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Transcript

Électrotechnique et électroénergétique DYNAMIC PROPERTIES OF THE ELECTROMAGNETIC FIELD ANDREI NICOLAIDE1, AURELIU PANAITESCU2 Key words: Energy, Dynamic properties of the electromagnetic field. In the present paper, there will be determined, in the framework of the macroscopic Electrodynamics, the dynamic properties of the electromagnetic field in the proper sense in bodies, a component of the physical system substance-field, as well as the dynamic properties of the system substance-field. For the proofs, only the general Maxwell equations and the relationships of inductions, strengths and polarizations will be used. There is also shown that the Maxwell equations expressed in terms of the state quantities of the field {E, B}, to which the four state quantities of the electromagnetic state of the substance have to be added, namely {ρ, J, P, M}, permit the dynamic characterization of the components of the substance-field system, like in the microscopic Electrodynamics of Lorentz. 1. INTRODUCTION One body and the electromagnetic field in which it is placed form a single physical system. A domain bounded by a closed surface located within the body is a part of this physical system. In the paper, we shall determine the momentum of the electromagnetic field and the flux of the momentum of the electromagnetic field through the surface, which bounds the domain, the electromagnetic field being a component of the considered physical domain. In the paper, we shall also determine the momentum and the flux of the momentum for the substance located within the domain and being in an electromagnetic state, as well as for the whole physical system substance-field. The electromagnetic material properties of the body substance may be whatever ones, the body may be at rest or in motion, and the regime of variation of the electromagnetic field will be generally variable. No material law will be used, in order to keep the general character of the equations of the electromagnetic field. In the proofs, we shall use only the definitions of the state electromagnetic quantities, the relationships between inductions, strengths and polarizations and the local forms of the general laws of electromagnetic phenomena (the general 1 2 "Transilvania" University of Braşov, E-mail: [email protected] "Politehnica" University of Bucharest, E-mail: [email protected] Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 53, 3, p. 239–252, Bucarest, 2007 240 Andrei Nicolaide, Aureliu Panaitescu 2 Maxwell equations, in the case of domains with continuity and smoothness, adding the extension of Minkowski for moving media [18]). The dynamic properties of the system substance-field may be deduced by using the Maxwell equations, in which occur the field state quantities E, D, B, H, and the electromagnetic state quantities (material quantities) of the body, ρv, J, P, M. The electromagnetic field is characterized by two state quantities, E and B. The primitive electromagnetic state quantities of bodies: q, p, i, m or their volume densities, ρv, P, J, M have been introduced from the mechanical (ponderomotive) actions exerted in vacuo by stationary electric or magnetic fields, upon certain small proof bodies (test bodies, check bodies) being in electromagnetic state (manifested by electric charge, electric polarization, electrokinetic condition, magnetization) also stationary [6, 8, 13, 16, 17]. We recall that these forces, as shown in the previous paper [19], are expressed as follows: F = qE; f dV = ρv dV E ; F = grad( p ⋅ E ) ; ↑ d F = id l × B; F = grad(m ⋅ B ) ; (1 a, b) f d V = grad( P d V ⋅ E ) ; ↑ f d V = J d V × B; (3 a, b) f d V = grad( M d V ⋅ B ) ; ↑ (2 a, b) ↑ (4 a, b) or also: F = ( p ⋅ ∇) E v , (5) F = (m ⋅ ∇) Bv . (6) In formulae above, the arrow indicates the quantity that is to be differentiated, as mentioned in Appendix. It is useful to add the following consideration. As the volume element around any point tends to zero, at that point, the influence of the field produced by the polarization of that element, on the resultant field may be neglected. The case differs from that of a molecule of finite dimensions. For this reason, the field state quantities to which the volume element is submitted will be considered as being the mean values (average values). For this reason, the field state quantities of the relations (1b, 2b, 3b, 4b) have not any subscript designing an external field or a field in vacuo. The volume densities of the preceding forces are: f = ρ v E ; f = grad( P ⋅ E ) ; f = grad( M ⋅ B ) ; f = J × B ; (7 a, …, d) ↑ ↑ 3 Dynamic properties of the electromagnetic field 241 or also: f = ρv E ; f = (P ⋅ ∇ )E ; f = (M ⋅ ∇ )B ; f = J × µ 0 H . (8 a, …, d) In electrostatic field, the forces of (7 b) and (8 b) are equal. Also, in magnetostatic field, the forces of (7 c) and (8 c) are equal, but only if the considered body is in vacuo. An analysis of the expressions of the mentioned type is not given in literature, but it could be carried out starting from references: [3, p. 192–193, 490–491], [7, p. 258], [17, p. 30, 92, 101, 107], [12], [14]. It is worth noting that Maxwell introduced certain expressions of local forces, partially similar to expressions above. Also, for more precision, it should be mentioned that the set of equations established by Maxwell, as referred in [10, vol. II, p. 180], contains 12 vector equations (called by him equations in quaternion form) designated in his work by capital letters from (A) to (L). These equations include 9 general equations, 3 material equations, to which there are also added 2 force equations. The expressions of (7) and (8) have been introduced subsequently. The density of the polarization electric charges and the densities of the amperian electric currents are [19]: ρ pv = − div P ; ρ ps = − div s P ; J m = curl M ; J ms = curl s M . (9 a, …, d) Like in the previous paper [19], we shall consider any closed surface bounding a domain with continuity within any body, situated in an electromagnetic field. 2. THE ELECTRIC FORCE ACTING ON A DIELECTRIC BODY SITUATED IN AN ELECTROSTATIC FIELD In the macroscopic theory, the mechanical actions on the substance of a volume element dV are produced either by the field, i.e., by ponderomotive actions (like the action of the gravitational field), or by direct contact with the neighbourhood volume elements (e.g., pressure force, elastic stress). If a body situated in vacuo has on its external surface an electric charge, the electric field acts on the surface elements dS, with a force that is transmitted by contiguity to all elements of the body, by action substance-substance. Therefore, a volume element dV, of the inside of the body, is submitted to a force even if the substance of the considered volume dV is not in an electromagnetic state. The forces produced by the electromagnetic causes acting on any volume element can be calculated using the relationships of the macroscopic Electromagnetic Field Theory, but the 242 Andrei Nicolaide, Aureliu Panaitescu 4 resulting distribution of forces have to be determined by using the relationships of the Theory of Plasticity. Let us consider any dielectric body with linear electric properties, situated in vacuo, in an electrostatic field. Let the body be not charged with electric charge. The external surface of the body will be denoted by Σext. The surface density of the polarization electric charge on this surface will be ρ ps = P ⋅ n . Let Σ be a closed surface, situated in the interior of the dielectric body in a domain with continuity. The polarization electric charge is distributed only in the volume bounded by this surface, with the volume density ρ pv = − div P . No surface polarization electric charges exist, the surface being in a domain with continuity. It is necessary to take into account that in the electrostatic field curl E = 0 and hence ( P ⋅ ∇) E = grad( P ⋅ E ) . ↑ Further on, several vector transformations will occur. There are several manners to perform them. For instance, it is possible to use the theorems of Appendix. Also, it is possible to use the basic transformation formula of a closed surface integral into a volume integral and multiply the relation by any constant vector that will be renounced, by simplifying, after performing the calculations, [10, vol. I, p. 151]. The force exerted by the electrostatic field on the substance of the inside of the surface Σ will be: Fe = ∫ ρ pv E d V = VΣ ∫ (− div P )E d V = VΣ ( ) = − ∫ ( P ⋅ n )E d S + ∫ grad P ⋅ E d V . Σ (10) ↑ VΣ The formula considered as valid for this case, in agreement with the measured values, is ∫ ( ) Fe = grad P ⋅ E d V . VΣ ↑ (11) The surface Σ has two faces, Σ+ outside, in contact with the substance of the outside the domain, with the normal n + = −n , and Σ− inside, in contact with the substance of the domain, with the normal n − = n . On the two faces of the surface element dS, there are two surface polarization electric charges, P ⋅ n d S on Σ− and − P ⋅ n d S on Σ+. The electric field acting on these charges and the resulting force will be zero. The total force acting on the substance of the domain bounded by the surface Σ− will be: 5 Dynamic properties of the electromagnetic field 243 Fe = ∫ ρ pv E d V + ∫ ρ ps E d S = Σ− VΣ = ∫ (− div P )E d V + ∫ (P ⋅ n)E d S = ∫ grad(P ⋅ E↑ )d V . Σ− VΣ (12) VΣ Therefore, the calculated force using the proposed model, with surface charges on both faces of the surface, is valid. Under certain conditions, formula (12) can be brought to a form corresponding to known relations [10, vol. I, p. 270]. 3. THE MAGNETIC FORCE ACTING ON A BODY SITUATED IN STATIONARY MAGNETIC FIELD (THE EFFECT OF AMPERIAN ELECTRIC CURRENTS) From the study of the magnetization state of a body the following results have been established: in stationary regime, the magnetization state of a homogeneous body, with magnetic permeability µ, situated in vacuo, is equivalent with the magnetic state associated to a distribution of amperian currents within the volume of the body, with the volume density J m = curl M and to a distribution on the external surface of the body Σext, with the density J ms = curl s M = M × n . It could be added that the amperian currents of the calculation model, like the polarization electric charges, are considered as distributed in vacuo, and the presence of these currents manifest by producing the permeability µ of that body. The force by which an exterior and non-uniform magnetic field acts on a portion of any magnetized body bounded by the surface Σ taken inside the body, hence having continuity, may be calculated by formulae above (4 a, b): ( ) FVΣ = ∫ grad M ⋅B d V . VΣ ↑ (13) The magnetic force can also be calculated using the Laplace formula: FVΣ = ∫ (curl M ) × B d V , VΣ (14) where J m = curl M represents the amperian (molecular), macroscopic electric current. We shall look for the relation that exists between expressions (13) and (14). The integrand of formula (13) may be expanded as follows: 244 Andrei Nicolaide, Aureliu Panaitescu ( ( ) ) grad M ⋅ B = grad( M ⋅ B ) − grad M ⋅ B = ↑ ↑ = grad(M ⋅ B ) − (B ⋅ ∇ )M − B × curl M . 6 (15) Also, taking into account that div B = 0 , we shall obtain: ∫ (B ⋅ ∇ )M dV = ∫ (B ⋅ n)M d S . Σ VΣ Therefore: ( ) (16) FVΣ = grad M ⋅B d V = grad(M ⋅ B ) d V − ∫ ↑ VΣ − ∫ VΣ ∫ (B ⋅ n)M d S − ∫ B × curl M d V , Σ (17) VΣ where: n (B ⋅ M ) − M (B ⋅ n ) = B × (n × M ) = (M × n ) × B . (17 a) The relation (17) can be rewritten as: ∫ ( ) FVΣ = grad M ⋅ B d V = VΣ ↑ ∫ (M × n) × B d S + ∫ curl M × B dV , Σ VΣ (18) which represents the relation we have looked for. 4. FUNDAMENTAL IDENTITIES IN ELECTROMAGNETISM As already shown in the previous paper [19], starting from the Maxwell equations, two identities may be derived, the energetic and the dynamic one, respectively, which are very important in the study of electromagnetic phenomena. In the previous paper, the energetic identity has been examined, further on, we shall examine the dynamic identity. Like in paper [19], the following symbols, in the extended sense, will be used in the equations of the electromagnetic field (a form of the Maxwell equations): ρ e = ρ v + ρ pv = div D − div P = div(ε 0 E ) , J e = J + J m + J p = J + curl M + ∂P . ∂t (19) (20) 7 Dynamic properties of the electromagnetic field 245 Therefore, the general equations of the macroscopic Electrodynamics, denoted by (a) will have the same form as the equations of the microscopic Electrodynamics, denoted by (b): ∂B ∂B ; curl E = − ; ∂t ∂t curl E = − curl H = J + ∂D ∂E 1 ; curl B = J e + ε 0 ; ∂t µ0 ∂t div E = ρe ρ ; div E = e ; ε0 ε0 div B = 0 ; div B = 0 . (21 a, b) (22 a, b) (23 a, b) (24 a, b) In the classical microscopic Electrodynamics (the theory of electrons), in which the microscopic quantities are used, the Lorentz equations, as shown in [19], with certain slight modifications of symbols, are: curl e = − curl ∂b , ∂t ∂e b = j + ε0 , µ0 ∂t (25) (26) ρv , ε0 (27) div b = 0 , (28) dive = to which the Lorentz law of ponderomotive action has to be added [19]: f = ρv e + j × b . (29) Further on, there will be shown that if beside the general equations of the electromagnetic field, (21)–(24), two densities of the forces acting on the substance will be used, namely a volume density and a surface density, then the theorem of the conservation of the momentum of the macroscopic Electrodynamics will have the same form as the corresponding theorem of the microscopic theory of Electrodynamics, i.e., (63) and (65). As already shown in paper [19], if we consider a closed surface Σ within a domain with continuity of the physical system substance-field, then on this surface with two faces Σ− and Σ+, the single densities that could exist are the surface 246 Andrei Nicolaide, Aureliu Panaitescu 8 densities of polarization electric charges and the densities of the amperian electric currents. This occurs because on the surface Σ, situated within a domain with continuity, the following relations are fulfilled: divs D = 0 ; curls H = 0 . (30 a, b) The mentioned densities are equal in absolute value and have opposite signs, and are given by relations: ρ s Σ − = ρ ps = P ⋅ n ; J s Σ − = J ms = M × n . (31 a, b) Therefore, like in [19], there follows the two force densities: ∂P f v = ρ e E + J e × B = [div(ε 0 E )]E + J + + curl M × B, ∂t f s = ρ s Σ − E + J sΣ − × B = ( P ⋅ n) E + ( M × n) × B. (32 a, b) 5. THE FUNDAMENTAL DYNAMIC IDENTITY 5.1. RELATIONS OF VOLUME AND SURFACE FORCES The fundamental dynamic identity may be derived from the general equations of the electromagnetic field (Maxwell equations), expressed in terms of vectors E and B. For obtaining this, the vector product of both sides of equations (21 a) and (22 a) will be performed by vectors ε0E and B, respectively, and then the two equations will be added side by side. There follows: (curl E ) × ε 0 E + curl ∂P 1 × B + B × B = J + curl M + µ0 ∂ t ∂ε E ∂B + 0 ×B− × ε0 E . ∂t ∂t (33) We shall put the last equation in a convenient form, for emphasizing the forces, which include gradient operators. We can write the relations: grad ε 0 E ⋅ E − (ε 0 E ⋅ ∇ )E = ε 0 E × curl E , ↑ (34 a) 1 1 1 grad B ⋅ B − B ⋅ ∇ B = B × curl B . ↑ µ0 µ0 µ0 (34 b) 9 Dynamic properties of the electromagnetic field 247 Also, from relation (19), we get: div(ε 0 E + P )E = ρ v E , (35 a) [div(ε0 E )]E = ρv E − E div P . (35 b) or: Replacing in equation (33) the first term by expression (34 a), the second by (34 b), and summing the obtained equation and the equation (35 a), side by side, we shall obtain: 1 B ⋅ ∇ B − µ0 (ε0 E ⋅ ∇ )E + (div ε0 E )E − grad ε 0 E ⋅ E↑ + 1 − grad B ⋅ B = ↑ µ0 ∂P × B + = ρv E + (− div P )E + J + curl M + ∂ t ∂ + (ε 0 E × B ); ∂t 1 grad ε 0 E ⋅ E = grad(ε 0 E ⋅ E ); ↑ 2 1 1 1 grad B ⋅ B = grad B ⋅ B . ↑ µ0 2 µ0 (36) (36 a) In the right-hand side of identity (36), there occurs the force acting on the substance and produced by the sources of the electromagnetic field, distributed in volume. Like previously, after integrating over the considered domain, there is to be added the force acting on the surface Σ− and due to the surface polarization electric charges and to the surface magnetization current sheets: FΣ = ∫ [(P ⋅ n )E + (M × n ) × B ]d S . Σ (37) By integrating both sides of relation (36) over the volume VΣ, and transforming the volume integral of the left-hand side into a surface integral, then adding expression (37) to both sides, and using relations (A.1)–(A.3), we shall obtain the following relationship: 248 Andrei Nicolaide, Aureliu Panaitescu 10 ⌠ 1 1 1 2 B 2 d S + (ε 0 E ⋅ n )E + (B ⋅ n ) B − n ε 0 E + 2 µ0 µ0 2 ⌡ Σ + ∫ (P ⋅ n)E d S + ∫ (M × n)× B d S = Σ Σ ⌠ ∂P = ρ v + ρ pv E + J + curl M + × B dV + ∂t ⌡ ( ) (38) VΣ + ⌠ ∂ ∫ (P ⋅ n)E d S + ∫ ( M × n) × B d S + ⌡ ∂t (ε E × B )d V ; Σ Σ 0 VΣ ∫ (B ⋅ ∇ )B d V = ∫ B B ⋅ d S − ∫ B div B d V ; VΣ Σ VΣ (38 a) ρ pv E = (− div P )E ; div B = 0 . In relation (38), we shall replace the volume density of the polarization electric charge. We can use the following relation: ∫ E (P ⋅ d S ) − ∫ (div P )E d V = ∫ grad(P ⋅ E↑ )d V − Σ VΣ VΣ − ∫ P × curl E d V . (39) VΣ Also, we shall replace in relation (38) the term containing a curl, by using the following relations: B × curl M = grad M ⋅ B − (B ⋅ ∇ )M = grad( M ⋅ B ) − ( ) − grad(M ⋅ B )− (B ⋅ ∇ )M , ↑ (40 a) ↑ ∫ (B ⋅ ∇ )M d V = ∫ M (B ⋅ n) d S − ∫ M div B d V ; Σ VΣ div B = 0 . VΣ (40 b) Therefore, we shall get: − B × curl M d V = − grad(M ⋅ B ) d V + ∫ VΣ ( ∫ VΣ ) + grad M ⋅ B d V + M (B ⋅ n ) d S . ∫ VΣ ↑ ∫ Σ (41) 11 Dynamic properties of the electromagnetic field 249 By replacing expressions (39) and (41) into relation (38), and taking also into account relation (21 b), after reducing the like terms, we shall get: ⌠ 1 1 1 B − n ε 0 E 2 + B 2 d S + (ε 0 E ⋅ n )E + (B ⋅ n ) 2 µ0 µ0 2 ⌡ Σ + ∫ ( P ⋅ n) E d S + ∫ ( M × n ) × B d S = Σ ( ) Σ ( ) ⌠ ∂ = ρ v E + J × B + grad P ⋅ E + grad M ⋅ B + (P × B ) d V + ↑ ↑ ∂t ⌡ (42) VΣ ⌠ ∂ + (ε 0 E × B ) d V . ⌡ ∂t VΣ Further, in order to get a certain symmetrical form, we shall write: grad P ⋅ E = grad P ⋅ E , ↑ ↑ grad P ⋅ E = grad( P ⋅ E ) − grad P ⋅ E , ↑ ↑ (43 a, b) where both sides of relation (43 a) are identical. By subtracting equations (43 a) and (43 b) side by side, we shall obtain: 1 1 grad P ⋅ E = grad P ⋅ E − grad P ⋅ E + grad( P ⋅ E ), ↑ ↑ ↑ 2 2 (44 a) and similarly for magnetic quantities: 1 1 grad M ⋅ B = grad M ⋅ B − grad M ⋅ B + grad(M ⋅ B ), ↑ ↑ ↑ 2 2 (44 b) and if D = εE , B = µH , then: 1 1 grad P ⋅ E − grad P ⋅ E = − E 2 grad ε , ↑ ↑ 2 2 1 1 grad M ⋅ B − grad M ⋅ B = − B 2 grad µ . ↑ ↑ 2 2 (45 a, b) By replacing expressions (44 a) and (44 b), into (42), and grouping the terms, we shall obtain the expression: 250 Andrei Nicolaide, Aureliu Panaitescu 12 ⌠ 1 1 1 B − n ε 0 E 2 + B 2 + (ε 0 E ⋅ n )E + (B ⋅ n ) µ0 2 µ0 2 1 1 dS = + (P ⋅ n )E − (B ⋅ n )M − n P ⋅ E + M ⋅ B + 2 2 + n (B ⋅ M ) ⌡ Σ [ ( ) ( )] ) ( )] 1 ⌠ − grad P ⋅ E + ρ v E + J × B + 2 grad P ⋅ E ↑ ↑ dV + = + 1 grad M ⋅ B − grad M ⋅ B + ∂ ( P × B ) 2 ↑ ↑ ∂t ⌡ [ ( (46) VΣ ⌠ ∂ + (ε 0 E × B ) d V . ⌡ ∂t VΣ By grouping the terms in relation (46), there follows: ⌠ 1 1 (D ⋅ n )E + (B ⋅ n )H − n 2 D ⋅ E + 2 B ⋅ H d S = ⌡ Σ ⌠ 1 1 − grad D⋅ E + ρ v E + J × B + grad 2 D ⋅ E ↑ 2 ↑ dV + = 1 1 + grad B ⋅ H − grad B⋅ H ↑ 2 2 ↑ ⌡ (47) VΣ ⌠ ∂ + (D × B ) d V . ⌡ ∂t VΣ Remark. It is worth noting that the two transformations (44 a, b), which contain terms similar to those of the Minkowski formulae, are not casual, but have the aim to obtain the expression of the right-hand side, which represent the single formula considered as experimentally valid for the volume density of the force acting on polarized or magnetized isotropic substances at rest [10, vol. I, p. 157]. By using the expression of the right-hand side of relation (42) or (46), for the same body in vacuo, we shall obtain the same result. However, for calculating the volume density of the force, without other precaution (e.g., adding some terms), we 13 Dynamic properties of the electromagnetic field 251 shall obtain correct results only by using the integrand of the right-hand side of (46), because, as mentioned, only in this case the experimental results are satisfied. It is to be mentioned that relation (47) could be derived directly from the general equations of the electromagnetic field (Maxwell equations) written in terms of the vectors of the set {E, B, D, H}, namely by using equations (21 a) and (22 a), correspondingly multiplied with factors D and B, and added side by side, and then by performing calculations similar to those of Sub-section 5.1. Therefore, we have derived two relations concerning the electromagnetic forces acting upon a domain bounded by a closed surface Σ, namely (46) and (47), corresponding to both manners of writing the general equations of the electromagnetic field, i.e., the Maxwell equations, using either the set {E, B} or {E, B, D, H}, [1, 4, 5, 11], [13–18]. These developments are not given in literature. APPENDIX Transformations of Gauss and Ostrogradski type and other relations, [9, p. 120], [10, p. 474], given below, may be used: ∫ [(a ⋅ ∇ )b + (div a )b]d V = ∫ (a ⋅ n) b d S = ∫ b (a ⋅ d S ), (A.1) ∫ grad f d V = ∫ f n d S , (A.2) Σ VΣ VΣ Σ Σ ( ) 1 grad a 2 , (A.3) 2 where a and b are two field vectors, and f a scalar function, all being continuous and differentiable. In several papers, parentheses containing terms with arrow subscript or with the subscript const, having before them a differential operator, like grad, are used. The letter with arrow subscript will be differentiated, while the other will be kept as constant as performing that operation. The letters with subscript const will be kept constant. These parentheses very useful for expressing certain differential operations have not to be used in integral transformations before expanding the parentheses, in order to avoid errors. grad a ⋅ a = grad(a const ⋅ a ) = grad(a c ⋅ a ) = ↑ Received on 18 December, 2007 REFERENCES 1. L. Kneissler, Die Maxwellsche Theorie in veränderter Formulierung, Springer, Wien, 1949. 2. R. Răduleţ, Bazele teoretice ale electrotehnicii, Electrostatica şi Magnetostatica, Institutul Politehnic din Bucureşti, Litografia Învăţământului, 1955. 252 Andrei Nicolaide, Aureliu Panaitescu 14 3. I. E. Tamm, Bazele teoriei electricităţii, Edit. Tehnică, Bucharest, 1957. 4. Al. Timotin, Proprietăţile dinamice ale câmpului electromagnetic macroscopic în medii oarecare, PhDThesis, Institutul Politehnic, Bucharest, 1957. 5. Al. Timotin, On the definition of the force volume density (in Romanian), Bul. Inst. Politehnic Bucharest, 20, 3, pp. 139–147 (1958). 6. R. 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