3P4_1191

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8/18/2019 3P4_1191


1192 PIERS Proceedings, Moscow, Russia, August 19–23, 2012

In a dispersive medium, there concepts of the phase and group velocities of modulated waves. Theconcepts of the group and phase velocities were used for description of propagation of the neareld of an electric dipole in vacuum [14]. This raises the question of whether application of theseconcepts to the case of antennas located in vacuum is justifable.

Our viewpoint is as follows. The concept of the phase velocity, which is rigorously valid for anisolated harmonic wave, is extended to a superposition of harmonic waves with variable amplitudesin the following form: it relates to the velocity of motion of phase fronts corresponding to the zerosof the components of the electric and magnetic elds.

The situation concerning the concept of the group velocity is more complicated. Radiationof a small antenna in vacuum can be treated as excitation of a spherical waveguide. From thisviewpoint, an antenna in vacuum is a structure with frequency dispersion. The group velocity of asuperposition of waves is usually introduced as the velocity of motion of the envelope. In this case,it is assumed that the envelope varies slowly as compared to the high-frequency lling [15]. Thisdenition is used in [14]. Near a point antenna, the envelope varies rapidly and the approach usedin [14] is, strictly speaking, is unjustied. The surprising thing is that, in [14], this approach wasused for the analysis of individual group velocities of electric and magnetic eld components andnot the Poynting vector.

We will analyze the instantaneous velocities of motion of the zeros and extrema of the electricand magnetic elds and the Poynting vector. Instantaneous velocities of the zeros of the electricand magnetic elds are the phase velocities (which coincides with [14]) and instantaneous velocitiesof extrema of the Poynting vector are the analogues of the group velocities.

3. A TECHNIQUE FOR THE ANALYSIS OF THE EVOLUTION OF THE FIELD LINESOF THE ELECTROMAGNETIC FIELD AND THE VECTOR LINES OF THEPOYNTING VECTOR FOR AN ELECTRIC DIPOLE

Let us consider an electric dipole excited by a pulsed current and directed along the z axis of theCartesian coordinate system. Length l of the dipole is small as compared to the duraction of thepulse exciting the the antenna and the dipole diameter is substantially lesser than the dipole length.Expressions for the elds excited by the dipole are well known [16]:

E (R,θ, t ) = cosθ2πε 0

p(t )R 3 +

p (t )cR 2 e R +

sin θ4πε 0

p (t )c2 R

+ p (t )

cR 2 + p(t )

R 3 e θ

H (R,θ,t ) = 14π

rot 1R

p t e z

(1)

In formula (1), p(t) = q (t)l is the time dependence of the dipole moment and primes above function p (t ) denote derivatives with respect to the argument t = t − R/c .

Let us introduce variable τ = ct − R and write eld components in the following form:

E R (R,θ, t ) = cosθ2πε 0 R 3 f 1 , f 1 = p (τ ) + Rp (τ ) , (2)

E θ (R,θ, t ) = sinθ4πε 0 R 3 f 2 , f 2 = p (τ ) + Rp (τ ) + R2 p (τ ) , (3)

H ϕ (R,θ,t ) =

c

4π sinθ

p (τ )

R +

p (τ )

R 2 (4)

The analysis of the dipole elds is simplied due to the fact that the time dependence of theseelds is parametric and the eld structure is axially symmetric. This allows us to use vector analysisin a plane and apply well-known methods of the qualitative theory of rst-order ordinary differentialequations. A qualitative analysis was performed for the electric eld and the Poynting vector inthe cylindrical coordinate system. The results of the qualitative analysis were reported in [17–20].

4. SOME RESULTS OF THE ANALYSIS OF THE NEAR-ZONE FIELDS OF ANELECTRIC DIPOLE

A feature of the applied analysis of nonstationary elds is the possibility of studying the eldevolution for sources with arbitrary time dependences of the charge and current. For this reason,we consider in detail harmonic excitation of a dipole.

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Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 1193

We will analyze evolution of eld component E R at θ = 0 and components E θ , H ϕ , and S R atθ = 90 ◦ . Instantaneous phase velocities of the eld components are determined by equating to zerothe eld components [17–20] and instantaneous velocities of the eld extrema and the Poyntingvector are determined from conditions ∂U

∂R = 0, where U are the corresponding components of theeld or the Poynting vector.

Trajectories of the zeros (dashed lines) and extrema (bold solid lines) of eld components E R ,E θ , and H ϕ , respectively, in plane ( ωt, k0 R) are shown in Figs. 1(a), 1(b), and 1(c). Thin straightlines depict corresponding velocities of light in vacuum.

As follows from Fig. 1(a), the zeros and local extrema of E R originate at the point of the dipolewith innite velocities. These velocities monotonically decrease with time and tend to the velocityof light in vacuum.

As follows from Figs. 1(b) and 1(c), the process of formation of the zeros and extrema of components E θ and H ϕ is qualitatively different. Local extrema of E θ and H ϕ arise at a nitedistance from the dipole with innite velocities. Then, the extrema are split into two extrema.One extremum moves toward innity with a decreasing velocity tending to the velocity of light invacuum. The other extremum moves toward the dipole. Zeros of E θ demonstrate similar behavior.Zeros of H ϕ originate at the point of the dipole with an innite velocity; then they move towardinnity and their velocities tend to the velocity of light in vacuum.

Trajectories of the zeros of E θ (dashed lines) and H ϕ (dotted lines) and the extrema of the

Poynting vector (bold solid lines) in plane ( ωt, k0 R) are shown in Fig. 1(d). It follows from theanalysis of Fig. 1(d) that formation and motion of the extrema of the Poynting vector are similarto those of the extrema of E θ and H ϕ . Thus, we can say that the instantaneous velocities of theextrema of component S R of the Poynting vector are larger than the velocity of light in vacuum ina certain time interval.

Comparison of our results with the results of study [14] shows that they coincide for the phasevelocities. As to the group velocities of eld components determined in [14], their behavior quali-tatively agrees with the velocities of motion of the extrema of E R , E θ , and H ϕ . Evolution of thePoynting vector was not analyzed in [14].

Along with the harmonic excitation of the dipole, we also analyzed particular variants of excita-tion of this dipole by pulses with a nite duration and a small number of oscillations of the chargeand current. It has been shown that evolution of the zeros and extrema of the eld components andthe Poynting vector is more complicated than in the case of harmonic excitation. In this case, localextrema of the elds and the Poynting vector in the near zone may move with velocities exceedingthe velocity of light in vacuum; however, they cannot leave behind the pulse fronts.

The results obtained can be extended to the case of radiation of a magnetic dipole or an electricloop with the use of the principle of the principle of permutational duality. Whence, the performedanalysis qualitatively explains the experimental results presented in [11–13].

(a)

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1194 PIERS Proceedings, Moscow, Russia, August 19–23, 2012

(c)

(d)

(b)

Figure 1: Evolution of the extrema and zeros of components E R , E θ , H ϕ , and S R in plane ( ωt, k0 R).

REFERENCES

1. Hertz, H., Ann. d. Phys. B , Vol. 36, No. 1, 1889.2. Brillouin, L., Wave Propagation and Group Velosity , Academic Press, New York, London,

1960.3. Wainstein, L. A., Sov. Phys. Usp. , Vol. 118, No. 2, 1976.4. Shvartsburg, A. B., Phys. Usp. , Vol. 50, No. 37, 2007.5. Enders, A. and G. Nimitz, J. Phys. I France , Vol. 2, No. 9, 1992.6. Ranfagni, A. and D. Murnai a.o., Phys. Rev. E , 1998.7. Ranfagni, A. and D. Murnai a.o., Phys. Rev. E , 2004.

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Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 1195

8. Ranfagni, A. and D. Murnai a.o., Phys. Letters A , Vol. 247, 1998.9. Ranfagni, A. and D. Murnai a.o., Phys. Letters A , Vol. 352, 2006.

10. Ranfagni, A. and D. Murnai a.o., Phys. Letters A , Vol. 372, 2008.11. Missevitch, O. V., A. L. Kholmetskii, and R. Smirnov-Rueda, J. Appl. Phys. , Vol. 101, 023532,

2007.12. Missevitch, O. V., A. L. Kholmetskii, and R. Smirnov-Rueda, J. Appl. Phys. , Vol. 102, 013529,

2007.13. Missevitch, O. V., A. L. Kholmetskii, and R. Smirnov-Rueda, EPL , Vol. 93, 64004, 2011.14. Walker, W., Lorentz Group, CPT and Neutrios , A. Chubucalo, V. Dvoeglazov, D. Ernst,

V. Kadyshevsky and Y. S. Kim, Eds., 327, World Scientic, Singapore, 2000.15. Crawford, Jr, F. S., Waves , Vol. 3, Mc Graw-Hill Book Company, 1968.16. Feynman, R. P., R. B. Leghton, and M. Sands, The Feynman Lectures on Physics , Vol. 2,

Addison-Wesley Publishing Company, Inc. Reading, Massachusetts, Palo Alfo, London, 1964.17. Permyakov, V. A., D. V. Sorokovik, and A. N. Korykin, The Third International Conference

Ultrawideband and Ultrashort Impulse Signals, UWBUSIS-2006 , Sevastopol, Ukraine, Sep. 18–22, 2006.

18. Permyakov, V. A. and D. V. Sorokovik, Nonlinear World , Vol. 5, No. 12, 2007 (in Russian).19. Permyakov, V. A. and D. V. Sorokovik, Nonlinear World , Vol. 6, No. 4, 2008 (in Russian).20. Permyakov, V. A., D. V. Sorokovik, and A. N. Korykin, “Qualitative analysis of dipole anten-

nas impulse radiation,” Progress In Electromagnetics Research Symposium Abstracts , Moscow,Russia, Aug. 18–21, 2009.

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