Physics
Maxwell's Equations
Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields. They form the basis of classical electromagnetism and are essential for understanding the behavior of light, electricity, and magnetism. The equations relate the electric and magnetic fields to the sources of these fields, such as electric charges and currents.
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12 Key excerpts on "Maxwell's Equations"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Research World(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter-1 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)
- 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)
- University Publications(Publisher)
____________________ WORLD TECHNOLOGIES ____________________ Chapter- 5 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. (For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.) The other two equations
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)
- 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)
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 - PDFEngineering Electromagnetics
Pergamon Unified Engineering Series
- David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
Thus began the age of communications. The ultimate test or proof of any physical law is the supporting evidence provided by scientific experiments. Maxwell's Equations have withstood without 40 The Physical Basis of Electromagnetics revision all theories and experiments performed since their inception. This includes relativity, which forced the revision of many other laws. It might even be said that Maxwell's Equations were instrumental in the conception of rela-tivity. They apply in all materials, for all possible variations of space and time, and for macroscopic problems (for which they were intended). This text will adopt Maxwell's Equations as the fundamental laws of electro-magnetics. While this is unusual in an elementary text, it is done for two reasons: 1. The general applicability of them to all problems. 2. The empirical nature of all laws which precludes arguments that one set of laws is more fundamental than any other. The more common chronological development does present simpler laws —but each law has limited applicability. Certainly, all possible simplifications will be made before each problem, but we will begin with Maxwell's Equations. This avoids the trap often encountered by students who apply a simple physical law to a problem beyond its scope. Also, it is hoped that the reduction of general laws to simple, usable models will be instructive in the delicate art of mathematical and physical modelling of complicated problems. Maxwell's Equations Maxwell's Equations are stated in terms of four field vectors E, D, B, and H. By traditional usage these four vectors are called E, the electric field intensity, D, the electric displacement vector, H, the magnetic field intensity, B, the magnetic induction or flux density. Precise definition of each field vector will be withheld until the appropriate experiments or problems are discussed which describe the properties of each.
eBook - ePubElectromagnetism
Maxwell Equations, Wave Propagation and Emission
- Tamer Becherrawy(Author)
- 2013(Publication Date)
- Wiley-ISTE(Publisher)
Chapter 9Maxwell's Equations
In 1865, Maxwell unified electricity and magnetism in a single theory, called electromagnetism . The fields E and B cannot be considered as independent, as the variation of one in time requires the presence of the other. Thus, they constitute a single physical entity, called the electromagnetic field . This theory is verified by all its consequences, particularly the existence of electromagnetic waves that propagate in vacuum with the speed equal to the speed of light. The existence of these waves with the same properties of polarization and propagation as light waves was verified experimentally by Hertz in 1884. Electromagnetic theory also solved the old problem of the nature of light: it is an electromagnetic wave of a very short wavelength. The formulation of electromagnetism was a very important event in the history of science.In this chapter, we write Maxwell's Equations and the equations of propagation for the fields and the potentials. We discuss the questions of energy and its transfer and the radiation pressure.9.1. Fundamental laws of electromagnetism
If electric and magnetic phenomena vary in time, all the relevant quantities (fields, free charge density q v , conduction current density j , polarization P , magnetization M , etc.) may depend on time, and we expect that some of the basic equations that we have derived in previous chapters will be modified.a) We have seen in section 3.6 that the conservation of electric charge implies that the charge flowing out of a closed surface per unit time (i.e. the total intensity) is equal to the rate of decrease of the total charge Q in the enclosed volume υ (Figure 9.1a
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- 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 .
- 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.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
306 Maxwell’s equations and electromagnetism in the absence of currents or other sources. In free space, the current density J ! ¼ 0 and the two curl equations become r ! E ! ¼ ∂ B ! ∂t ¼ μ o ∂H ! ∂t (14.54) r ! H ! ¼ ∂D ! ∂t ¼ ϵ o ∂ E ! ∂t (14.55) thus indicating the presence of an electric field supported by the time-varying magnetic field, and the presence of a magnetic field supported by the time-varying electric field, even in the absence of any local sources. This creates the possibility of self-propagating electromagnetic waves in free space, thus making possible electromagnetic wave propagation from radio (and lower) frequencies to optical (and higher) frequencies. The effects of materials on the optical properties – or the properties of an electromag- netic wave at any frequency – are the result of ε(ω), μ(ω), and σ(ω) as specified in the constitutive relations. We will examine the phenomena of propagating electromagnetic waves in more detail in the next chapter. Question 14.10 The displacement current can be expressed as I D ¼ ∂ ∂t Ð S D ! d S ! . Find the magnitude of the displacement current for an electric field E ! ¼ 1 cos ð2πf t ÞV=m at a frequency f ¼ 60 Hz normal to a 1 cm2 cm surface. Assume that the dielectric medium is free space with ϵ o ¼ 8.85 10 12 F/m. Does this give you a hint as to why, although the connection between currents and magnetic field was known by Jean-Baptiste Biot and Félix Savart in 1820 and Michael Faraday discovered that currents could be induced by a magnetic field in 1831, the displacement current term in the Ampere–Maxwell Law wasn’t conceived by James Clerk Maxwell until 1861? 14.9 Applications of Faraday’s Law 14.9.1 Eddy currents We will conclude this chapter by examining three more applications of Faraday’s Law, the phenomenon of eddy currents, the operation of an electric generator, and voltage- level conversion with a transformer.
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