An insightful problem involving the electromagnetic radiation from a pair of dipoles

Transcription

1 IOP PUBLISHING Eur. J. Phys. 31 (2010) EUROPEAN JOURNAL OF PHYSICS doi: / /31/4/011 An insightful problem involving the electromagnetic radiation from a pair of dipoles Glenn S Smith School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA , USA glenn.smith@ece.gatech.edu Received 5 April 2010, in final form 4 May 2010 Published 26 May 2010 Online at stacks.iop.org/ejp/31/819 Abstract The time-average power radiated by a pair of infinitesimal dipoles is examined as their spacing is varied. The results elucidate the effect of the interaction of the dipoles on their radiation. (Some figures in this article are in colour only in the electronic version) 1. Introduction Sometimes a great deal can be learned from a simple example, and we believe that this is the case for the following problem. Consider an infinitesimal electric dipole p 1 with harmonic time dependence (e iωt ). The current moment for this dipole is j 1 = iω p 1 = I 0 lẑ, (1) in which I 0 is the current and l the infinitesimal length. When this dipole is alone in free space, it radiates the time-average power P rad 0 P rad 0 = ck4 0 p 1 2 = ζ 0k0 2 I 0 l 2, (2) 12πɛ 0 12π in which k 0 = ω/c = 2π/λ and ζ 0 = μ 0 /ɛ 0 are the wave number and wave impedance, respectively [1, 2]. Now we add a second dipole identical to the first with the moment j 2 = j 1.Asshownin figure 1(a), the two dipoles are on the x axis and are separated by the distance d. What is the time-average power radiated by the pair of dipoles? Students confronted with this question often try to use their intuition to quickly obtain an answer. A few typical answers are as follows /10/ $30.00 c 2010 IOP Publishing Ltd Printed in the UK & the USA 819

2 820 GSSmith Dipole z r Field Point 2 d 1 (a) x z y r (b) x Figure 1. (a) Pair of dipoles and the the coordinates used when analysing the field. (b) Coordinates used in evaluating the integral for the time-average power radiated. (i) The current in each dipole is fixed, so the radiation from each dipole is independent of the other, making the total time-average power radiated by the pair P rad 1+2 = 2 P rad 0. (Wrong) (ii) When the two dipoles are very close together, they radiate essentially as one dipole with the current 2I 0. So the total time-average power radiated by the pair for any spacing is P rad 1+2 = 4 P rad 0. (Partially right) (iii) When the two dipoles are very far apart, they radiate essentially as if they were isolated. So the total time-average power radiated by the pair for any spacing is P rad 1+2 = 2 P rad 0. (Partially right) The correct solution to this problem, which is presented in the next section, can provide insight into the process of electromagnetic radiation. This material is suitable for use as an example or a homework problem in classes on electricity and magnetism, undergraduate or graduate, in which students first encounter the radiation from an infinitesimal dipole. It illustrates important points with regard to the interaction between the two dipoles and between current distributions in general. For example, the analysis shows that the total power radiated by the dipoles is essentially unaffected by an increase in their spacing once it is greater than a few wavelengths. However, the distribution for the radiated power in space, the radiation pattern, is significantly affected by their spacing, no matter how large. 2. Time-average power radiated A straightforward way to determine the the time-average power radiated is to recognize that it is equal to the time-average power supplied by the two dipoles to the electromagnetic field.

3 An insightful problem involving the electromagnetic radiation from a pair of dipoles 821 The power supplied to the field by dipole 1 in the presence of dipole 2 is [3] P rad 1 = 1 2 Re( E j 1 ) = 1 2 Re[( E 1 + E 2 ) j 1 ] = P rad Re[ E 2 ( r = ˆxd/2) j 1 ], (3) in which the superscript indicates the complex conjugate, and we have introduced the electric field of dipole 2 at dipole 1 [1, 2]: E 2 ( r = ˆxd/2) = iζ 0k0 2I [ 0 l 1 4π k 0 d + i (k 0 d) 2 1 ] (k 0 d) 3 e ik 0dẑ. (4) Here we have arranged the result so that the three terms normally associated with the electric field of a dipole are evident, that is, terms proportional to 1/R,1/R 2 and 1/R 3, where R is the distance from the dipole. Very close to the dipole, the last of these terms (1/R 3 ) is dominant, and it is the one that we associate with the electrostatic field [4] (for comparison with the electrostatic case, the current moment ii 0 l/ω is replaced by the traditional dipole moment p). When the spacing between the dipoles is small, we might expect this term to be the most important in determining the power supplied; however, we will find this is not the case. After substituting (4) into(3) and rearranging terms, we obtain the power supplied by dipole 1: P rad 1 = 1+ 3 [ sin k0 d + cos k 0d P rad 0 2 k 0 d (k 0 d) sin k ] 0d. (5) 2 (k 0 d) 3 Because of symmetry, the power supplied by dipole 2 is the same as that supplied by dipole 1, so the total power supplied or radiated by the pair of dipoles is just twice (5): [ P rad 1+2 sin k0 d = cos k 0d P rad 0 k 0 d (k 0 d) sin k ] 0d 2 (k 0 d) 3 = 2+3(p 1 + p 2 + p 3 ), (6) in which we have identified the contributions of the aforementioned three terms in (4) asp 1, p 2 and p 3. Figure 2 is a graph of the time-average power radiated by the pair of dipoles (6) versus their spacing in terms of the wavelength d/λ. From this graph we can determine the validity of the intuitive answers given in the introduction. Contrary to answer (i), the interaction of the dipoles clearly affects the power radiated, even though the currents in the dipoles (dipole moments) remain fixed. Answers (ii) and (iii) are partially correct because when the dipoles are very close, P rad P rad 0, and when the dipoles are very far apart, P rad P rad 0. The transition between these two levels is accompanied by an oscillation with a period of about one wavelength. This oscillation shows that the interaction of the dipoles can either increase or decrease the power radiated above that when the dipoles are isolated (2.0 on the graph). The use of a physical model may help with understanding this result. Consider each of the dipoles to be an electrically short linear antenna driven at its terminals by an ideal current source. The ideal current source produces a fixed current in the antenna (fixed dipole moment), but the voltage across the source is free to vary. The presence of the electric field of antenna 2 at antenna 1 changes this voltage and therefore the time-average power supplied to the field by antenna 1. Another way to view this phenomenon is to say that the radiation resistance, R rad, of antenna 1 (the resistance seen by the source) is changed by the presence of antenna 2[5]. For conservation of power, the power supplied by the source must equal that radiated; hence, (1/2) I 0 2 R rad = P rad 1 = (1/2) P rad 1+2 or R rad = P rad 1+2 / I 0 2. Thus, the graph in figure 2 can be viewed as one for the normalized radiation resistance of one of the antennas; it shows how the radiation resistance of the antenna varies with the spacing.

4 822 GSSmith 4.0 rad 1+2 rad 0 < P > / < P > d Figure 2. The time-average power radiated by the pair of dipoles versus their spacing in terms of the wavelength. When the spacing between the dipoles is small, k 0 d = 2πd/λ 1, the three terms in the last line of (6) behave as p 1 1+ p 2 1 (k 0 d) p 3 1 (k 0 d) (7) 6 Note that, when these terms are summed as in (6), the factors 1/(k 0 d) 2 in p 2 and p 3 cancel, so all three terms contribute comparably to the time-average power supplied when the dipoles are closely spaced. As the spacing between the dipoles is increased, oscillations occur in P rad 1+2.Theyare caused by the phase of the field E 2 changing with respect to that of the current j 1 and vice versa (see (3)). When E 2 and j 1 are referred to the same direction, say ẑ, maxima [minima] occur when the difference in their phases is ±(2n +1)π [±2nπ] with n = 0, 1, 2, 3,...The oscillations that occur for d/λ > 1.0 are almost entirely due to the term p 1. This can be seen from figure 3, in which p 1, p 2 and p 3 are graphed versus d/λ. For the p 1 term in (6), the maxima [minima] occur when sin(k 0 d) 1, or d/λ 1.25, 2.25,... [sin(k 0 d) 1 or d/λ 1.75, 2.75,...]; hence, the period is about one wavelength for the oscillation in figure 2. Another, perhaps more intuitive and yet more involved, method for obtaining the timeaverage power radiated by the pair of dipoles is to enclose them in a spherical volume and calculate the time-average power passing through the surface of the sphere. To simplify the calculation, we make the radius of the sphere r very large (lim r ), so that the field is the radiated field, that is, the part of the field that behaves as 1/r. For a single dipole located at the origin, the radiated (superscript r) electric field is [1, 2] ( ) E 0 r ( r) = iωμ0 I 0 l e ik 0 r sin θ ˆθ. (8) 4π r When a dipole is displaced from the origin, the additional phase accrued for a wave propagating from the dipole to the field point must be taken into account. For the two dipoles shown in figure 1(a), this can be done with the array factor

5 An insightful problem involving the electromagnetic radiation from a pair of dipoles 823 p i p 1 p 2 p d Figure 3. The three elements that make up the time-average power radiated by the pair of dipoles versus their spacing in terms of the wavelength. AF( r) = e ik 0(d/2)ˆx r +e ik 0(d/2)ˆx r = e ik 0(d/2) sin θ cos φ +e ik 0(d/2) sin θ cos φ = 2 cos [(k 0 d/2) sin θ cos φ]. (9) Now the product of (8) and (9) is the radiated electric field for the pair of dipoles: ( ) E r ( r) = AF E 0 r = iωμ0 I 0 l e ik 0 r sin θ cos [(k 0 d/2) sin θ cos φ] ˆθ. (10) 2π r The complex Poynting vector for this field is S c r ( r) = 1 E r ( r) 2ˆr, (11) 2ζ 0 so the time-average power radiated is P rad 1+2 = 2π π φ=0 = 3 2π P rad 0 Re [ˆr S r c ( r)] r 2 sin θ dθ dφ θ=0 2π π φ=0 θ=0 sin 3 θ cos 2 [(k 0 d/2) sin θ cos φ]dθ dφ. (12) With a change in the angular variables from (θ,φ)to(ψ, χ), see figure 1(b), we have sin θ cos φ = cos ψ,sin 2 θ = 1 sin 2 ψ cos 2 χ,sinθ dθ dφ = sin ψ dψ dχ, and (12) becomes P rad 1+2 P rad 0 = 3 2 = 12 2π π χ=0 ψ=0 π/2 1 χ=0 ξ=0 (1 sin 2 ψ cos 2 χ)cos 2 [(k 0 d/2) cos ψ]sinψ dψ dχ (sin 2 χ + ξ 2 cos 2 χ)cos 2 [(k 0 d/2)ξ]dξ dχ. (13) The integrals in (13) are of standard form [6], and they are easily evaluated to give our final answer: [ P rad 1+2 sin k0 d = cos k 0d P rad 0 k 0 d (k 0 d) sin k ] 0d. (14) 2 (k 0 d) 3 As expected, this is the same as our earlier result (6) obtained by calculating the time-average power supplied to the electromagnetic field by the two dipoles.

6 824 GSSmith d/ = 5.24 d/ = Deg Figure 4. Patterns for the time-average power radiated per unit solid angle in the horizontal plane, [d P rad 1+2 /d ] θ=π/2, for two different spacings of the dipoles. 3. Radiation patterns From our preceding observations, we can conclude that the two dipoles behave more-or-less independently once their spacing is greater than d/λ 2.0 because the power radiated then varies less than ±10% with an increase in the spacing. It is important to realize that the total power radiated being unaffected by the spacing does not imply that the distribution of the power in space is unaffected. This point is illustrated in figure 4, where patterns are plotted for the horizontal plane (x y plane); that is, the time-average power radiated per unit solid angle, [ ] d Prad 1+2 d θ=π/2 = Re ( r 2ˆr S c r ) 1 = r E r 2 2ζ 0 = 3 2π P rad 0 cos 2 [(k 0 d/2) cos φ], (15) is plotted as a function of the angle φ. The results are for the two spacings d/λ = 5.24 and d/λ = 5.74, which are the spacings for an adjacent maximum and minimum of P rad 1+2,see the arrows in figure 2. Note that these precise values differ slightly from our earlier estimates of d/λ 5.25 and d/λ 5.75, which were based on the maxima and minima of sin(k 0 d) and not on the entire expression for P rad 1+2. At the two spacings used for figure 4, the time-average powers radiated differ by less than 10%, but the patterns are quite different, particularly at angles near φ = 0, 180. For these large spacings, we see that the pattern changes significantly with the spacing, but it changes in a manner that keeps the total power radiated nearly constant. Acknowledgments The author is grateful for the support provided by the John Pippin Chair in Electromagnetics that furthered this study.

7 An insightful problem involving the electromagnetic radiation from a pair of dipoles 825 References [1] Smith G S 1997 An Introduction to Classical Electromagnetic Radiation (Cambridge: Cambridge University Press) section 7.1.2, pp [2] Jackson J D 1999 Classical Electrodynamics 3rd edn (New York: Wiley) section 9.2, pp [3] Smith G S 1997 An Introduction to Classical Electromagnetic Radiation (Cambridge: Cambridge University Press) section 1.7.3, pp 94 8 [4] Clemmow P C 1973 An Introduction to Electromagnetic Theory (Cambridge: Cambridge University Press) section 2.1.5, pp 27 9 [5] Smith G S 1997 An Introduction to Classical Electromagnetic Radiation (Cambridge: Cambridge University Press) section 7.2, pp [6] Dwight H B 1961 Tables of Integrals and other Mathematical Data 4th edn (New York: Macmillan)