Lienard Wiechert Potential

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  eBook - ePub

  Classical Electromagnetic Radiation

  • Jerry Marion(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER 7

  The Liénard-Wiechert Potentials and Radiation

  Publisher Summary

  This chapter discusses the ultimate sources of all electromagnetic radiation—namely, moving charges. Radiation can be produced only if a charge undergoes acceleration. There are many interesting applications of accelerating charges—the production of X-rays, the acceleration of charged particles to velocities approaching the velocity of light, the radiation from antennas, etc. If the currents are steady, the equation for the vector potential is valid. The calculation of the potential requires a retarded-time integration over the entire volume containing charge that contributes to the potential Φ. The chapter describes radiation from a charged particle confined to a circular orbit and from an accelerated charged particle at low velocities.

  7.1 Introduction

  In this chapter we shall be concerned with the ultimate sources of all electromagnetic radiation, viz., moving charges. We shall find that radiation can be produced only if a charge undergoes acceleration. There are many interesting applications of accelerating charges—the production of X-rays, the acceleration of charged particles to velocities approaching the velocity of light, the radiation from antennas, etc. We shall study the radiation fields associated with these processes and will find a close similarity in the results. In Chapter 13 some of the results obtained here will be derived from the standpoint of relativity theory.

  7.2 Retarded Potentials

  We found in Section 4.6 that the scalar and vector potentials could be calculated from the expressions [cf. Eqs. (4.38) and (4.39) ]

  In these equations we have explicitly indicated that the potentials are to be computed at a position designated by the radius vector r by integrating ρ and J throughout V by considering these quantities to be functions of the radius vector of integration r ′. The distance between the integration point r ′ and the point at which Φ and A are computed is |r’ − r′ |, and dv ′ is the volume element at r

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  Waves in Complex Media

  • Luca Dal Negro(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  2.3.1 Electromagnetic Potentials Maxwell’s equations (2.2) and (2.3) are automatically satisfied if we introduce an aux- iliary vector field A(r,t) and an auxiliary scalar field φ(r,t) defined by the following: 5 B(r ,t ) = ∇ × A(r ,t ) (2.12) E(r ,t ) = − ∂ A(r ,t ) ∂t − ∇ φ(r ,t ). (2.13) The vector field A (r,t) is called the vector potential and the scalar field φ(r,t) the scalar potential of the electromagnetic field. Inserting the preceding definitions into the two remaining Maxwell’s equations (2.1) and (2.4), and using the well-known vector identity ∇ × ∇ × A = −∇ 2 A + ∇(∇ · A), we obtain the following: 4 This equation is valid only in the nonrelativistic limit, i.e., for small velocities compared to the speed of light. Moreover, by using either the Lagrangian or the Hamiltonian formalism, the Lorentz equation can be rigorously derived as the equation of motion for a charged particle interacting with the electromagnetic field. 5 This fact follows from the well-known vector identity ∇ · ∇ × A = 0 valid for any vector field A(r,t) and ∇ × ∇ φ = 0 for any scalar field φ (r,t). 2.3 The Lorentz Equation 27 ∇ 2 φ + ∂ ∂t (∇ · A) = −ρ/ 0 (2.14) ∇ 2 A − 1 c 2 0 ∂ 2 A ∂t 2 − ∇  ∇ · A + 1 c 2 0 ∂ φ ∂t  = −μ 0 J, (2.15) where we have written J tot ≡ J to simplify the notation (and similarly for the charge density). The two preceding equations form a system of second-order coupled partial differential equations that is entirely equivalent to Maxwell’s equations. Furthermore, we can uncouple these equations by exploiting the arbitrariness involved in the defini- tions of the potentials. To achieve this goal, we recognize that the fields E and B are invariant under the following gauge transformation of the potentials: A → A  = A + ∇ χ (2.16) φ → φ  = φ − ∂ χ ∂t , (2.17) where χ(r , t) is a scalar function, known as the gauge function.

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  Waves and Oscillations in Nature

  An Introduction

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

    (Publisher)

  The expres-sion (equation 2.131), also referred to as Lorentz force density, provides the connection between classical mechanics and electromagnetism. The Lorentz force is utilized in Hall effect devices, in the focusing and deflection of elec-tron beams in cathode-ray tubes, and in galvanometer movements. If an electric current, i , passes through a specimen (conductor or semicon-ductor) placed in a magnetic field, B , a potential proportional to the current and to the magnetic field is developed across the material in a direction per-pendicular to both the current and to the magnetic field, which is referred to as the Hall effect[40]. The charge carriers experience a Lorentz force (see equation 2.131), which would deflect them toward one side of the slab. The field required to achieve equilibrium is E H = v d B (2.132) where v d is the drift velocity of the carriers in the conductor. Electromagnetic Waves 107 2.3 Time-Varying Fields Both the electric field E ( r , t ) and the magnetic field B ( r , t ), in which r (= x, y, z ) is the position vector and t the time, are time-dependent which can be specified at every point in space. In what follows, the fundamental laws governing time-varying fields are enumerated in brief. 2.3.1 Faraday’s Law In 1831, Faraday found experimentally, what is referred to as Faraday’s law on induction, that the work done by an electric force in moving a unit positive charge around a closed path is proportional to the time rate of decrease of the flux of magnetic force through any open surface. When a changing magnetic field is linked with a coil, an electromotive force (EMF) is induced in it. This change in magnetic field may be caused by changing the magnetic field strength by moving a magnet towards or away from the coil or by moving the coil into or out of the magnetic field as desired. In other words, the magnitude of the EMF induced in the circuit is proportional to the rate of change of flux.

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  Electromagnetic Radiation, Scattering, and Diffraction

  • Prabhakar H. Pathak, Robert J. Burkholder(Authors)
  • 2021(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  (5.148) 5.5 Potentials and Fields of a Moving Point Charge Consider an ideal point charge of strength Q 0 , which moves along a prescribed trajectory as shown in Figure 5.11. The charge has a nonuniform velocity v at r ¤ 0 and at time t ¤ . It is of interest here to find the electromagnetic fields E , H produced by this moving point charge. The results obtained below for the ideal point charge can be readily extended via a superposition integral to deal with a classical charge distribution moving together with the same velocity. The charge in Figure 5.11 defines the following 186 ELECTROMAGNETIC POTENTIALS AND FIELDS OF SOURCES IN UNBOUNDED REGIONS Figure 5.11 Point charge Q 0 moving with a velocity v and producing the electromagnetic fields E , H which are observed at the point P . charge and current densities: L v ( r ¤ , t ¤ ) = Q 0 r ¤ -r ¤ 0 ( t ¤ ) J v ( r ¤ , t ¤ ) = Q 0 v ( t ¤ ) r ¤ -r ¤ 0 ( t ¤ ) M ; r ¤ ¸ V , (5.149) where, V is the volume of external space consisting of a homogeneous, isotropic medium ( , ) . The results developed below are of interest in predicting the interactions of high-speed charged parti-cles with dielectric media, as well as in some particle physics applications. One also notes that electro-magnetic radiation resulting from the emission of charged particles from the sun may cause disruptions of electronic communications systems. The starting point for evaluating the EM fields radiated by a moving charge is the use of the space-time integrals of (5.116) and (5.119), rather than just the spatial integrals in (5.128) and (5.129). The latter equations are actually obtained from the former (i.e., (5.116) and (5.119)) after evaluating the temporal integrals in closed form thus leaving only the spatial integrals over the source region.

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  Foundations of Classical Mechanics

  • P. C. Deshmukh(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  Maxwell’s equations integrate the electromagnetic phenomena, and thus constitute the first unified field theory, which is the forerunner of subsequent unification formalisms. Maxwell’s equations also laid the seeds of the special theory of relativity, as we shall see in the next chapter, developed by Albert Einstein about four decades after Maxwell’s work. 437 Basic Principles of Electrodynamics 12.3 THE ELECTROMAGNETIC POTENTIALS Solutions to the Maxwell’s equations provide the electromagnetic fields. These may then be used in the Lorentz force law (Eq. 12.23) for applications in solving the equation of motion for a charged particle in an electromagnetic field. From the two theorems we discussed in Section 1, we know that the electromagnetic vector fields can be determined, using appropriate boundary conditions, from the Maxwell’s equations which provide the curl and the divergence of both the vector fields. Furthermore, from the Helmholtz theorem, we can write each field as a sum of a solenoidal field, and an irrotational field. We know that the divergence of the magnetic flux density field B itself is always zero. It can therefore be written as the curl of a vector field, as the divergence of the curl of a vector is necessarily solenoidal. Thus, B (r , t) = ∇ × A (r , t). (12.28) The magnetic vector induction field is B (r , t) therefore derivable from the vector field A (r , t). The vector field is called the magnetic vector potential field. Using Eq. 12.28 in the Maxwell’s equation (Eq. 12.16c) for the curl of the electric field intensity, we get:            ∇ × = − ∂ ∂ = − ∂ ∇ × ∂ = −∇ × ∂ ∂ E r t B r t t A r t t A r t t ( , ) ( , ) [ ( , )] ( , ) , (12.29a) i.e.,       ∇ × + ∂ ∂         = E r t A r t t ( , ) ( , ) . 0 (12.29b) In other words,     E r t A r t t ( , ) ( , ) + ∂ ∂           constitutes an irrotational vector field.

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