Retarded Potential

3 Key excerpts on "Retarded Potential"

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  eBook - ePub

  Electromagnetism

  Maxwell Equations, Wave Propagation and Emission

  • Tamer Becherrawy(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  They show that we must take into account the time delay due to the propagation if the charge q or its position varies. We note that equation [ 15.77 ], which determines t ', may be very complicated. It may have no roots; then, the potentials vanish at the considered position and time. If it has several roots / i, the potentials are the superpositions of the potentials produced by the particle at the corresponding positions and arriving at M at the same time t. To understand the physical meaning of the Retarded Potentials and fields, let us consider the following simple example: assume that a charge q is initially at rest at the origin O and it is suddenly accelerated at t = 0 to have a velocity v q. At t < 0, the field configuration is that of a particle at rest, i.e. B = 0 and E = q r /4πεr 3 (spherically symmetric). After / = 0, the field is deformed. It is evident that a test charge placed close to O will feel this deformation before a test charge placed far away. A test charge located at a distance R from O feels this deformation only after the time / = R/v. At this time, the source charge q is no longer at O but a point O ' such that 00' = v q t. At a given time /, the sphere of radius R = ct divides the space into two regions: outside the sphere, where there is only the spherical symmetric electric field E = q r /4πεr 3 of the charge q at rest and inside the sphere, where we have both electric and magnetic fields with the electric field having no spherical symmetry. If the charge velocity is small, compared to the propagation speed of the wave (P q << 1), the correction factor g(t') is almost equal to 1

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  eBook - PDF

  Introduction to Electrodynamics

  • David J. Griffiths(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  y z x r r w(t r ) q Particle trajectory Retarded position Present position FIGURE 10.8 454 Chapter 10 Potentials and Fields Then r 1 − r 2 = c(t 2 − t 1 ), so the average speed of the particle in the direction of the point r would have to be c—and that’s not counting whatever velocity the charge might have in other directions. Since no charged particle can travel at the speed of light, it follows that only one retarded point contributes to the potentials, at any given moment. 14 It follows, then, that V (r, t ) = 1 4π 0 qc (rc − r · v) , (10.46) where v is the velocity of the charge at the retarded time, and r is the vector from the retarded position to the field point r. Moreover, since the current density is ρ v (Eq. 5.26), the vector potential is A(r, t ) = μ 0 4π  ρ(r  , t r )v(t r ) r d τ  = μ 0 4π v r  ρ(r  , t r ) d τ  , or A(r, t ) = μ 0 4π qcv (rc − r · v) = v c 2 V (r, t ). (10.47) Equations 10.46 and 10.47 are the famous Liénard-Wiechert potentials for a moving point charge. 15 Example 10.3. Find the potentials of a point charge moving with constant velocity. Solution For convenience, let’s say the particle passes through the origin at time t = 0, so that w(t ) = vt . We first compute the retarded time, using Eq. 10.44: |r − vt r | = c(t − t r ), 14 For the same reason, an observer at r sees the particle in only one place at a time. By contrast, it is possible to hear an object in two places at once. Consider a bear who growls at you and then runs toward you at the speed of sound and growls again; you hear both growls at the same time, coming from two different locations, but there’s only one bear. 15 There are many ways to obtain the Liénard-Wiechert potentials. I have tried to emphasize the ge- ometrical origin of the factor (1 − ˆ r · v/c) −1 ; for illuminating commentary, see W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2d ed. (Reading, MA: Addison-Wesley, 1962), pp. 342-3. A more rigorous derivation is provided by J.

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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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