Electromagnetic Four Potential

9 Key excerpts on "Electromagnetic Four Potential"

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  Semiconductor Lasers and Herterojunction LEDs

  • Henry Kressel(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 4 Relevant Concepts in Electromagnetic Field Theory 4.1 Introduction This chapter deals with some of the more fundamental aspects of classical electromagnetic theory, with the emphasis placed on those results applicable to wave phenomena in the optical frequency spectrum. Since wave phenome-na can be derived from Maxwell's equations it is appropriate to consider some of the more fundamental aspects of these equations. For the most general case, the electric and magnetic fields are generally written as a function of both spatial variables, x, y, z and time t as follows Because mathematical tools such as Fourier analysis can be applied to an arbitrary time varying signal, we can assume without loss of generality the harmonic time varying signals where Re means the real part of the term in brackets. In these expressions, the quantities E and H are complex functions. To simplify the field expressions, the Re designation is dropped, and the resulting field expressions become e = e (x,y,z;t) h = h (x,y,z;t) (4.1.1) (4.1.2) e = Re[E(x, y, z)e i(0t ] h = Re [H (x, y, z)e i0 ><] (4.1.3) (4.1.4) e = E(JC, y, z)e icot h = H(x, y, z)e icot (4.1.5) (4.1.6) 117 118 4. RELEVANT CONCEPTS IN ELECTROMAGNETIC FIELD THEORY The complex notation illustrated here is very useful in analysis; however, certain basic precautions should be taken with some algebraic manipulations such as multiplication of two complex quantities. This situation arises when energy and power are derived from products of the electric and magnetic fields. To discuss the complex number manipulations, consider the electric / i d ) I v(t) Z L . _y FIG. 4.1.1 Power considerations in an electric circuit. The instantaneous voltage and current are shown in the circuit. circuit shown in Fig. 4.1.1. The time average power flowing to the load Z L is given by [1] = ±fi(t)v(t)dt (4.1.7) where the brackets enclosing p(t) indicate the time average.

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  Quantum World Unveiled By Electron Waves The

  • Akira Tonomura(Author)
  • 1998(Publication Date)
  • World Scientific

    (Publisher)

  This reference plane can be replaced by a plane parallel-transported or rotated along the vertical magnetic line. All these apparently different flows of vector potentials represent the same dis-tribution of the uniform magnetic field shown in Fig. 74(a). There are many more 92 The Quantum World Unveiled by Electron Waves (a) (b) (c) Fig. 74. Vector potentials for a uniform magnetic field. (a) Uniform magnetic field (b) Vector potential — vortex flow (c) Vector potential — laminar flow choices of vector potentials for this physical situation, and all the distributions of A represent a single physical arrangement. The different vector potentials can be changed from one to another by gauge transformations and are regarded as physi-cally equivalent. You may not understand why vector potentials are so unintelligible. In order to help your understanding, I will show you later that vector potentials can be related to the number of magnetic lines of force having ever passed through at a point. The gauge freedom of vector potentials may be compared to the ambiguity in the number of the accumulated magnetic lines of force. It is shown later on that this freedom of A provided a key to the deeper truth of nature. Electrotonic State It was Lord Kelvin and Maxwell who introduced and developed the concept of vector potentials A. Here I would like to show you how the concept of vector potentials came to be devised. The story begins with Faraday, who discovered electromagnetic induction in 1831: when a magnet comes close to a metal ring, a current is induced to flow in the ring in such a direction that the value of the magnetic flux flowing through the ring is kept unchanged [see Fig. 75(a) and (b)]. Faraday thought that the ring became in a peculiar electrical condition of matter when a magnet came close to it.

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  Classical Electrodynamics

  • Tung Tsang(Author)
  • 1998(Publication Date)
  • WSPC

    (Publisher)

  94 Vector and Scalar Potentials Sec. 4.6 with only one component instead of the electric field E with three components. Similarly, we have introduced the magnetic vector potential A in magnetostatics (Chap. 3). We will now examine these potentials for dynamic (time-dependent) problems. In elec-trostatics,we have VxE=0, hence E=-V since the curl of gradient is zero. In magneto-statics, we have VB=0, hence B =VxA since the divergence of curl is zero. For dynamic problems, we still have V-B=0 (eq. 4.10b), hence B=VxA (4.41) is still valid. However,VxE*0 because of Faraday law of induction (eq. 4.10d), hence E=-VO is no longer valid. Substitution of eq. (4.41) into eq. (4.10d) gives* C dt C dt I C dt J (4.42) Since curi of divergence is zero, we can set the square bracket in eq. (4.42) to -VO hence-E = ~ V O — c dt (4.43) Eq. (4.41) and (4.43) are the general definitions for the scalar potential and vector potential A. For electrostatics,the second term in eq.(4.43) vanishes,and we have E=-V4>. In eqs. (4.41) and (4.43), we differentiate 4> and A to obtain E and B. Usually, only E and B can be measured experimentally,while O and A are derived from E and B. Hence O and A may not be unique. There may be integration constants in and A. We can choose new potentials 4>' and A': A ' -A + Vg <*>' =<|>-1^I c dt /A ... (4.44) where § is any arbitrary function. It can be readily seen that the same E and B fields can be obtained if we use 3>' and A' in eqs. (4.41) and (4.43): B = VxA' E = -V 4 > ' ---^ -C ^ (4.45) The choice between 4>, A and A' is somewhat arbitrary. The transformation from ,A-^4>',A' according to eq. (4.44) is known as the gauge transformation. For many dynamical problems such as electromagnetic (EM) wave propagations in the presence of sources, it is often convenient to use and A instead of B and E. We can substitute eqs. (4.41) and (4.43) into the Maxwell equations (4.10).

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  Modelling of Molecular Properties

  Theoretical Principles and Numerical Simulations

  • Maria Emilova Velinova(Author)
  • 2019(Publication Date)
  • Arcler Press

    (Publisher)

  SECTION II ELECTROMAGNETIC FIELDS AND SYMMETRY ELECTROMAGNETIC POTENTIALS BASIS FOR ENERGY DENSITY AND POWER FLUX CHAPTER 4 H E Puthoff Institute for Advanced Studies at Austin, 11855 Research Blvd., Austin, TX 78759, USA ABSTRACT In rounding out the education of students in advanced courses in applied electromagnetics it is incumbent on us as mentors to raise issues that encourage appreciation of certain subtle aspects that are often overlooked during first exposure to the field. One of these has to do with the interplay between fields and potentials, with the latter often seen as just a convenient mathematical artifice useful in solving Maxwell’s equations. Nonetheless, to those practiced in application it is well understood that various alternatives Citation : H E Puthoff, “Electromagnetic potentials basis for energy density and power flux” 2016 Eur. J. Phys. 37 055203 https://doi.org/10.1088/0143-0807/37/5/055203 Copyright : © Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Modelling of Molecular Properties: Theoretical Principles and Numerical Simulations 66 in the use of fields and potentials are available within electromagnetic (EM) theory for the definitions of energy density, momentum transfer, EM stress– energy tensor, and so forth. Although the various options are all compatible with the basic equations of electrodynamics (e.g., Maxwell’s equations, Lorentz force law, gauge invariance), nonetheless certain alternative formulations lend themselves to being seen as preferable to others with regard to the transparency of their application to physical problems of interest. Here we argue for the transparency of an energy density/power flux option based on the EM potentials alone.

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  Scientific Foundations of Engineering

  • Stephen McKnight, Christos Zahopoulos(Authors)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  By convention, the negative of the electric potential, measured in volts, is defined as r ! φ ¼ E ! þ ∂ A ! ∂t : (14.33) Since the gradient of a constant is zero, we can add any constant to the electric potential φ 0 ¼ φ + φ o without changing any physical effect. This corresponds to the fact that the zero of potential (the electrical ground of the system) is arbitrary. 292 Maxwell’s equations and electromagnetism At low frequencies where we can ignore the time dependence of A ! , ∂A ! ∂t ¼ 0, and we have the well-known electrostatic relationship: E ! ¼  r ! φ (14.34) where φ is the familiar electric potential or voltage (as in a 12-volt battery, for example) of any point with respect to a ground point where φ ¼ 0. This equation allows us to define the units of electric field to be volts/m. 14.8 Maxwell’s equations in integral form Maxwell’s equations as we have written them are strictly local relationships. The vector derivatives of the magnetic and electric fields at any point are related to the sources (charges and currents) and the time derivatives of the fields at the same point. We can, however, apply Gauss’s and Stokes’ theorems to Maxwell’s equations in differential form and express them in integral forms. We note that this transformation to integral form will result in non-local relationships – the fields at one point will be related to sources and fields elsewhere. 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.

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  Principles of Optics

  Electromagnetic Theory of Propagation, Interference and Diffraction of Light

  • Max Born, Emil Wolf(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Eqs. (27) and (28) have a simple interpretation. Consider the electrostatic potential of two fixed electric charges — e and + e situated at points whose position vectors with respect to a fixed point Q 0 (r Q ) are — a and a respectively. The Coulomb potential φ {Β) due to the charges is (see Fig. 2.1): (30) 2.2] E L E C T R O M A G N E T I C P O T E N T I A L S A N D P O L A R I Z A T I O N 79 If a is sufficiently small, we may expand l/R 2 in powers of the components of a and obtain i - = i - + a . g r a d i - + . . . > (31) where the operator grad is taken with respect to the coordinates of the point at which the charge — e is situated. If we neglect the higher-order terms in (31), (30) becomes <£ = e a . g r a d -^ = e ^ . (32) Suppose now that a is gradually decreased whilst e is increased without limits in such a way that ea approaches a finite value p : Urn ea = p. (33) Α-»Ό Then R 1 approaches R and (32) becomes in the limit 0 = p . g r a d o ^ . (34) This expression is identical with (27) in the special case when p is independent of time. If the charges depend on time but at each instant of time differ only in sign, we have, in place of (30), e(t - R 2 /c) e(t - Rjc) φ = ^ j — (35) and obtain by the same limiting process <^ = [ p ] . g r a d 0 ^ + ^ i Ä . [ p ] , which is (27). Thus (23) may be interpreted as the scalar potential of a distribution of electric dipoles of moment Ρ per unit volume. The last term in (28) may be shown to be the magnetic potential arising from these dipoles. In a similar way (28) may be shown to be the vector potential of a magnetic dipole of moment m(t), such a dipole being equivalent, of course, to an infinitesimal closed circuit of area A normal to m, carrying the current* cm/A. Hence the first two terms in (24) may be interpreted as the vector potential of a distribution of magnetic dipoles of moment M per unit volume.

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  Introduction to Classical Electrodynamics

  • Y K Lim(Author)
  • 1986(Publication Date)
  • WSPC

    (Publisher)

  178 The fact that Q.. are the components of the electric quadrupole moment tensor defined by Eq. (3.28) shows that the field is that of an electric quadrupole. The electric vector is given by Eq. (4.63) and a calculation similar to that set out in Eq. (4.91) gives E = _ L ? x H ..,-i£H : rad U E Q rad u £ o The average Poynting vector of the radiation f i e l d is therefore < N > = I R e ( E x H * ) --L -H d Hj ad k = -i <*— |k*Q| 2 k , 2 raa r a d 2ue„ r a a r a a 4ire„ 288irr2 O 0 (4.94) where we have made use of the fact that H . is transverse to the direction of propagation. The angular distribution of the quadrupole field is rather complicated and will not be discussed here. The interested reader may refer to the more advanced texts. It suffices to remark that while the 4 dipole radiation intensity varies as w , the quadrupole radiation intensity varies as u . The higher order terms in the expansion of A correspond to the fields of the higher multipoles. In general, each A can be similarly divided into electric and magnetic multipole potentials. Mathematical manipulation becomes increasing complex, however, and other methods must be used for the calculation of the higher multipole radia-tions. In this and the previous chapter we have taken a survey of the electromagnetic fields under the various approximations. Under static conditions, the electric and the magnetic fields may be treated separately, often along parallel lines. This is no longer possible when time variation is involved as the interaction between the tuso fields sets in. In the study of quasistationary fields we are concerned mostly with currents and applied electromotive forces. The rapidly 179 varying fields are important in that they are the sources of electro-magnetic radiation. At large distances from the source and for limited regions of space the radiation may be considered as plane electromag-netic waves. In the following chapters we shall consider the propaga-tion of such waves in matter.

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  The Power and Beauty of Electromagnetic Fields

  • Frederic R. Morgenthaler(Author)
  • 2011(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  It bears repeating that of the multitude of possible choices, those based upon Eqs. (8.7) and (8.10) represent two extremes worthy of special attention: the first because it expresses the field representation, the second because it emphasizes the circuit representation. Naturally, there are similar expressions involving dual Alternate energy-momentum tensors. CHAPTER 9 DIELECTRIC AND MAGNETIC MATERIALS 9.1 INTRODUCTION In Chapter 1, Section 1.7, dielectric and magnetic materials are characterized by elec-tric and/or magnetic dipoles that are generated by polarization currents and Amperian currents. Because we are using a vector potential formulation, it is convenient to model both the polarization and magnetization in terms of electric charges and currents. When these are added to J u , the current associated with the free (unpaired) charge, the result (repeated for convenience) is J total = J u + J polarization + J amperian = J u + ∂ P o ∂ t + ∇ × M o (9.1a) ρ total = ρ u + ρ polarization + ρ amperian = ρ u − ∇ · P o + 0 (9.1b) The total four-vector electric current is then J total = [ J total , i c ρ total ] (9.2) As previously mentioned, each component of electric charge is separately conserved, as is their total. The Power and Beauty of Electromagnetic Fields , First Edition. F. R. Morgenthaler. c 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 129 130 DIELECTRIC AND MAGNETIC MATERIALS 9.2 MAXWELL’S EQUATIONS WITH POLARIZATION AND MAGNETIZATION We continue to employ a four-dimensional formulation and introduce four-vectors: E , B , P , M , and V used to represent the electric field, magnetic induction, polarization, magnetization, and the continuum velocity of the material.

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  Waves and Oscillations in Nature

  An Introduction

  • A Satya Narayanan, Swapan K Saha(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  Therefore, the electric field is expressed as E = Q 4 π 0 r 2 ˆ r (2.66) In a 2-dimensional system, the curves characterized by constant V ( x, y ) are called the equipotential curves. In 3D, surfaces such that V ( x, y, z ) = constant are called the equipotential surfaces. Since E = −∇ V , the direction of E at a point is always perpendicular to the equipotential through that point. The properties of equipotential surface are • the electric field lines are perpendicular to the equipotentials and point from higher to lower potentials; • the equipotential surfaces produced by a point charge form a family of concentric spheres and, for constant electric field, a family of planes perpendicular to the field lines; • the tangential component of the electric field along the equipotential surface is zero; and • no work is needed to move a particle along an equipotential surface. Electromagnetic Waves 93 As stated in Section 2.1.3.3, the electric field caused by a continuous charge distribution ρ ( r ) is given by E ( r ) = 1 4 π 0 V ρ ( r ) r − r | r − r | 3 d V (2.67) where d V is the volume element and ρ ( r , t ) the local charge density at the position vector r (= x, y, z ). A given charge distribution creates the electrostatic potential scalar field, the gradient of which is the electric field. The electrostatic potential ϕ ( r ), which gives rise to by a continuous charge distribution ρ ( r ), is ϕ ( r ) = 1 4 π 0 V ρ ( r ) | r − r | d V (2.68) By differentiating equation (2.67), the electric field intensity is derived as the gradient of a potential E ( r ) = − ∇ V ρ ( r ) | r − r | d V = − ∇ ϕ ( r ) (2.69) = − ˆ i ∂ϕ ∂x + ˆ j ∂ϕ ∂y + ˆ k ∂ϕ ∂z (2.70) Equation (2.69) for the electric field is provided in terms of the potential gradient, which implies that the electric vector field is always normal to the equipotential surface and is directed along the direction of the decreasing potential.

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