Electromagnetic Field Tensor

6 Key excerpts on "Electromagnetic Field Tensor"

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  Essence of Electromagnetism

  • (Author)
  • 2014(Publication Date)
  • Academic Studio

    (Publisher)

  This is called the Electromagnetic Field Tensor, and it puts the electric and magnetic forces on the same footing. In matrix form, the tensor is as below. where E is the electric field B the magnetic field and c the speed of light. When using natural units, the speed of light is taken to equal 1. ________________________ WORLD TECHNOLOGIES ________________________ There is actually another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing and , to get the dual tensor G μν . In the context of special relativity, both of these transform according to the Lorentz transformation like , where the are the Lorentz transformation tensors for a given change in reference frame. Though there are two such tensors in the equation, they are the same tensor, just used in the summation differently. Maxwell's equations in tensor notation Using this tensor notation, Maxwell's equations have the following form. In the above, the tensor notation f ,α is used to denote partial derivatives, . The four-vector J α is called the current density four-vector, which is the relativistic analogue to the charge density and current density. This four-vector is as follows. The first equation listed above corresponds to both Gauss's Law (for α = 0 ) and the Ampère-Maxwell Law (for α = 1,2,3 ). The second equation corresponds to the two remaining equations, Gauss's law for magnetism (for α = 0 ) and Faraday's Law (for α = 1,2,3). This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors.

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  The Theory of Electromagnetism

  • D. S. Jones, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  THE REPRESENTATION OF THE ELECTROMAGNETIC FIELD 49 STRESS AND ENERGY 1.22 The stress tensor in free space Consider a bounded portion of free space which contains a distribution of charge and current. It has already been explained in § 1.1 (see also § 2.18) that the force on this distribution is ρ E + J Λ B and we now wish to express this fact in a different way. It is shown in § 2.20 that, if we introduce the tensor S defined by ay = - (E -D + H · B) ôij + EtDj + HiBj (using Cartesian tensors with the suffixes referring to the three space coordinates), / s . n d S = Π ρ Β + J Λ B + -^- (D Λ B) ) d r (132) s T in free space. Suppose, firstly, we consider fields which are independent of time. Then (132) states that the force exerted on the charge and current inside T can be expressed as a certain surface integral. A relation connecting the surface integral of a force and a volume force occurs in the theory of stress in elasticity 1 and the tensor is called the stress tensor. For this reason S is known as the electromagnetic stress tensor. The tensor was introduced by Maxwell and Faraday to avoid dealing with forces acting at a distance. They postulated the existence of an ether which filled the whole of space and was in a state of stress determined by S». This state of stress was respon-sible for the transmission of force from one charge to another in a way similar to that in which force is transmitted by an elastic medium. Subsequent research showed that this view was untenable (see Chapter 2) and that the stress components of S have no physical reality. The most that can be said is that the forces can be calculated on the basis that there exists a fictitious state of stress given by 8.

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  Waves, Particles and Fields

  Introducing Quantum Field Theory

  • Anthony C. Fischer-Cripps(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  15

  15.1  Introduction

  The electric field E and the magnetic field B at a point are, in general, functions of t , x , y , z .

  E = E t , x , y , z ) B = B t , x , y , z )

  (15.1)

  That is, at every point in space (x , y , z ) there may be an electric and/or a magnetic field. Both the E field and the B field are vector fields. In the case of the electric field, the vector points in the direction in which a positive test charge would move if placed at that point. In the case of the magnetic field, the direction of the vector is perpendicular to the direction of motion of the test charge and the direction from which the force acts upon the test charge by the right-hand rule. The magnitude of each field is usually written as E and B . Both magnitude and direction of the E and B fields may be a function of time as well as position.

  When fields vary with time, the change in magnitude or direction can be determined by taking the derivative of the function that describes the field. That is, the gradient. We might take the derivative with respect to time, or with respect to a direction – a spatial rate of change of a function. The value of a field may change at a different rate depending on which direction the derivative is taken. The spatial derivative can be found from the differential operator

  ∇

  .

  In this chapter, we deal with the electromagnetic field which is either stationary or moving with a constant velocity. The electric field arises from the presence of a charged particle. The magnetic field arises from the uniform motion of a charged particle. It is only when we have an accelerated charged particle that we obtain an electromagnetic wave

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  Dynamics of Classical and Quantum Fields

  An Introduction

  • Girish S. Setlur(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 3 The Electromagnetic Field and Stress Energy Tensor The most familiar example of a relativistic field theory is the electromagnetic field. We wish to think of the Maxwell equations as the Lagrange equations of a suit-able Lagrangian. The aim is to find this suitable Lagrangian. Next by examining the symmetries of the Lagrangian we may write down conserved quantities via Noether’s theorem. But first we begin with a description of the relativistic na-ture of the electromagnetic field. We wish to convince the reader that hidden in the equations that describe everyday phenomena are the seeds of special relativity. Historically, the observation that these equations are not compatible with Galilean relativity is what led to the development of special relativity. 3.1 Relativistic Nature of the Electromagnetic Field We show in this section that the equations of electromagnetism (Maxwell’s equa-tions) are compatible with Lorentz transformations but not Galilean transforma-tions. We purposely avoid using four-vector notation from the start as it hides some details about how the transformations actually work. Many students find these quite mysterious and it is hoped that an explicit demonstration component-wise, though clumsy is nevertheless illuminating. The electromagnetic field is described by two vectors E and B . These are determined by the solutions of Maxwell’s equation that involve sources of these field, namely, current and change densities j and ρ , respec-tively. First we show that j μ = ( ρ , j x , j y , j z ) transforms as a four-vector in the sense of 55 56 Field Theory Figure 3.1: James Clerk Maxwell (13 June 1831 to 5 November 1879) was a Scottish mathematician and the leading figure of one of the greatest revolutions in physics—the theory of electromagnetism. He unified the equations governing electricity and magnetism into one framework and studied the properties of elec-tromagnetic waves.

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  Mathematical Foundations of Imaging, Tomography and Wavefield Inversion

  • Anthony J. Devaney(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  11 The electromagnetic field 11.1 Maxwell equations We work in the frequency domain where the “electromagnetic” (EM) field consists of the electric E(r, ω) and magnetic H(r, ω) field vectors and the electric D(r, ω) and magnetic B(r, ω) flux vectors, where ω is the temporal frequency. The time-dependent fields and fluxes are obtained, in the usual way, via an inverse temporal Fourier transform so that, for example, e(r, t) = 1 2π  ∞ −∞ dω E(r, ω)e −iωt , where, of course, E(r, ω) =  ∞ −∞ dt e(r, t)e iωt . From this point onward we will not include the temporal frequency ω in the arguments of the various field quantities except in special cases where its exclusion can result in confusion or we wish to emphasize frequency dependence. We emphasize, however, that all of the EM field quantities depend to some extent on ω and this dependence must be accounted for except in narrow-band applications such as occur in optics, where the use of lasers is common. The four field vectors are coupled by the famous Maxwell equations, which, in the SI system of units, assume the form ∇ · D(r) = ρ (r), ∇ · B(r) = 0, ∇ × E(r) = iωB(r), ∇ × H(r) = −iωD(r) + J(r), where J and ρ are the current density and charge density, respectively. These two quantities are coupled via the charge–current conservation equation ∇ · J(r) = iωρ (r), (11.1) which can be inferred directly from the Maxwell equations. The four field vectors are also coupled via the so-called constitutive relations which, in the case of linear and isotropic media (which we will assume throughout this chapter) are given by 459 460 The electromagnetic field D(r) =  (r)E(r), B(r) = μ(r)H(r), where  and μ are the dielectric “constant” and permitivity of the medium, respectively. These two parameters are generally complex and are directly related to the complex index of refraction n(r) of the medium via the equation n(r) = c  μ(r) (r), where c is the velocity of light in vacuum.

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  Trends in Electromagnetism

  From Fundamentals to Applications

  • Victor Barsan, Radu P. Lungu, Victor Barsan, Radu P. Lungu(Authors)
  • 2012(Publication Date)
  • IntechOpen

    (Publisher)

  Eur. J. Phys. 31, L25–L27. Wikipedia, http://en.wikipedia.org//wiki/Shell_theorem/; last accessed 30 August 2011. 20 Trends in Electromagnetism – From Fundamentals to Applications J apan 1. Introduction Classical electromagnetism is a well-established discipline. However, there remains some confusions and misunderstandings with respect to its basic structures and interpretations. For example, there is a long-lasting controversy on the choice of unit systems. There are also the intricate disputes over the so-called EH or EB formulations. In some textbooks, the authors respect the fields E and B as fundamental quantities and understate D and H as auxiliary quantities. Sometimes the roles of D and H in a vacuum are totally neglected. These confusions mainly come from the conventional formalism of electromagnetism and also from the use of the old unit systems, in which distinction between E and D , or B and H is blurred, especially in vacuum. The standard scalar-vector formalism, mainly due to Heaviside, greatly simplifies the electromagnetic (EM) theory compared with the original formalism developed by Maxwell. There, the field quantities are classified according to the number of components: vectors with three components and scalars with single component. But this classification is rather superficial. From a modern mathematical point of view, the field quantities must be classified according to the tensorial order. The field quantities D and B are the 2nd-order tensors (or 2 forms), while E and H are the 1st-order tensors (1 forms). (The anti-symmetric tensors of order n are called n -forms.) The constitutive relations are usually considered as simple proportional relations between E and D , and between B and H . But in terms of differential forms, they associate the conversion of tensorial order, which is known as the Hodge dual operation.

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