Physics
Maxwell's Equations Differential Form
Maxwell's equations in differential form are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They are used to understand the relationship between electric charge and current, and the resulting electric and magnetic fields. The equations are essential for understanding the behavior of electromagnetic waves and the principles of electromagnetism.
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12 Key excerpts on "Maxwell's Equations Differential Form"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter-4 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications tech-nologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 11 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications technologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Maxwell's Equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications technologies. Maxwell's equations have two major variants. The microscopic set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents. Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, On Physical Lines of Force, which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper. It is often useful to write Maxwell's equations in other forms; these representations are still formally termed Maxwell's equations. A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred. Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.
eBook - PDF- Joseph J. S. Shang, Sergey T. Surzhikov(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
The Maxwell equations in the time domain consist of a first-order divergence-curl system that is classified as hyperbolic partial differen- tial equations. In essence, Maxwell equations are the foundation for describing all electromagnetic phenomena, including electromagnetic wave propagation, reflec- tion, refraction, and interaction with reflective surfaces. In practical aerospace engineering applications, this system of equations has been widely used for plasma generation, microwave energy transfer, scattering, absorption, and emission, as well as antenna and integrated circuit design. From the previous derivations, the system of linear, partial differential equations can be presented as: ∂ ∂ + ∇ × = B E t 0 (3.6a) 3.2 Maxwell Equations in the Time Domain 77 7 ∂ ∂ - ∇ × + = D H J t 0 (3.6b) ∇⋅ = B 0 (3.6c) ∇⋅ = D ρ e (3.6d) The first equation, Faraday’s law, relates the time rate of change of the magnetic flux density B (Weber/m 2 ) to the curl of the electric field intensity E (Volt/m). Physically, it means a changing magnetic flux density in time will induce a gradient of electric field intensity in space. The second equation is the generalized Ampere’s circuit law relating the time rate of the electric displacement D (coulomb/m 2 ) to the curl of the magnetic field strength H (Ampere/m) and electric current density J (Ampere/ m 2 ). The most significant fact is that the generalized Ampere’s law introduces the displacement current ∂ ∂ D t , so there is no ambiguity that electromagnetic waves can propagate in a vacuum. The last two equations are the Gauss laws for mag- netic and electric fields. Equation (3.6c) simply requires that the divergence of B must vanish in any electromagnetic field to preclude the existence of a magnetic dipole. The second Gauss law defines the direct relationship between the divergence of the electric displacement D and the electric charge density ρ e (Coulomb/m 3 ).
eBook - PDF- Y K Lim(Author)
- 1986(Publication Date)
- WSPC(Publisher)
Chapter II MAXWELL'S EQUATIONS Maxwell's equations are based on Faraday's law of induction and Ampere 's circuital law with the extension and re-interpretation introduced by Maxwell. Also incorporated are the physical ideas of the inseparability of magnetic poles and charge conservation. For the immediate applica-tions, the integral Maxwell's equations are used to deduce the boundary conditions for the field vectors, and the differential Maxwell's equations are used to obtain the Poynting theorem and the wave equations. Scalar and vector potentials are introduced to facilitate the general solution of Maxwell's equations. 2.1 Maxwell's Equations in Integral Form A consistent and logical description of the structure and properties of the electromagnetic field together with its relation to the sources can be formulated on the basis of the integral equations which express Faraday's law of induction and Ampere's circuital law. These laws, with the extension and generalized interpretation of Maxwell, serve as the fundamental axioms in Maxwell's formulation of an electromagnetic theory. These fundamental axioms are 36 B«dS= - I) E-dr , (2.1) (D+J)-dS = | H-dr , (2.2) where S is an open, two-sided surface bounded by a closed loop C. For convenience, partial derivatives with respect to time is indicated by an overhead dot. For mathematical manipulations to be possible it is assumed that the field vectors involved and their derivatives are continuous, single-valued and bounded in the domain of integration. It should be noted also that these relations are assumed to be valid only for stationary media, to which we shall confine ourselves throughout this book. In addition to the fundamental axioms two supplementary axioms are required. They are obtained from the former by endowing the fields with certain physical properties which are found to be true from our empirical experience. Consider the special case where S is a closed surface.
eBook - PDFComputational Electromagnetism
Variational Formulations, Complementarity, Edge Elements
- Alain Bossavit, Isaak D. Mayergoyz(Authors)
- 1998(Publication Date)
- Academic Press(Publisher)
CHAPTER 1 Introduction: Maxwell Equations 1.1 FIELD EQUATIONS Computational electromagnetism is concerned with the numerical study of Maxwell equations, (1) -3td + rot h = j, (2) 3tb + rot e = 0, (3) d = toe + p, (4) b = G (h + m), completed by constitutive laws, in order to account for the presence of matter and for the field-matter interaction. This introductory chapter will explain the symbols, discuss constitutive laws, and indicate how a variety of mathematical models derive from this basic one. The vector fields e, h, d, b are called electric field, 1 magnetic field, magnetic induction, and electric induction, respectively. These four 2 vector fields, taken together, should be construed as the mathematical represen- tation of a physical phenomenon, that we shall call the electromagnetic field. The distinction thus made between the physical reality one wants to model, on the one hand, and the mathematical structure thanks to which this modelling is done, on the other hand, is essential. We define a model as such a mathematical structure, 3 able to account, within some 1Italics, besides their standard use for emphasis, signal notions which are implicitly defined by the context. 2Two should be enough, after (3) and (4). Reasons for this redundancy will come. 3The structure, in this case, is made of the equations and of the framework in which they make mathematical sense: Euclidean three-dimensional space, and time-dependent entities, like scalar or vector fields, living there. There are other possible frameworks: the algebra of differential forms ([Mi], Chapter 4), Clifford algebra [Hs, Ja, Sa], etc. As Fig. 1.1 may suggest, Maxwell's theory, as a physical theory, should not be confused with any of its mathematical descriptions (which are historically transient; see [Cr, Sp]). 2 CHAPTER 1 Introduction: Maxwell Equations reasonably definite limits, for a class of concrete physical situations.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
On the other hand, the integral relations involve the electric and magnetic fields themselves, not the spatial derivatives of the fields as in the differential form. The integral forms are particularly useful in situations of high symmetry where the integrals can be evaluated easily to find the fields. In nearly all cases except for a handful of special symmetries, however, the electric and magnetic vector fields need to be calculated by solving the vector differential equations that result from the differential form of Maxwell’s equations with known boundary conditions – given values of the fields (or the field derivatives) on the boundary of the volume where the solution is to be obtained. Exact solutions of Maxwell’s equations involve complex programming or software packages, and full three-dimensional solu- tions are significantly limited in the size of the volume that can be addressed. Because of their historical interest and because of the relation between fields and sources given by the high symmetry cases, we will review the integral forms of Maxwell’s equations and some of the simple solutions below. 14.8.1 Coulomb’s Law: r ! D ! ¼ ρ Taking the integral of the first Maxwell equation over a volume V inside a surface S, we find ð V r ! D ! dV ¼ ð V ρ dV : (14.35) Applying Gauss’s theorem, and recognizing the volume integral of the charge density is just the charge enclosed inside the surface of integration, we find þ S D ! dS ! ¼ Q enclosed : (14.36) 293 14.8 Maxwell’s equations in integral form where dS ! has the dimension of a small element of the surface with a direction normal to the surface at the given point. As the simplest example, consider a charge Q in vacuum with the surface S a sphere centered on Q.
eBook - ePubTeaching Electromagnetics
Innovative Approaches and Pedagogical Strategies
- Krishnasamy T. Selvan, Karl F. Warnick, Krishnasamy T. Selvan, Karl F. Warnick(Authors)
- 2021(Publication Date)
- CRC Press(Publisher)
Differential forms are used by physicists in general relativity [ 3 ], quantum field theory [ 5 ], thermodynamics [ 6 ], mechanics [ 7 ], as well as electromagnetics. A section on differential forms is commonplace in mathematical physics texts [ 8, 9 ]. Differential forms have been applied to control theory by Hermann [ 10 ] and others. 7.1.2 Differential Forms in EM Theory The laws of electromagnetic field theory as expressed by James Clerk Maxwell in the mid-1800s required dozens of equations. Vector analysis offered a more convenient tool for working with EM theory than earlier methods. Tensor analysis is in turn more concise and general, but is too abstract to give students a conceptual understanding of EM theory. Weyl and Poincaré expressed Maxwell’s laws using differential forms early this century. Applied to electromagnetics, differential forms combine much of the generality of tensors with the simplicity and concreteness of vectors. General treatments of differential forms and EM theory include papers [ 4, 11 – 15 ]. Ingarden and Jamiolkowksi [ 16 ] is an electrodynamics text using a mix of vectors and differential forms. Parrott [ 17 ] employs differential forms to treat advanced electrodynamics. Thirring [ 18 ] is a classical field theory text that includes certain applied topics such as waveguides. Bamberg and Sternberg [ 6 ] develop a range of topics in mathematical physics, including EM theory via a discussion of discrete forms and circuit theory. Burke [ 2 ] treats a range of physics topics using forms, shows how to graphically represent forms, and gives a useful discussion of twisted differential forms. The general relativity text by Misner, Thorne and Wheeler [ 3 ] has several chapters on EM theory and differential forms, emphasizing the graphical representation of forms
eBook - PDFTheoretical Concepts in Physics
An Alternative View of Theoretical Reasoning in Physics
- Malcolm S. Longair(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Maxwell’s second equation is (C). Maxwell’s third equation is (G). 11 Maxwell’s fourth equation is found from the divergence of (B). (A) is absorbed into (C). (D) contains the expression for the force on unit charge, f = (v × B) , normally referred to as the Lorentz force. (E) is the constitutive expression relating E and D. (F) is Ohm’s law, which Maxwell recognised was an empirical relation. (H) is the continuity equation for electric charge. Heaviside and Hertz are given credit for putting Maxwell’s equations into their conventional vector form but it is apparent that only the simplest modifications were needed from Maxwell’s original version. 12 This would include a consistent use of Gaussian units, to be replaced by the SI system in 1960. 114 Maxwell (1865): A Dynamical Theory of the Electromagnetic Field There are some intriguing features of Maxwell’s presentation of the set of equations: • Whereas the electrical displacement ( D x , D y , D z ) appeared awkwardly in his papers of 1861–62, it is now deeply embedded into the structure of electromagnetism as: the opposite electrification of the sides of a molecule or particle of a body which may or may not be accompanied with transmission through the body. The idle-wheels were an unnecessary artefact – the phenomenon of electrical displace- ment must necessarily occur and its time variation contributes to the total current. • The electromagnetic momentum A ≡ [ A x , A y , A z ] is what we now call the vector poten- tial. The origin of this identification is apparent from (6.2) and (6.3) and presages the four-vector notation of special relativity in which the four-vector for the electromagnetic potential is written A ≡ [φ/c, A] = [φ/c, A x , A y , A z ] with the dimensions of momentum divided by electric charge. Maxwell makes liberal use of the vector potential in his development of the equations, in contrast to contemporary practice of working initially with the fields E, D, B, H and J .
eBook - PDFElectrodynamics
Lectures on Theoretical Physics, Vol. 3
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
We now write down the two principal axioms which relate the 1 In the title of his comprehensive paper: La théorie analytique des phénomènes électrodynamiques, uniquement déduite de Vexpérience. 3.4 MAXWELL'S EQUATIONS IN INTEGRAL FORM 13 quantities defined in Eqs. (1) and (2) in this completely general sense. They are: | | Β η άσ= -^ Β . Λ , (3) >H.rfs (4) f C n da = ΦI In words: Every change in the number of magnetic lines of force which traverse a given surface σ produces in its boundary s an electric loop tension which is numerically equal to the rate of change, but opposite in sign (Faraday's law of induction) and The number of electric current lines, which traverse an arbitrary surface σ is accompanied by a magnetic hop tension in the bounding curve s of σ which is equal to it in both magnitude and direction (Ampere's law relating mag-netic field and electric current). Let us convince ourselves first that this equating of electric and mag-netic quantities is dimensionally proper. The two surface integrals defined in (1) (in spite of their dimensionally incorrect designation as numbers of force lines or current lines) have, according to (2.8) and 2.4a), the dimensions newton MS joule S , Q .. , £T = —pr— and ^»respectively. According to (2.9a) the latter dimension agrees with the dimension of the line integral in (4). The time rate of change of the first expression yields joule/Q, i.e. the dimension of an electric tension (expressible in volts), in agreement with the right side of Eq. (3). From this dimensional check our fundamentally different conception of B and H becomes apparent, and it is clear that our special introduction of the symbol Q for the dimen-sion of charge is unavoidable. Next we concern ourselves with the signs in Eqs. (3) and (4). They correspond to the rules of Lenz and Ampère. Ampere's rule is simply the right-handed screw rule, by which we correlated the positive normal of the surface σ with the sense of travel along the boundary s.
- Francis Piriou, Stephane Clenet(Authors)
- 2024(Publication Date)
- Wiley-ISTE(Publisher)
Maxwell’s Equations: Potential Formulations 135 It is possible to extend the domain of definition of the current density to the entire domain Ω by extending J over Ω 0 and posing J = 0 in Ω 0 . It can be noted that divJ = 0 on Ω 0 . The extension of J to the entire domain Ω raises no problem for the continuity of the normal component since J.n| Γj = 0. On the contrary, as will be noted in the following, it allows for the definition of a source term without dealing with the non-connected domain and the introduction of cuts. The function space of the current density, defined by relation [3.310], is then written as: [3.311] As for the fields H and B, they are defined throughout the domain Ω. The properties of the magnetic field H are governed, at the beginning, by equations [3.302] in Ω c and [3.207] in Ω 0 . Nevertheless, due to the extension of current density J to the entire domain Ω, the field H verifies equation curlH = J on Ω. As for the magnetic flux density B, it is defined, in the entire domain Ω, by equation [3.303]. Taking into account the boundary conditions on the boundary Γ (see equation [3.308]), it can be noted that, considering equation [3.305], the normal component of the magnetic flux density is zero. The function spaces of the fields H and B can be introduced as follows: 0 H( , ), H (div0, ) ∈ Ω ∈ Ω H curl B [3.312] In order to solve the magnetodynamic equations, two potential formulations can be used. The first one, known as “electric formulation”, is based on the magnetic vector potential A and the electric scalar potential V. The second, known as “magnetic formulation”, uses the electric vector potential T and the magnetic scalar potential ϕ. For these two formulations, it can be seen that, considering the choice of potentials, the coupling with magnetostatics can be readily made. 3.6.1.1. Electric formulation A-V The development of the A-V formulation, when an electromotive force e(t) is imposed, is quite natural.
- Vladimir Okhmatovski, Shucheng Zheng(Authors)
- 2024(Publication Date)
- Wiley-IEEE Press(Publisher)
32 1 Foundations of Electromagnetic Theory where we used to denote the length of loop as well. The voltage produced by the forces of extraneous nature (e.g. chemical reaction in a battery) is ext = ∮ E ext ⋅ d. (1.183) Often the solution of static Maxwell equations (1.179) is obtained for verification of digital and mixed-signal design through computation of DC current flow in the circuit. The general proce- dure for evaluation of the time-invariant electric and magnetic fields can be formulated as follows. First, for the given extraneous sources E ext the current distribution j e is found in the conductor as a solution of VIE (1.149) under condition of zero frequency = 0. Once the conductivity current density j e,cond is determined using the numerical solution of (1.149) (e.g. using MoM described in Chapter 3), vector potential A e can be evaluated using integral representation (1.148) and then magnetic field is determined from the vector potential as H = 0 ∇ × A e . According to Ohm’s law the electric field inside the conductor mimics the current vector field j e,cond and is calculated as E = j e,cond ∕ e . The potential e 0 (r) remains constant in the cross-section of a wire (including its boundary) and changes linearly along its longitudinal coordinate in accordance with for- mula 14 E = −∇ e = −d e ∕d ⋅ , since E remains constant along the wire. 15 Since the static scalar potential satisfies Poisson equation the electric field outside the conductor can be found as the solu- tion of the homogeneous satisfies Poisson equation ∇ 2 e (r) = 0 satisfying the boundary condition e (r) = e 0 (r) on the surface of the conductor. We will revisit these solutions in Chapter 3 in more detail and demonstrate numerical algorithms allowing for solution of static equations (1.179).
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