Physics
Maxwell's Equations Integral Form
Maxwell's Equations in integral form are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They encompass Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of electromagnetic induction, and Ampère's law with Maxwell's addition. These equations provide a concise and powerful framework for understanding the behavior of electromagnetic fields.
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12 Key excerpts on "Maxwell's Equations Integral Form"
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 - PDF- Joseph J. S. Shang, Sergey T. Surzhikov(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
7 1 3 Maxwell Equations Introduction Three-dimensional, time-dependent Maxwell equations consist of four funda- mental laws of electromagnetics: Faraday’s induction law, the generalized Ampere’s electric circuit law, and the two Gauss laws for electric displacement and magnetic flux density. The original formulations are based on experimental observations, but all are developed from the phenomenon that electromagnetic waves interact with transmitting mediums on molecular/atomic scales. In fact, all comply with the rig- orous theory to become the most basic law of physics. These equations govern all electromagnetic phenomena and are equally applicable to all electromagnetic fields in free space and in all other media. These fundamental equations were established by James Clerk Maxwell in 1873 and verified experimentally by Heinrich Hertz in 1888. Albert Einstein’s special theory of relativity further affirmed the rigorousness of the Maxwell equations in 1905 (Kong 1986). For these reasons the Maxwell equations occupy the widest range of applicability and a unique position in plasma physics. It is worthy of note that all these laws follow a single physical concept. An under- standing of these concepts is invaluable in order to apply these equations. Faraday’s law of induction simply states that a changing magnetic flux density will induce electric field intensity in the path surrounding it. The generalized Ampere’s law on a varying time frame defines the displacement electric current and is also a partial definition of magnetic intensity and magnetic force. Gauss’s law for magnetic flux describes that it has no source; the lines of magnetic flux have no beginning or end in an electromagnetic field. In contrast, Gauss’s law for electric displacement is that an electric field must be originated and terminated on an electric charge, and it is a partial definition of the electric flux density.
eBook - PDFPermanent Magnet and Electromechanical Devices
Materials, Analysis, and Applications
- Edward P. Furlani(Author)
- 2001(Publication Date)
- Academic Press(Publisher)
Review of Maxwell’s Equations 2.1 INTRODUCTION Maxwell’s equations govern the interaction of charged matter, and the behavior of electromagnetic fields. They provide a fundamental under-standing of a wide range of phenomena including the magnetic interac-tions that are of interest to us. In this chapter, we review electromagnetic field theory and Maxwell’s equations. This review is not intended to provide a working knowledge of the theory because it is assumed that the reader already possesses such knowledge. Instead, it is written to provide an overview of the theory, with an emphasis on the reduction and simplification of the field equations for the applications that we intend to study. We start with Maxwell’s equations in their most general form. These are presented in both differential and integral form along with constitut-ive relations and boundary conditions. We then introduce the scalar and vector potentials, and show that they provide an alternate, and often more tractable formulation of electromagnetic field theory. Next, we specialize to the case of quasi-static field theory . Here, there is a partial uncoupling of the field equations that simplifies their solution. Finally, we study static field theory in which all time dependence is negligible. We find that the magnetic and electric fields uncouple into separate magnetostatic and electrostatic field equations. Permanent magnet and electromechanical devices are analyzed using magnetostatic and quasi-static field theory, respectively. We study the methods and techniques for analyzing such devices in Chapters 3 and 5, respectively. Throughout this text we use SI units in the Sommerfeld convention. These and other units are discussed in Appendix D. CHAPTER 2 73 2.2 MAXWELL’S EQUATIONS Electromagnetic fields arise from sources of charge and current, and are governed by Maxwell’s equations. We begin with the field equations in differential form, Maxwell’s Equations H J D t (2.1) · B 0 (2.2) E B t (2.3) · D .
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- Paul G. Huray(Author)
- 2011(Publication Date)
- Wiley-IEEE Press(Publisher)
Maxwell’s equations represent the vector field quantities: E = Electric field intensity (volts/meter). B = Magnetic flux density (weber/meter 2 or tesla) H = Magnetic field intensity (ampere/meter) D = Electric flux density (coulombs/meter 2 ) J = Electric current density (ampere/meter 2 ) ρ V = Electric charge density (coulomb/meter 3 ) Units of the field quantities in SI units are shown in parentheses. Table 7.2 Maxwell’s Equations Differential form Integral form Name ∇ × E = − ∂ B / ∂ t arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp integralloop E dl t ds B C S ⋅ = -∂ ∂ ( ) ⋅ ∫ ∫∫ Faraday’s law ∇ × H = J + ∂ D / ∂ t arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp integralloop H I dl t ds D C S ⋅ = + ∂ ∂ ( ) ⋅ ∫ ∫∫ Ampere’s law ∇ · D = ρ V arrowrightnosp harpoonrightnosp doubleintegralloop D Q ds S ⋅ = ∫∫ Gauss’s law for electric charge ∇ · B = 0 arrowrightnosp arrowrightnosp doubleintegralloop B ds S ⋅ = ∫∫ 0 Gauss’s law for magnetic charge 7.5 Magnetic Vector Potential 209 7.5 MAGNETIC VECTOR POTENTIAL We have shown in Chapter 3 that the divergence of the curl of any vector field is identically zero; that is, arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp ∇ ∇ ⋅ × ( ) = A A 0 for any vector field . (7.22) Because the fourth of Maxwell’s equations states that B is solenoidal, as given by Equation 7.20d ( ∇ · B = 0), we can thus assume that B may be written in terms of another vector field, A , that we will call the magnetic vector potential : harpoonrightnosp arrowrightnosp arrowrightnosp B A = × ∇ . (7.23) NOTE We can see from Equation 7.23 that, given a magnetic flux density , B , there will be an infinite number of vector fields, A , that can satisfy the identity; for example, adding a constant to A will also satisfy Equation 7.23. This means that, to specify a unique definition of the vector field, A , we will need to make an additional restriction on A .
eBook - PDFEngineering Electromagnetics
Pergamon Unified Engineering Series
- David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
By way of introduction, we are fortunate to find many linear, isotropic, homo-geneous materials. In addition to air and free space these include most ordinary dielectrics (styrofoam, polystyrene, etc.) and most gases. Any material whose composition varies with position is inhomogeneous. At high field levels many materials are non-linear, although at low fields few are. All ferromagnetic metals (iron), are non-linear and anisotropic, while ferrites (ground iron and ceramic mixed) are anisotropic. Finally, ionized gases (ionosphere) and liquids (plasma) exhibit strong anisotropic, frequency dependent properties. INTEGRAL FORMS OF MAXWELL'S EQUATIONS Maxwell's Equations as they appear in Eqs. (2.1) thrqugh (2.4) are in differ-ential form. Historically, each equation first appeared in integral form since point by point measurements of the fields were beyond the first electromagnetic pioneers. The integral forms will now be presented. Gauss's Law (Maxwell's third equation) Consider the integral over a volume, K, of both sides of the equation V • D = p, which is, (2.29) The right-hand side simply gives the total charge in the region. The left-hand side is perfect for applying the divergence theorem described in Chapter 1. Applying these two ideas results in Gauss's Law, which is, (2.30) In so many words, integrating the displacement vector, D, over any closed surface, 5, must result in a value equal to the total charge, Q, enclosed within that surface. Conservation of Magnetic Flux (Maxwell's fourth equation) Analogous to the derivation of Gauss's Law, integration of Maxwell's fourth equation leads to, (2.31) 50 The Physical Basis of Electromagnetics where again S is a closed surface surrounding a volume, V. The magnetic flux, t//, which will be discussed in detail in Chapter 7, is defined as, (2.32) This surface, S, is not closed, so that ifj is the total magnetic flux flowing through the surface S — analogous to the total current, /, flowing through the surface.
- K Umashankar(Author)
- 1989(Publication Date)
- WSPC(Publisher)
504 INTRODUCTION TO ENGINEERING ELECTROMAGNETIC FIELDS MAXWELL'S EQUATIONS Faraday's Law: ) (14.5.17) £ £ J v (F,co)*ds =-jcoJJJp v (F,co)dv (14.5.18) Table 14.2 Frequency Domain Maxwell's Equations in Integral Form. ELECTROMAGNETIC BOUNDARY CONDITIONS 505 In the above two tables, the basic electromagnetic field equations are given by the Faraday's law and the Ampere's law. The expressions (14.5.1) and (14.5.2) are a set of coupled partial differential equations. These two curl equations, in fact, form an independent set of coupled relationships between time harmonic varying electric field and magnetic field quantities. The additional field relationships stated by the Gauss's law, the expressions (14.5.4) and (14.5.5), do not form independent set of equations. They can be directly deduced from the two Maxwell's curl equations. 14.6 ELECTROMAGNETIC BOUNDARY CONDITIONS The various frequency domain electromagnetic field equations derived in the previous sections are applicable for the case of a very large three dimensional region which is linear, homogeneous and isotropic medium. The medium parameters, such as, the permeability, permittivity and conductivity are assumed to be constant. In many practical situations, this may not be the case, and the three dimensional region can consist of two or more media having different media parameters separated by boundary layers.
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 - PDF- Tung Tsang(Author)
- 1998(Publication Date)
- WSPC(Publisher)
88 Maxwell Equations Sec. 4.2 where e is dielectric constant, i is magnetic permeability, P is polarization, and M is magnetic moment density. The equation of continuity is given by eq. (3.2), a t (3.2) Eq. (4.10a) (same as eq. 2.140) is Coulomb law (or Gauss law) where D is related to the free charge density p e . Eq. (4.10b) (same as eq. 3.16) is the magnetic analog of Coulomb law except for the absence of free magnetic poles. Eq. (4.10d) (same as eq. 4.3) is Faraday law of electromagnetic induction: changing magnetic flux will produce EMF (induced potential) in the circuit. Eq. (4.10c) is the modified Ampere's law. In magnetostatics, both Biot-Savart law and Ampere's law state that magnetic fields rotate about current segments. Ampere's law is given by eq. (3.25) in vacuum and by eq. (3.63) for materials. Because of equation of continuity (3.2), we have shown in Sec. 3.3 that in vacuum, the displacement current density [l/(4jt]d t E (where dt=d/dt) should be added to the current density J. In materials, the displacement current density is [l/(4jr)]d t D since the displacemnt current density is associated with p e in eq. (3.22). A simple demonstration of the displacement current is charging or discharging of a parallel plate capacitor filled with a dielectric sheet as shown in Fig. 2.6. The charges are ±Q on the plates of area A. Inside the dielectric between the plates, we have D=4JCQ/A. During the charging or discharging of the capacitor, the current and current density are I=dQ/dt and J=I/A. Hence we get J=[l/(4jt)]dD/dt which is the displacement current density. 4.3. Introduction to Electromagnetic Waves A very important application of the Maxwell equations (4.10) is the propagation of electromagnetic (EM) waves in isotropic dielectric media which are insulators. Examples are light propagation in air, water, glass, etc.
- Harald J W M??ller-Kirsten(Author)
- 2004(Publication Date)
- WSPC(Publisher)
Chapter 7 The Maxwell Equations 7.1 Preliminary Remarks Now that we have dealt with electrostatics and magnetostatics also for macro- scopic objects, the next step is to introduce time dependence. Proceeding in our phenomenological and historical approach we are led to consider next Faraday’s law of induction. With this we can complete the equations of macroscopic electrodynamics with the addition of Maxwell’s displacement current. The result is the full set of Maxwell’s equations. 7.2 Time-Dependent Fields and Faraday’s Law of Induction Faraday discovered in 1831 that an electric current arises in a closed wire loop i f the wire is moved through a magnetic field, in other words when the position or orientation of the wire with respect to the magnetic field is changed, or if the magnetic field varies with time. We consider two situations in which Faraday’s observation applies. In the first case the field B is maintained constant in time. (a) In an electric field E the charge q experiences the force = E. F=qE, - dF dq This force F results from a nonvanishing potential difference in the conductor. On the other hand (cf. Lorentz-force), the field B acting on a charge dq moving with velocity v = dl/dt implies that the latter experiences the force 151 152 CHAPTER 7. THE MAXWELL EQUATIONS dF given by dl dt dF = dq- x B, so that dF dl dq dt - = -x B . Identifying the forces of Eqs. (7.1) and (7.2) in order to arrive at an expla- nation of Faraday’s observation, we obtain dl dt dqE = - x d q B , i.e. Fig. 7.1 (a), (b) The moving current loop. dl dt provided the right side (or one component) is parallel to E. This means, the electric force acting on the charge dq is equal to the force which the field B exerts on the charge dq moving with velocity v. Put differently: The right side of Eq. (7.3), i.e. the force that B exerts on dq, induces the electric force Edq, i.e.
- Ozgur Ergul(Author)
- 2021(Publication Date)
- Wiley(Publisher)
99 Note that potentials depend on the selected gauge, while electric and magnetic fields do not. ¯ E (¯ r, t ) = − ¯ ∇ Φ(¯ r, t ) − ∂ ¯ A (¯ r, t ) ∂t (4.270) ¯ H (¯ r, t ) = 1 μ ¯ ∇ × ¯ A (¯ r, t ) . (4.271) 4.4.3 Time-Harmonic Sources and Helmholtz Equations When time-harmonic sources are involved in a dynamic case, Maxwell’s equations can be written in the phasor domain. Then they can be combined to derive higher-level equations (similar to wave equations) to analyze the given dynamic problem. In a homogeneous medium, the result is Helmholtz equations, 100 which are among the most fundamental tools of electromagnetics. 100 In general, Helmholtz equations are in the form ( ¯ ∇ 2 + k 2 ) ¯ f = ¯ g or ( ¯ ∇ 2 + k 2 ) f = g . Instead of starting from Maxwell’s equations, one can also convert wave equations into Helmholtz equations via proper transformations of time derivatives. Now, we assume that the volume electric current density and the volume electric charge density can be written ¯ J v (¯ r, t ) = ¯ J 0 (¯ r ) cos( ωt + φ j 0 ) (4.272) q v (¯ r, t ) = q 0 (¯ r ) cos( ωt + φ q 0 ) , (4.273) where ω = 2 πf is the angular frequency and f is the frequency. In these expressions, if ω is known, the amplitudes ( ¯ J 0 and q 0 ) and phases ( φ j 0 and φ q 0 ) provide all the information regarding the sources. When the sources are time-harmonic, the resulting electric and magnetic fields are also time-harmonic with the same frequency (Figure 4.33 ).
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
302 Maxwell’s equations and electromagnetism A final high-symmetry application of Ampere’s Law is the field inside an infinitely long solenoid, a coil of wire wrapped in a cylindrical shape around an open core. A small section of a solenoid is shown enlarged in Figure 14.12 with the magnetic fields curling around the wire. Since the figure is a cross-section of a single wire, the current is the same in each loop coming out of the page and going in. As is clear, the fields between the loops will cancel out, whereas the fields in the bore of the solenoid will add. By symmetry, the field inside the coil is horizontal and uniform across the cross-section of the solenoid. Outside the coil, the field is zero from the cancelation of the fields of the wire segments going in and coming out. The field can be determined by applying Maxwell’s equation around the path shown in Figure 14.13. The only segment of the closed-loop integral that contributes is the segment down the center of the solenoid – there is no field outside and on the perpendicular segments H ! d ℓ ! ¼ 0 because H ! ⊥ d ℓ ! . The total current through the loop is NI where N is the number of wire segments inside the loop and I is the current in the wire. Ampere’s Law gives: þ ℓ H ! d ℓ ! ¼ Hw ¼ NI ) H ¼ I N w (14.50) H H a b H Figure 14.11 Magnetic field created by an infinite sheet of current. By symmetry the field is horizontal and is independent of distance from the plate since the same current is enclosed by path a and path b. Figure 14.12 The magnetic field created by solenoid coil is the sum of the field from each coil with current going into or out of page. 303 14.8 Maxwell’s equations in integral form where N w is the number of turns per unit length (turns/m) of the solenoid.
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