Mechanics and classical Electrodynamics when applied for the explanation of phenomena produced at atomic scale lead to results which are in contradiction with the experimental results. So, for the study of phenomena at atomic scale, the Quantum Mechanics also called Undulatory Mechanics has to be used. The bases of these Mechanics were built up by Louis-Victor-Pierre-Raymond de BROGLIE (1892 – 1987) in 1924 and Erwin SCHRÖDINGER (1887 – 1961) in 1926. It is recalled that in Quantum Mechanics, the notion of trajectory of a particle does not exist. This circumstance is expressed by the non-determination principle (incertitude principle) that was formulated by Werner HEISENBERG (1901 – 1976) in 1927 and is one of the fundamental principles of Quantum Mechanics. General Theory of the Electromagnetic Field 30 It can be added that certain models proposed by R.L. VALLÉE 20, based on the analysis of trapped electromagnetic waves and some principles, allowed the find of certain results as by means of Quantum Mechanics. It is to be noted that Quantum Mechanics is very useful for both theoretical and practical purposes. Indeed, most of the phenomena that occur in semiconductors and magnetic materials may be explained only by the use of this mechanics. Also, the achievement of new materials (semiconductors and magnetic materials) is related to Quantum Mechanics. 4. THE SYSTEM OF UNITS OF MEASURE There are several systems of units that can be used in electromagnetism 42-47. For practical purposes at the “Congrès International des Électriciens” held in Paris in 1881, the following units of measure have been adopted: ohm, volt, ampère, farad. Giovanni GIORGI (1871 – 1950) proposed in 1902 a system of units from which the International System of Units, SI (abbreviation from the French denomination Système International d’Unités), utilised at present, was derived. In 1935, the International Electrotechnical Commission (IEC or in French CEI - Commission Électrotechnique Internationale) recommended that preparation be made for the transition to the system of units suggested by Giorgi. In this system the basic units were the following: unit of length – the metre, unit of mass – the kilogram, unit of time – the second, magnetic permeability of free-space – µ . This proposal was not universally accepted and instead of the 0 magnetic permeability, it was suggested that the fourth basic unit was to be the ampere. The corresponding system of units was denoted MKSA. The International System of Units denoted SI was adopted by the eleventh Conférence générale des Poids et Mesures (General Conference of Weights and Measures) in 1960. The base (basic, fundamental) units of this system are the following: metre, kilogram, second, ampere, kelvin (for thermodynamic temperature), mole (for amount of substance), candela (for luminous intensity). In respect to electrical quantities, the SI system differs from the MKSA system by the denomination of the unit of magnetic induction that is tesla (T) in the SI system and 2 Wb/m in the MKSA system. It is useful to add the concept of rationalization. The rationalization of the equations of the electromagnetic field means the presentation of the main equations, i.e., the Maxwell equations, in a form not containing the factor 4S . In this way, certain symmetry in the equations connecting electric and magnetic quantities appears. In fact, the rationalization consists in adopting for İ and µ appropriate values in order to ensure the symmetry 0 0 mentioned above. The SI system ensures the rationalization of the equations of the electromagnetic field, hence it is a rationalized system of units. 1. GENERALITIES ON THE THEORY OF THE ELECTROMAGNETIC FIELD AND ON THE STRUCTURE OF SUBSTANCE 1.1. FIELD AND SUBSTANCE Field and substance are fundamental forms strictly connected in which matter exists. There are many varieties of fields. For the electromagnetic field, the following definition can be used. The electromagnetic field is a different form of existence of the substance of bodies, and exists in the regions of space in which ponderomotive actions (forces or torques) of electromagnetic nature can act on the bodies. By ponderomotive actions of electromagnetic nature we understand forces and torques exerted on bodies and which have not a cause of mechanical or thermal nature. The major part of the properties of the electromagnetic field is indirectly studied by the effect that it produces (for instance, mechanical and thermal effects), because most of the manifestation manners of the electromagnetic field are not directly accessible to the human senses. Only the electromagnetic waves of certain wave-length within 0.4 µm and 0.76 µm are directly perceptible as light waves. It has been established that the electric and magnetic phenomena are transmitted in space at a finite velocity even in vacuo, from a body to another. It follows that, in space, a physical system, termed field, exists and allows the transmission of ponderomotive actions in space and time. This statement is in accordance with the principle of continuity 38, p. 230, 231, 1, Vol. I, Arts. 7, 59, 60; namely two distinct bodies can act to one another only by an inter-medium. Thus, the action is exerted not at distance, but through a medium. Therefore, all the laws could be expressed in a differential form between infinitely close points. At the same time, according to the principle of causality, the laws can contain only quantities that can be observed directly or indirectly. In the study of the properties of bodies and generally of their substance, hence media presented by various bodies, we can distinguish: a – homogeneous media and non- homogeneous media; b – isotropic and anisotropic media. Homogeneous medium shows the same properties at all points. Isotropic medium shows the same properties along any direction. To any field of physical nature characterized by scalar or vector quantities, there corresponds a field with mathematical meaning, namely a field of scalars or vectors, respectively. It means that, to any point of the physical field, there corresponds a scalar or a vector. The physical field under consideration will be called scalar or vector field, respectively. The vectors may be of polar or axial types; thus there are polar vectors and axial vectors. To any rotation motion of a body about an axis, a vector having the axis direction is attributed for representing the angular rotation velocity of the body about the axis. In this case, the vector designates the axis about which the rotation is accomplished. Such a vector is called an axial vector. The case is similar for the vector representing the torque acting on a body. To the gravitational force acting on a body, a vector is attributed forGeneral Theory of the Electromagnetic Field 32 representing the force. Such a vector is called a polar vector. Starting, for instance, from the expression of a torque it follows that a vector product c au b yields an axial vector if each vector from the right-hand side is a polar vector. From the mathematical form of a polar vector, it follows that its components, in a three-orthogonal rectilinear (Cartesian) right-handed system of co-ordinates, change their sign if the positive direction of every co-ordinate axis is inverted. From the mathematical form of the vector product, it follows that its components, in a system of co-ordinates described above, do not change in sign if the factors are polar vectors. Consequently, in a vector equation, all the terms must be vectors of the same type. The scalar function of a scalar field or the vector function of a vector field depends on the position vector: r i x j y k z , (1.1) and may be written M r and G r , respectively: M r M x, y , z , (1.2 a) G G r i G j G k G . (1.2 b) x y z In the study of vector fields, the use of Vector Calculus is very convenient and will be utilized further on. It is worth noting that the terms of direction have to be used in many sentences. Direction is the course taken by a moving person or thing. The word direction is used especially for straight-line ways (paths), but it is not compulsory. A direction normally shows two senses but the word sense will be considered to have the same meaning as direction. Each of them can be used. Certain concepts that often occur in a study of the electromagnetic field will be further considered. 1.2. LINES OF FIELD. TUBES OF LINES OF FIELD. EQUIPOTENTIAL SURFACES. FLUXES. 1.2.1. Lines of Field A line of field also termed a line of force, is a curve that is tangent at any point to the vector of the field strength at that point (Fig. 1.1). Hence, starting from any point of a vector field, and adding up, from this point, along the direction of the field strength an infinitely small straight-line segment, we obtain a next point, after the starting point, of the line of field. Continuing in this manner, we shall obtain the curve representing the line of field that passes through the given starting point. In accordance with the definition given for the lines of field, it follows that the element of a line of field is: d l i d x j d y k d z. (1.3)Generalities on the Theory of the Electromagnetic Field and on the Structure of Substance 33 G G a b Fig. 1.1. Explanation concerning the lines of field: a – line of field; b – tube of field lines. The element of a line of field and the field strength vector at the same point are homo- parallel (i.e., parallel and of the same direction). It follows that in a three-orthogonal rectilinear system of co-ordinates we have: d x d y d z . (1.4) G G G x y z In the case of a scalar field, by line of field we understand the line of field of the vector field, the vector at any point being the gradient of the scalar field at the same point. In the case of a vector field of the velocities of the particles of a fluid in motion, the vector at any point being the instantaneous velocity of the particle of the moving fluid, the field lines are called lines of current. In the case of a force field, they are called lines of force (denomination also used in the case of any vector field). We may imagine each line of field represented by a thread (thin cord, thin rope) having at every point the direction (sense) of the field vector at that point. Moreover, the lines can also be supposed to be elastic. Then, they can suggest the forces that could appear in the considered field. As an example we may consider the lines of force between two bodies charged with electricity of opposite sign. The lines of force can suggest the attraction forces exerted between the two bodies. Further on, we shall make some remarks concerning mathematical aspects. The lines of field of a potential field with sources (i.e., the case of a field caharacterized by the curl equal to zero and the divergence different from zero, at any point) are open lines. These lines diverge from the positive sources of the field (i.e., the points where the divergence is positive) and converge towards the negative sources of the field (i.e., the points where the divergence is negative). The lines of field of a curl solenoidal field (i.e., a field with the curl different from zero and the divergence equal to zero, at any point) can be: a – lines of a relatively simple form which are closed at finite distances or at infinity; b – lines of relatively complicated form, which may be closed lines as well as open lines, as certain conditions are fulfilled or not.General Theory of the Electromagnetic Field 34 A case of type b given in literature 7 is that of the magnetic field lines corresponding to the following configuration of two electric currents: One current flows along a circumference, the other flows along a straight line perpendicular to the plane of the circumference at its centre. The lines of field are also termed flux lines. 1.2.2. Tubes of Field Lines A tube of field lines is called a surface formed by a set of field lines (of a vector field) which pass through all the points of a closed simply curve, and has the form of a tube. The concept of tube of field lines has a geometrical meaning only in the case of the field lines of type a. 1.2.3. Equipotential Surface Equipotential surface is called a surface, in a scalar field, formed by the set of points for which the field scalar has the same value. For a given scalar field M x, y, z , it is possible to deduce a gradient vector as a x, y, z r gradM x, y, z , where the plus or minus sign has to be taken according to the adopted convention. Therefore, the vector field a x, y, z derives from the potential M x, y, z of the scalar field. This is the case in which the concept of equipotential surface is important. In the case above, equipotential surface is thus the surface formed by the set of points for which the potential has the same value: M x, y, z const. (1.5) The equipotential surfaces serve to the study of a scalar field (e.g., the field of an electrostatic potential) from the qualitative point of view, because it permits to follow the directions along which the scalar function increases, decreases or remains constant. In addition, from a quantitative point of view, it permits to appreciate the rate of variation of the scalar field function. The equipotential surfaces, as stated above, are surfaces on which the scalar potential (from which the vector field derives) shows the same value at any point. In this situation, the lines of the vector field are the orthogonal trajectories of the equipotential surfaces. Let us give some examples according to the nature of the vector field: Surfaces of equal electric potential (in electrostatic or in electrokinetic stationary fields); surfaces of equal magnetic potential (in magnetostatic fields); surfaces of equal potential of velocity (in fluids, in non-turbulent flow); surfaces of equal gravitational potential (in gravitational fields). The equipotential surfaces are also called level surfaces. 1.2.4. Flux The concept of flux is generally used to characterize the transmission rate of a conservative quantity (for instance a liquid) through a surface. The flux through (or on) a surface is equal to the conservative quantity (e.g., quantity of fluid) that passes through the surface per unit of time.Generalities on the Theory of the Electromagnetic Field and on the Structure of Substance 35 Consequently, the flux is expressed by the surface-integral of a quantity also called flux density, and characterizes the transmission rate of the quantity through the surface under consideration. This quantity can be a conservative one. It may be a scalar or a vector one. Let us give examples of scalar and vector conservative quantities. Examples of scalar quantities: Mass, volume of incompressible fluids, energy. Examples of vector quantities: Momentum (also called quantity of motion), electric field strength, electric displacement (electric induction), magnetic induction. The flux of an incompressible volume of liquid is termed flow rate. Let us calculate the flux ) of any conservative scalar quantity W characterized by the flux density vectors P , through a surface Ȉ : ) P˜ d S P˜ n d S , ³³ (1.6 a) 66 where the surface 6 may be an open or a closed one, according to the considered case, and n is the unit vector of the normal at any point of the surface. Further on, the closed surfaces will be denoted by capital Greek letters, e.g., 6 , whereas the open surfaces by capital Latin letters and sometimes with an index that denotes the curve bounding the open surface, e.g., S , where denotes the curve that bounds the surface S. If no mention is made, by a closed curve we shall understand a simple closed curve. If a point or a point-like body moving along a curve (also expressed around a curve, especially if the curve is a closed one) always in the same direction, after having started from any point of that curve, will arrive at the starting point passing only once through each point of the curve, this curve will be considered as a simple closed curve. Also, if no mention is made, by an open surface bounded by a closed curve we shall understand a simply connected open surface (i.e., without holes). Generally, in a field of vectors G r , the flux is called the surface-integral of the component of the vector G along the direction of the normal to the surface6 : ) G d S n˜ G d S G˜ d S n G n G n G d S , n x x y y z z ³³ ³³ (1.6 b) 66 66 where the unit vector of the oriented normal also referred to as the unit vector of the positive normal, namely n at any point of the surface is adopted as explained below. For a closed surface, the positive normal is oriented outwards the surface. For an open surface bounded by a simple closed curve, the positive direction of the normal is associated, according to the right-handed screw rule, with the positive direction of travelling along the curve, the latter being adopted arbitrarily. Although the adoption of the travelling direction along the curve is arbitrary, however imprecision cannot appear, because in the concerned equations, both directions simultaneously occur (direction of travelling along the curve and direction of the normal to the surface bounded by the curve). In Fig. 1.2, there are represented the directions that occur in the calculation of the flux for three cases: Closed surface, open surface, manifold open surface (i.e., formed by several sheets). Fig. 1.2 c concerns the calculation of the flux of a vector through an openGeneral Theory of the Electromagnetic Field 36 S n" " " n n n S " . ab c Fig. 1.2. The reference directions to the calculation of fluxes for: a – closed surface; b – open surface; c – manifold open surface. surface bounded by a curve forming several near loops (the case of a helix). The surface is a helical one. We shall recall the generation of a helical surface. Consider a straight-line segment having one end at any point on one axis with which the segment forms a constant angle. Let the segment rotate about the axis, and simultaneously the point representing the end above to move along the axis with segments proportional to the arc of rotation of the segment. The curve described by each point of the segment will be a helix. The surface described by the segment will be a helical surface. In the case of the helical surface, it follows that this flux is, in fact, equal to the sum of fluxes through every loop. The flux corresponding to all loops is referred to as linked flux or flux-linkage. The flux through a single sheet is referred to as flux-turn. Each tube of field lines containing a flux equal to the unit may be associated with a central line of field. This line may be referred to as unit field line or unit flux line. Then, the flux through any surface will be equal to the number representing the algebraic sum (i,e., taking into account the sense of the lines) of the unit field lines that pass through the surface. 1.3. PHYSICAL QUANTITIES. LAWS AND THEOREMS. The characterization of physical states and phenomena is achieved by means of physical quantities. A detailed analysis referring to physical quantities has been made inGeneralities on the Theory of the Electromagnetic Field and on the Structure of Substance 37 several works 1, 8, 11, 21, 22. Further on, some principal aspects will be explained. A kind of a physical quantity (in French, espèce de grandeurs physiques) is a class of physical properties susceptible of quantitative determination. For defining the kind of a physical quantity, it is necessary to know the measuring procedure and the unit of measure. The choice of the unit of measure is arbitrary. As examples, the following three kinds of quantities utilized in electromagnetism will be given: Electric charge, electric field strength, magnetic field strength. A kind of physical quantity characterizes a common property of the elements (objects) of a set of physical objects. To identify a common property of a set of physical objects, it is necessary that a relation of order should exist between these objects. A measure procedure is a repeatable experimental operation, by which, to each physical quantity it is possible to associate a mathematical quantity called value or magnitude in respect to a physical quantity termed unit. According to the manner of introducing, the kinds of physical quantities can be divided into the following ones: Kinds of primitive quantities and kinds of secondary (derived) quantities (in French, espèces de grandeurs primitives et espèces de grandeurs secondaires ou dérivées), also termed primitive and secondary (derived) quantities, respectively (in French, grandeurs primitives et, respectivement, grandeurs secondaires ou dérivées). The kinds of secondary (derived) quantities can be defined by means of other ones supposed as being known, hence introduced previously. The kinds of primitive quantities have to be introduced directly, by experimental way, and described by the measurement procedure, because they can no more be defined by means of quantities introduced previously. In electromagnetism, apart from the primitive quantities of mechanics (length, time, mass, force), a series of new primitive quantities is necessary for characterizing from an electromagnetic point of view the state of bodies and of the electromagnetic field. A system of units contains fundamental and derived units (in French, unités fondamentales ou de base et unités dérivées). The fundamental units have to be determined directly, experimentally (e.g., the metre). The derived units are derived by using the fundamental units (e.g., the square metre). The fundamental units must not be those of the primitive quantities, but those of the quantities that frequently appear in practice. The laws express relations that are essentially necessary and repeatable between phenomena. In physics, laws are called the relations that express the most general knowledge on the phenomena of a research domain. They reflect the objective properties (of phenomena) that cannot be deduced by logical analysis (in the framework of the respective research field) from more general relations. The laws are established by the generalization of a great number of experimental results. In the theory of the electromagnetic field, there are general laws and material laws also called constitutive laws. The material laws differ from the general ones by the fact that they contain in their expression quantities specific to various materials, called material quantities. The relations that can be deduced by logical analysis from other more general ones, and finally from laws, are called theorems. It is useful to mention that there are relations that, at the time at which they were established, had law character but subsequently, after the progress of science, more general relations were discovered and the first ones could be derived or have representedGeneral Theory of the Electromagnetic Field 38 particular cases. Therefore, many relations that have a theorem character are called, by historical reason, laws. Further, for more clarity, when necessary, each denomination will be mentioned. The examination of various quantities may be made in two ways, as the structure of bodies is taken into consideration. Consequently, two manners of studying quantities and phenomena naturally appear: The microscopic manner of the study and the macroscopic one. Correspondingly, there are a microscopic theory of the electromagnetic field and a macroscopic theory of the electromagnetic field. 1.4. MANNERS OF STUDYING THE THEORY OF THE ELECTROMAGNETIC FIELD 1.4.1. The Macroscopic Study of the Electromagnetic Field The character of the macroscopic study (from the Ancient Greek; µȢ (makros) means long and (skopeo) means look at or examine) results from the fact that, in the framework of the study, the atomic structure of bodies is not taken into consideration; it is assumed that the substance is continuously distributed throughout the whole space. In the case of macroscopic study, all relations are obtained by an analysis of the mode in which the phenomena manifest themselves at the scale of the human senses. 1.4.2. The Microscopic Study of the Electromagnetic Field The character of the microscopic study (from the Ancient Greek; µȢ (mikros) means small and ıțȠʌȑȦ (skopeo) means examine) results from the fact that, in the framework of the study, the atomic discontinuous structure is taken into account. 1.4.3. Generalities Concerning the Microscopic Study of the Electromagnetic Field The microscopic study of the electromagnetic fields takes into account the atomic discontinuous structure of bodies. It is recalled that all bodies are constituted of atoms, and an atom of each body is composed of relatively light bodies called electrons that have negative charge, and of a relatively heavy nucleus. The nucleus is essentially constituted of protons, particles that have a positive electric charge, and neutrons, particles that have no electric charge. The electron is the smallest material particle with an indivisible negative charge. The  19 electric charge of an electron is  e  1.602u 10 C, and its mass is  30 m 0.9108u 10 kg . e The proton is a particle with a positive electric charge equal in absolute value to that  27 of electron, and its mass is m 1.672u 10 kg, hence approximately 1836 times p greater than the mass of electron. The neutron is a particle with zero electric charge, but with the mass approximately equal to that of the proton. Țțȡȩ ıțȠʌȑȦ ĮțȡȩGeneralities on the Theory of the Electromagnetic Field and on the Structure of Substance 39 The substance is constituted of molecules that are formed of atoms. According to the Rutherford-Bohr-Sommerfeld atomic model, the substance presents itself in the form of planetary systems, each atom consisting of a nucleus with positive electric charge and of one or more electrons that turn about nucleus around closed orbits. Concerning the form of a nucleus and of an electron, the simplest proposal is to consider them as being of spherical form. However, this supposition is not satisfactory and it is necessary to use the Quantum Mechanics. Also, there are possibilities of using models that permit to obtain results close to the various ones obtained experimentally 20, p. 64. The dimensions of nucleus and electrons, in the framework of the simple physical model above, are so small, that in many phenomena, the atomic nuclei and the electrons can be considered of negligible dimensions. Hence, they can be considered as material points or point-like bodies with electric charge and mass. The study of the nucleus structure does not enter into the frame of the present work, and belongs to the domain of Nuclear Physics. It is useful to be mentioned that, generally, one of the aims of Physics is the determination of the number, repartition and character of the particles charged with electricity which characterize the nature of bodies. Let us give the following example: The derivation of the laws of chemical and physical phenomena, by the aid of interaction laws of particles with electric charge. The single important exception is represented by the phenomena in which an important function is represented by forces of mechanical nature (gravitation forces, elastic forces, capillary forces, friction forces, etc.) and nuclear forces, because only these forces cannot be reduced to the action of electric charges. The characterization of the local state of the electromagnetic field in the macroscopic theory may be done by the help of local state quantities of the field: The electric field strength, the electric displacement (electric induction), the magnetic field strength, and the magnetic induction that will be further studied. 1.4.4. Macroscopic Average (Mean) Values The macroscopic properties have to be described by means of macroscopic quantities. Macroscopic quantities are the quantities obtained by determining the average (mean) values of microscopic quantities for space domains and time intervals that are physically infinitesimal quantities (physically infinitesimal is in French, infiniment petit au sens physique 3, p. 408). These mean values are called macroscopic average (mean) values. By a physical infinitesimal domain of space (also called of volume) we understand a domain small enough, from a macroscopic point of view for, within it, the macroscopic quantities show a negligible variation with distance, and at the same time, great enough, from a microscopic point of view. The last condition means that the domain must contain a very great number of particles, i.e., molecules, atoms and elementary particles. By a physical infinitesimal interval of time we understand a time interval small enough, from a macroscopic point of view for, within it, the macroscopic time-dependent quantities show a negligible variation with time, and at the same time great enough, from a microscopic point of view. The last condition means that the time interval must have the duration much greater than the duration of processes occurring at microscopic scale, i.e., molecular or atomic scale. General Theory of the Electromagnetic Field 40 The value of a physically infinitesimal volume depends on the nature of the substance  5 3 and can be considered of about (10 cm) . It can also be mentioned that the processes occurring at microscopic scale (e.g., the variation of the electric field strength) are depending on the period of the orbital motion of electrons 18, p. 50. This period is of  16 about 10 s. In fact, infinitesimal means infinitely small. 1.4.5. Manner of Studying Adopted in the Present Work In the engineering practice, the study of phenomena at macroscopic scale is of a particular interest, however, in many cases, it is necessary to know the phenomena at the microscopic scale, as is the case of devices with semiconductor elements, elements of integrated circuits, devices with discharge in air, etc. Taking into account the physical model, often relatively simple in the microscopic study, the following procedure will be used: Firstly, the various quantities and phenomena at microscopic scale will be studied, then, by calculating the average (mean) values, the phenomena at macroscopic scale will be considered. As starting point, the expression of the Coulomb force acting between two material points with electric charge will be taken into account. Considering material points with electric charge, in motion with respect to various systems of reference (i.e., reference frames), and taking into account the transformation relations of the components of a force from the Special Theory of Relativity, the general relations of the Theory of the electromagnetic field can be obtained. It is often stated that the observation of relativistic effects requires great velocities of moving bodies and measurements of high precision. However, in Electromagnetism, relativistic effects are encountered even in the case of small velocities compared to the velocity of light. The component of the force referred to as being of magnetic nature and which acts upon a moving material point with electric charge, is a relativistic component. It represents a supplementary confirmation of the importance presented by the theory of relativity. The study of phenomena encountered in electromagnetism at microscopic scale requires the Quantum Mechanics. It is to be noted that by certain improvements of the models of various particles, the same results can be obtained for certain cases without utilizing the Quantum Mechanics 20. 1.4.6. Laws of the Theory of Electric and Magnetic Phenomena From the explanation concerning the laws, it results that they can be grouped (classified) as follows. The macroscopic theory of electric and magnetic phenomena is considered to have twelve important laws, nine of them being general laws and three of them material laws. The general laws are the following: 1. The law of electromagnetic induction. 2. The law of magnetic circuit (the magnetic circuital law). 3. The law of electric flux (the Gauss law). 4. The law of magnetic flux (the law of flux conservation).Generalities on the Theory of the Electromagnetic Field and on the Structure of Substance 41 5. The law of the relationship between electric displacement (electric induction), electric field strength and electric polarization. 6. The law of the relationship between magnetic induction, magnetic field strength, and magnetization. 7. The law of conservation of free (true) electric charge. 8. The law of the energy transformation in a body carrying conduction electric current. 9. Law of electrolysis. The most important material laws are the following: 1. The law of temporary electric polarization. 2. The law of temporary magnetization. 3. The law of electric conduction. To the general laws, the law of ponderomotive action upon a charged particle at rest can be added, and is referred to as law of ponderomotive action. However, it is included in the definition of the electric field strength. The law of electrolysis will not be examined in this work. An explanation of this law can be found in works containing sections devoted to Electrochemistry 23, 48, 49. The material laws exist only in the macroscopic theory. The material laws can be deduced from microscopic general laws under certain assumptions. Four types of fundamental quantities macroscopically characterize the electromagnetic state of bodies: Electric charge q, electric moment p , density of the conduction electric current J , magnetic moment m . The state of the electromagnetic field is macroscopically characterized by the following types of quantities: Electric field strength E , electric displacement (electric induction) D , magnetic field strength H , magnetic induction (magnetic flux density) B . These kinds of state quantities are introduced by the help of two kinds of fundamental quantities: Electric field strength in vacuo E and magnetic induction in vacuo B . The electromagnetic state of bodies and of the electromagnetic field is microscopically characterized by three kinds of fundamental quantities: Electric charge q , electric field strength E , magnetic induction B . The microscopic theory of electric and magnetic phenomena has five general laws: 1. The law of electromagnetic induction. 2. The law of magnetic circuit or magnetic circuital law. 3. The electric flux law (Gauss law). 4. The magnetic flux law (the magnetic flux conservation law). 5. The law of ponderomotive action upon a moving electrically charged particle.

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