A New Formulation of Maxwell’s Equations

symmetry S S Article A New Formulation of Maxwell’s Equations Simona Fialová* and František Pochylý   Citation: Fialová, S.; Pochylý, F. A New Formulation of Maxwell’s Equations. Symmetry 2021,13, 868. https://doi.org/10.3390/sym13050868 Academic Editor: Radu Abrudan Received: 1 April 2021 Accepted: 7 May 2021 Published: 12 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Victor Kaplan Department of Fluids Engineering, Brno University of Technology, Technická2896/2, 61669 Brno, Czech Republic; [email protected].cz *Correspondence: [email protected].cz Abstract: In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms. Keywords: Maxwell’s equations; divergence theorem; integral form; magnetism; optimization; analysis 1. Introduction Magnetic and magnetorheologicalfluidshavebeenincreasinglyusedinrecent years [1–3]. Their properties, especially viscosity, can be significantly changed by the effects of the magnetic field [ 4 – 10 ]. The mathematical model is, in such cases, composed of Maxwell’s and Navier–Stokes equations [ 11 ], which are mostly solved by numerical methods [ 12 – 15 ]. One of these models is presented at the end of this paper. The solution behavior of the mentioned models can be assessed on the basis of the symmetry or asymmetry of the operators forming the mathematical model [ 16 , 17 ]. Let us write, for example, Maxwell’s equations in the following operator form: −∂ ∂tcurl div 0· D H= j ρ(1)  ∂ ∂tcurl div 0· B E=0(2) From here, the symmetry of the problem for a nonconductive environment , where j=0,ρ=0, is quite obvious. Furthermore, it follows that the non-stationary field is rotational, because it is not possible to set E=gradφ . It is therefore necessary to use new methods for non-stationary problems solution. The operator equations, written in the summation symbolics, also show that (curlE)i=εijk ∂Ek ∂xj (3) The expression ∂Ek ∂xjcan be decomposed into a symmetric Ekj =1 2∂Ek ∂xj+∂Ej ∂xkand Symmetry 2021,13, 868. https://doi.org/10.3390/sym13050868 https://www.mdpi.com/journal/symmetry Symmetry 2021,13, 868 2 of 12 an antisymmetric part c Ekj =1 2∂Ek ∂xj−∂Ej ∂xk. While the Levi–Civit tensor εijk is antisymmetric, only the antisymmetric part c Ekj is used in Expression (3). Thus: (curlE)i=εijk c Ekj (4) These modifications could be deepened by a more detailed study of the symmetry of the mathematical model operators depending on the boundary conditions. A new formulation of Maxwell’s equations, which is essentially based on Gauss’s divergence theorem [17], can also contribute to this analysis. According to the authors, the mathematical formulation of Gauss’s divergence theorem is underestimated in terms of the physical substantiality of the problem. Gauss’s divergence theorem explains the following important finding: every spatial change in the physical variable f(x,t) , regardless of its tensor character, has its response at the boundary of a closed region: Z V ∂f(x,t) ∂xi dV =ZS f(x,t)nidS (5) where f can be understood as a symbolic variable. Of course, the term (5) has the opposite meaning. If we act on the boundary of the region Swith the effect of the variable f , this variable changes within the volume V. In brief, every change within the volume Vcan be measured at the boundary S, where it is reflected in its functional value; see Figure 1. Figure 1. General volume of the liquid Vsurrounded by continuous closed area Sthat consists of open areas ˆ S (outlined by closed curve k ). Volume Vcan be split into control volumes ∆V , closed by control areas ∆S. This principle can be used in qualitative analysis, numerical methods, and optimization of the electromagnetic field. The main reason is that this principle gives a very good idea, both qualitative and quantitative, of the influence of boundary conditions on the problem, which assignment is often in hands of the researcher. After all, in electromagnetic fields solution, the engineering practice successfully uses integral identities both on the principle of Gaussian divergence theorem for the closed area Sand Stokes theorem for the open area ˆ Sbounded by a curve k[15–19]. For example: curlE+∂B ∂t=0;Rk Edk+Rˆ S ∂B ∂t·ndˆ S=0(6) Symmetry 2021,13, 868 3 of 12 and others [ 20 ]. Since this paper is focused on solving non-stationary problems, it is appropriate to modify Maxwell’s equations into a more suitable form for optimization, qualitative analysis, and numerical methods. This can be achieved by using Gauss’s divergence theorem and the symmetry of the Kronecker delta operator δij =δji . The adjustment applies to all non-stationary terms of Maxwell’s equations for non-conductive media and ∂B ∂t for conductive media, where j6=0 , ρ6= 0. The modification, the proof of which is given in Section 2, can be expressed by a non-stationary term in a more suitable form for the application of Gauss’s divergence theorem: ∂Bi ∂t=∂ ∂xj∂Bi ∂txi(7) The scalar variant of Maxwell’s equations (Section 4) is also presented in the work, where new functions are introduced: Modified intensity of imprinted forces: b EhV·m−1i=∂B ∂tx(8) Modified stress of printed forces: b U[V]=∂B ∂t|x|2(9) Based on Gauss’s divergence theorem, they are again modified to a form suitable for analysis: ∂Bi ∂txi=∂ ∂xj∂Bi ∂txixi(10) Based on (7), by integrating (6) over the domain Vsurrounded by the surface S, new integral identities can be found, this time over the closed surface, where all boundary conditions appear. ZS (n×E)dS +ZS∂B ∂t·nxdS =0(11) The proof is given in Section 2. It follows from the above-mentioned equations that the article focuses on the appropriate modification of Maxwell’s equations so that the influence of non-stationary terms in the field Vis expressed by their values on the boundary Sof the closed region. The solution is based on the use of symmetry conditions and Gauss’s divergence theorem. The use of this procedure for qualitative analysis, numerical methods, and optimization are presented in the individual Sections for both the vector variant and the scalar variant of Maxwell’s equations. Both conductive and non-conductive environments are considered in the solution. The last Section presents a mathematical model of the interaction of a magnetorheological fluid with a magnetic field. Even for this interdisciplinary problem, Gauss’s divergence theorem can be used to redefine the mathematical model of Navier–Stokes equations. An example is given in Section 6. Aspecialpartisdevoted tothefinitevolumemethod fornon-stationary problems [13–22] . In the classical method, a non-stationary term is identified through the control volume in terms of the mean values of the integral calculus. This method does not allow the use of the finite volume method, while the new variant will allow it; see Section 2. 2. Symmetry in Principles of the Solution In the technical sciences, symmetry conditions play a special role, especially in the stability conditions of the system [ 13 , 14 , 23 ]. They have the same importance in the solution Symmetry 2021,13, 868 4 of 12 of electromagnetism tasks, where we can find in the term ∂Bi ∂xj and ∂Di ∂xj . These terms conclude the symmetric Sij and antisymmetric Aij part. It can be written: ∂Bi ∂xj =Sij +Aij, where (12) Sij =1 2 ∂Bi ∂xj +∂Bj ∂xi!and Aij =1 2 ∂Bi ∂xj −∂Bj ∂xi! Based on the symmetry principles, it is possible to find new shapes of Maxwell’s equations using the Gauss divergence theorem. The principle can be explained, for example, on magnetic induction. Let us consider the following equations: curlE=−∂B ∂t;divB=0. (13) The same formulated in the index symbolic (Einstein summation symbolics): εijk ∂Ek ∂xj =−∂Bi ∂t;∂Bi ∂xi =0. (14) Einstein’s summation symbolic is used in the mentioned relation and the following text. We note that these relations depend on the antisymmetric operator εijk and the expression ∂Ek ∂xj , which can be decomposed into symmetric and antisymmetric parts (see Section 1). Therefore, in the left part of Equation (14), only the antisymmetric part of the expression ∂Ek ∂xjmanifests. After these remarks, let us proceed to a derivation of a new variant of the equation εijk ∂Ek ∂xj=−∂Bi ∂t by modifying its right part containing a non-stationary term. We start from the validity of the equation ∂Bi ∂xi=0. For this purpose let us put: ∂Bj ∂xjxi=0; RV ∂Bj ∂xjxidV =RS BjxinjdS −RV Bj∂xi ∂xjdV =0(15) The relationship uses per partes integration in 3D space. After using a Kronecker delta symmetry adjustment: ∂xi ∂xj =δij =δji (16) Thus, Bj∂xi ∂xj=Bi. It is possible to write Equation (15) in the shape: Z V BidV =ZS BjxinjdS; or in the vector form (17) Z V BdV =ZS (B·n)xdS (18) Because it also holds that ∂ ∂xj∂Bj ∂t=0, it can be derived by analogy: Z V ∂Bi ∂tdV =ZS ∂Bj ∂txinjdS; or in the vector form (19) Symmetry 2021,13, 868 5 of 12 Z V ∂B ∂tdV =ZS∂B ∂t·nxdS (20) Expressions (19) and (20) are very important, because they point out the fact that unsteady states of the magnetic flux density are generated on the borders of the area and vice versa. From the Expression (17) follows the next important result based on the divergence theorem: ∂Bi ∂t=∂ ∂xj∂Bj ∂txi(21) Using Expression (21), it is possible to correct the form of Equation (14) on the principle of symmetry. After the substitution of the above-mentioned criteria, we obtain a new shape of Maxwell equations. ∂ ∂xjεijkEk+∂Bj ∂txi=0 (22) In Equation (22), the first term in bracelets represents the antisymmetric tensor of second grade, and in the second term, it is possible to decompose into the symmetric and antisymmetric parts. Equation (19) can be used to modify the control volume method. In the current control volumes method, the integration of the non-stationary term given in Maxwell’s equations is expressed on the basis of the mean value of the integral calculus [5,11–14,20,22,24]. Thus, in the form (see Figure 1): Z V ∂B ∂td(∆V)=∂Bc ∂t∆V(23) where ∆V=1 3Z ∆S (x·n)d(∆S)(24) In the newly proposed method, the integration is performed directly by using Relation (21) as follows: Z ∆V ∂Bi ∂td(∆V)=Z ∆V ∂ ∂xj∂Bj ∂txid(∆V)=Z ∆S∂Bj ∂tnjxid(∆S)(25) The same in the vector form: Z ∆V ∂B ∂td(∆V)=Z ∆S∂B ∂tnxd(∆S)(26) After the integration over the volume V, (22) can be easily written in the vector variant: ZS(n×E)+∂B ∂t·nxdS =0 (27) Note that from the obtained results (20), it is observable that the non-stationary change in magnetic induction within the field Vcan be determined by integration only at the system boundary. Conversely, time changes in the magnetic induction in the field Vare generated at the system boundary. This fact can be advantageously used for both qualitative analysis of non-stationary boundary conditions and optimization of non-stationary tasks by selecting a suitable target function depending on the boundary conditions. For the non-conductive space, the derivations are described in Section 3. Symmetry 2021,13, 868 6 of 12 3. A Non-Conductive Environment We assume σ= 0, ρe= 0. In this case, Maxwell’s equations will have the index symbolic form [1,3]: εijk ∂Hk ∂xj=∂Di ∂t,∂Di ∂xi=0(28) εijk ∂Ek ∂xj=−∂Bi ∂t,∂Bi ∂xi=0. (29) Due to the validity of Equations (17) and (18) using transform (21), new forms of Maxwell’s equations can be written without evidence, since: ∂Di ∂t=∂ ∂xj∂Dj ∂txi ∂Bi ∂t=∂ ∂xj∂Bj ∂txi(30) RV ∂Di ∂tdV =RS ∂Dj ∂txinjdS RV ∂D ∂tdV =RS∂D ∂t·nxdS (31) RV ∂Bi ∂tdV =RS ∂Bj ∂txinjdS RV ∂B ∂tdV =RS∂B ∂t·nxdS (32) ∂ ∂xj∂Dj ∂txi−εijk Hk=0 ∂ ∂xj∂Bj ∂txi+εijkEk=0(33) RSh∂D ∂t·nx−n×HidS =0 RSh∂B ∂t·nx+n×EidS =0(34) From the above, it is visible that in the case of a non-conductive environment, it is possible to derive a new variant for all Maxwell’s equations. All the conclusions given in Section 3remain valid, including the control volumes method. All these results can be used to solve the interdisciplinary problem of the motion of an incompressible fluid with the effects of a nonconductive magnetic field using the Maxwell stress tensor for this case; see Section 6. 4. The Scalar Variants of Maxwell’s Equations By the scalar variant of Maxwell’s equations [ 1 , 3 ], we mean the product of the multiplication of Equation (13) and the position vector x. curlE·x=−∂B ∂t·x, εijk ∂Ek ∂xjxi=−∂Bi ∂txi(35) The results of the solution of the scalar variant can be used again for the boundary conditions analysis and in the optimization area. For the solution, the divergence theorem (13) is beneficially used: divB=0⇒∂Bj ∂xj xixi=0 (36) By analogy to (15), we apply the multiplication: ∂Bj ∂xjxixi=0; RV ∂Bj ∂xjxixidV =RS BjxixinjdS −2RV Bj∂xi ∂xjxidV (37) Symmetry 2021,13, 868 7 of 12 From (37) follows the important knowledge: Z V BjδijxidV =Z V BixidV =1 2ZS BjxixinjdS (38) In the vector form written as: Z V B·xdV =1 2ZS (B·n)(x·x)dS (39) Considering the divergence theorem in the shape div ∂B ∂t=0, Equations (38) and (39) can be written: Z V ∂Bi ∂txidV =1 2ZS ∂Bj ∂txixinjdS (40) Z V ∂B ∂t·xdV =1 2ZS∂B ∂t·n(x·x)dS (41) where (x·x)=xixi=|x|2 From Equation (40) and using the divergence theorem, it is possible to derive the following important dependence that allows one to reformulate Maxwell’s Equation (35): ∂Bi ∂txi=1 2 ∂ ∂xj (∂Bj ∂txixi)(42) When we implement (42) into (35), we obtain: ∂ ∂xjεijkEkxi+1 2 ∂Bj ∂txixi=0 (43) The term in the bracelet can again be divided into the symmetric and antisymmetric parts as well as in (22). In the integral form: ZS1 2 ∂Bj ∂txixi+εijkEkxinjdS =0. (44) For the vector form, it holds: ZS∂B ∂t·n(x·x)+2(n×E).xdS =0. (45) One of the scalar variants of Maxwell’s equations was again derived under the assumption of Gauss’s divergence theorem validity. Derived relationships can be used to evaluate the results obtained by numerical methods. Even in this case, non-stationary changes in magnetic induction are reflected at the boundary of the region, and, here, it is possible to determine their values as a function of time. The resulting equations can also be easily used for optimization because the target function is scalar in this case. 5. A Non-Conductive Environment—Scalar Variant If we return to the problem of a non-conductive environment, we can rewrite Equations (28) and (29) in a differential form: Original equations: ∂Di ∂txi−εijk ∂Hk ∂xjxi=0 ∂Bi ∂txi+εijk ∂Ek ∂xjxi=0(46) Symmetry 2021,13, 868 8 of 12 ∂Bi ∂xi =0 (47) New variant: ∂ ∂xj∂Dj ∂txixi−2εijk ∂Hk ∂xjxi=0 ∂ ∂xj∂Bj ∂txixi+2εijk ∂Ek ∂xjxi=0. (48) Integral form: Original equations: RV ∂D ∂t·xdV =RS (n×H)·xdS ; RV ∂B ∂t·xdV =−RS (n×E)·xdS.(49) New variant: RS∂D ∂t·n(x·x)dS =2RS (n×H)·xdS; RS∂B ∂t·n(x·x)dS =−2RS (n×E)·xdS.(50) The new form of Maxwell’s equations is useful, whether for analysis or numerical solution. Comparing the left sides of Equations (49) and (50), it is obvious that nonstationary variables D(x,t) , B(x,t) , are generated only at the boundary of the system, and, therefore, it is possible to influence their process within the volume V. This can be essential in optimizing the non-stationary problems of electromagnetism. Here, scalar variants of Maxwell’s equations were also derived under the assumption of Gauss’s divergence theorem validity. Derived relations can be used to evaluate the results obtained by numerical methods. Even in this case, non-stationary changes in magnetic induction and electrical induction are reflected at the boundary of the region, and here, their values can be determined as a function of time. The resulting equations can also be easily used for optimization because the target function is scalar in this case. 6. An Interaction of a Non-Conductive Magnetic Liquid with a Magnetic Field In Section 3, it is shown that in the case of a non-conductive environment, it is possible to derive a new variant for all Maxwell’s equations. All the conclusions given in the Section remain valid, including the control volumes method. All these results can be used in solving the interdisciplinary problem of the motion of an incompressible fluid with the effects of a non-conductive magnetic field using the Maxwell stress tensor for this case [ 3 ]. In the presented case, we assume: ρe=0 , B=µ0H+M,M=χH.(51) The density of the volumetric magnetic force that acts on the elementary volume can be written in the form [8]: f=1 2χgradH2;H2=H·H.(52) Navier–Stokes equations of the magnetic liquid in the presented case are in the form [2,11,25–27]: ρ∂vi ∂t+∂ ∂xjvivj−σij=ρgi+1 2χ∂ ∂xiH2. (53) Considering gi=∂ ∂xi (gkxk), (54) Symmetry 2021,13, 868 9 of 12 then, Equation (53) can be written in a more transparent form: ρ∂vi ∂t+∂ ∂xjρvivj−σij −ρδijgkxk−1 2χδijH2=0. (55) Because the liquid is considered to be incompressible, the continuity equation is in the form: ∂vi ∂xi =divv=0. (56) Now, if we consider Equation (21), the Navier–Stokes equation for the incompressible magnetic liquid can be written in a new form: ∂ ∂xjρ∂vj ∂txi+ρvivj−σij −ρδijgkxk−1 2χH2=0. (57) This equation can be, using the Divergence theorem [ 16 , 17 ], rewritten in the new integral form: ZSρ∂v ∂t·nx+ρ(v·n)v−σ−g·x−1 2χ(H·H)·ndS =0 (58) σ=(σ1,σ2,σ3);σi=σij nj. (59) By comparing the original equation, Equation (53), and the new equation, Equation (57) , the advantage of the new variant is evident both for the numerical solution by the finite volume method and for the analysis of the influence of boundary conditions. Since in the above-mentioned case, assuming diva=0, with respect to (20), it holds: Z V ∂v ∂tdV =ZS∂v ∂t·nxdS (60) Z V ∂B ∂tdV =ZS∂B ∂t·nxdS (61) and concurrently for the result of the continuity equation: ZS ∂v ∂t·ndS =0 , ZS ∂B ∂t·ndS =0 7. Discussion A new formulation of Maxwell’s equations was derived, both in differential and integral variants. The basis for the derivation was Gauss’s divergence theorem, used for magnetic flux density B and electric flux density D . By the use of Gauss’s divergence theorem, Maxwell’s equations were transformed. This resulted in a tool that can be used in the numerical finite volume method and optimization. The obtained equations will also allow the qualitative analysis of the influence of boundary conditions. The mentioned changes concern the non-stationary terms of the type ∂B ∂t , resp. ∂D ∂t . This resulted in a new form of Maxwell’s equations that can be used in solving the interdisciplinary problem of the motion of an incompressible fluid with the effects of a non-conductive magnetic field using the Maxwell stress tensor for this case. For example: