THE UNIVERSITY OF MICHIGAN 7322-1-F Topics in Electrodynamics of Moving Media Final Report 1 May 1965 - 1 May 1966 By C. T. Tai, V Pyati and RM..Kalafu anrit NGR-23-O005-107 Prepared for National Aeronautics and Space Administration Langley Research Center Langley Station Hampton, Virginia 23365

THE UNIVERSITY OF MICHIGAN 7322-1-F TABLE OF CONTENTS Page I INTRODUCTION 1 II RADIATION DUE TO AN OSCILLATING DIPOLE OVER A LOSSLESS SEMI-INFINITE MOVING DIELECTRIC MEDIUM 2 Im DEFINITE AND INDEFINITE FORMS OF MAXWELL-MINKOWSKI EQUATIONS 3 IV ELECTRODYNAMICS OF MOVING, CONDUCTING MEDIA 4 V CERENKOV RADIATION TOPICS 8 VI DRIFTING, MAGNETO-IONIC MEDIUM - FIRST ORDER THEORY 14 VII THE TIME-DEPENDENT DYADIC GREEN'S FUNCTION FOR A MOVING ISOTROPIC MEDIUM 18 VIII REFERENCES 21 iii

THE UNIVERSITY OF MICHIGAN 7322-1-F I INTRODUCTION In preparing the final report, we have given some technical outline of those works which have not yet appeared in a technical report or been submitted for publication. Work performed on the project that has been published is indicated in the references. The latter includes a detailed study of the problem of a small antenna over a moving dielectric medium (Section 2), and a paper concerning the present views on the theory of electrodynamics of moving media (Section 3). In Section 4 are discussed some of the complexities introduced into the theory by extending it to conducting media. Several applications are presented in Section 5 which involve the Cerenkov radiation effect. In Section 6, earlier work on a drifting, magneto-ionic plasma is extended to finding the characteristic indices of refraction for some configurations of plasma parameters. The last section deals with the formulation of the time-dependent Green's function for moving, lossless media. ----------------------- ~ ~1 —----------

THE UNIVERSITY OF MICHIGAN 7322-1-F II RADIATION DUE TO AN OSCILLATING DIPOLE OVER A LOSSLESS SEMI-INFINITE MOVING DIELECTRIC MEDIUM A thesis bearing the above title was completed by Pyati (1966) during this period, and copies of this were distributed to NASA on March 25, 1966. An oral paper describing this work was presented at the Washington URSI meeting, April 19 - 21, 1966. It is expected that a written paper will soon be submitted for publication in a technical journal. The main contribution resulting from this study is the exact determination of the effects of the motion on the radiation pattern of a Hertzian antenna. 2

THE UNIVERSITY OF MICHIGAN 7322-1-F III DEFINITE AND INDEFINITE FORMS OF MAXWELL-MINKOWSKI EQUATIONS The basic equations which govern the field vectors in a moving medium are now well understood. These equations may be presented in various forms but they are all equivalent. All of them can be derived from the original theory of Minkowski. A paper concerning the present views on electrodynamics of moving media was presented at the International Convention of the Institute of Electrical and Electronics Engineers (Tai, 1966). Twenty copies of this paper were sent to the Langley Research Center. It is hoped that this work has clarified some of the recent controversies since the publication of the book by Fano, Chu and Adler (1960). L —----------------------- 3

THE UNIVERSITY OF MICHIGAN 7322-1-F IV ELECTRODYNAMICS OF MOVING, CONDUCTING MEDIA 4.1 Ohm's Law Upon the introduction of finite conductivity into the study of isotropic moving media, one is immediately beset by the existence of two apparently different forms of Ohm's law: 1) = 7yc' (E+V xB) (Weyl, 1922) and j(2) =. 7. (E + V x B) (Sommerfeld, 1952, and Cullwick, 1959) c 7 where JT is the conduction current density c o the rest-frame conductivity of the medium V = vz is the velocity of the medium (1 - 2)-1/2 0y 0 7= 0 7 0 O 0 1_ P=v/c (2) Schlomka (1950) claimed that J )is correct on the basis of an electronc theoretic model. His crucial step, however, involves attaching a physical meaning to the relative velocity of two bodies as seen by a third observer. This is dubious I —------------------— 4 —----------

THE UNIVERSITY OF MICHIGAN 7322-1-F and unsatisfying, and warranted further investigation. A study of the derivations showed that the decomposition of current density into conduction and convection terms is not unique. An expression was derived for the Joule heat loss per unit volume per unit time, ( [T(E + V xB]2) that is independent of the decomposition, using elementary relativistic thermo - dynamics. Since both field and energy terms are uniquely specified, either decomposition may be used, and the choice of Ohm's law is immaterial. 4. 2 Characterization of Charge Densities In extending the work of Tai (1965a) on point sources in isotropic, moving media to conducting media, the characterization of charge density requires careful interpretation. Denoting quantities measured in the rest system of the medium by primes, the total current density J is made up of convection and conduction terms in general: J =p i +a E s!! where p is the density of the charges introduced into the medium with velocityu The quantity p cannot be considered to be the total charge density, for this leads to inconsistencies; there must be an additional partial charge density caused by the presence of the source charge density p. Furthermore, the current density viewed in the stationary (unprimed) system has two convection currents, one due to the motion of the source charge (ps i), and one due to the motion of the medium -- Pr

THE UNIVERSITY OF MICHIGAN 7322-1-F where pr p -p, and p is the total charge density. With this interpretation, the introduction of source charges has no effect on Ohm's law, as would be required. The charge density 7 p is determined completely by p, and in the special case where p is a point charge s P = q 6(x) 6(y) 6(z) then 0, z < 0 p''r= l e v [ e r - {'y q 6(x) 6(y) exp - a z z > 0 The scalar potential 0 for the time-independent case (measured in the stationary system) must satisfy the following differential equation: a2+ 2+ i a i V X I 2,2 a a 2 - az ax ay az 1 " l' rO, z < 0 - + 6(x)6(y) (z)+2 6(x) 6(y) exp -,] *'2 a e v v 1, z>0 where a= [2(1 -n 2 )]2' l n = p / O ~ 0 o! i and C and ~ are the permittivity and permeability, measured in the rest-frame of the medium. 6

THE UNIVERSITY OF MICHIGAN 7322-1-F A partial solution has been found in closed form for the scalar potential, that is, a solution found by ignoring the second term on the right of the equation. By Fourier transform analysis, the partial scalar potential function 01 can be shown to be: 0, |al z < r -bz 2 r2 I/2] 01 =- -q e cosh b (z -7;[) / ] 2 —-----—? - -- I at z > r 22 r 1/2 27rEla| (z r)/ t t where b = 2 —a-, 2 al and it is assumed that n 1 > 1. The surface described by | al z = r is a cone, and 01 exhibits an exponential decay for large z. In the rest system of the medium, the radiation properties are determined by the vector potential A.This in turn can be shown to depend only on the partial solution 01, so that it is unnecessary to evaluate the total scalar potential 0. The results are identical with those of section 5. 3 ------------------- 7 m —-----------

THE UNIVERSITY OF MICHIGAN 7322-1-F V CERENKOV RADIATION TOPICS 5. 1 Unbounded, Lossless Media The well-known Cerenkov effect, arising from charged particles travelling with velocities exceeding the speed of light in the medium, has many interesting properties. The assumption of a point charge as a source results in infinite energy densities in the calculation of the fields, while the assumption of a dispersionless medium results in infinite energy. For a point charge, the shock wave has the form of a cone (see Fig. 1); fields are zero ahead of the cone, and behind the cone behave as 1 f_1'2 2 2 12 -Z /(z +v t) (n2 1) r i! in the rest system of the medium, or 1 2 2 ja z - r in the rest system of the charge. For a line charge moving perpendicular to its axis with velocity v, while the wedge angle ce is the same as before (see Fig. 1), there is no wake. That is, the fields are confined to a small region around the shock wave, and vanish in front and behind. This was worked out for the case of lossless, non-dispersive media. ------------------ 8

THE UNIVERSITY OF MICHIGAN 7322-1-F Shock wave travels with velocity c/n, where < v Charge travels with velocity v sin a = 1/ni3 ^^^ __ __Y \_r=0O In front: Zero Fields Behind: Wake for point charge y Zero fields for line charge FIG. 1: CERENKOV SHOCK WAVE GEOMETRY 9

THE UNIVERSITY OF MICHIGAN 7322-1-F 5.2 Bounded Media The introduction of perfectly conducting boundaries causes some interesting effects. An observer in the rest system of the medium finds the radiation spectrum discrete, instead of continuous. It is immaterial whether or not the boundary is moving or stationary if it is perfectly conducting. For the case of a point charge moving along the axis of a perfectly conducting cylinder of radius "b" having an appropriate medium (n B > 1), the solution has three regions, indicated in Fig. 2. In regions I and II, the solution is the sam as for the unbounded case; in region mI the fields have the behavior -(z +vt) (- j N (jr sinb221) l b m o m E= ib(n2/2 1)1/2 J1 (j2m)' /x'2'2' where r = x + y, and j is the m-th zero of J (x), the Bessel function. Work on the energy distribution was found to have been discussed in detail by Russian authors (Bolotovskii, 1961). For the corresponding two-dimensional case, that of a line charge moving between two parallel conducting plates, the fields are still restricted to the shock wave front which forms the zig-zag pattern shown in Fig. 3. The frequency spectrum seen by an observer in the rest-frame of the medium is again discrete, the frequencies excited depending on the eigen-values of the parallel-plate geometry 10

THE UNIVERSITY OF MICHIGAN 7322-1-F FIG. 2: SHOCK WAVE GEOMETRY, CYLINDRICAL BOUNDARY r=b c/n Zero fields between >X/ III Wwaveguide Mode Regiont v < a \__ r =O V/ - x==-b FIG. 2: SHOCK WAVE GEOMETRY, CYLINDRICAL BOUNDARY 11 A A Zero fields between X, X,wave fronts x= -b FIG. 3: SHOCK WAVE GEOMETRY, PARALLEL PLATE BOUNDARY 11

THE UNIVERSITY OF MICHIGAN 7322-1-F 5.3 Unbounded Conducting Media The case of a point charge moving in a conducting medium was considered in the program, tying in with the general study of conductive, moving media. As one would expect, the introduction of conductivity in the medium does not change the shape of the shock wave. The fields ahead of the shock wave are zero, and behind are attenuated in comparison with the lossless case. The behavior of the vector potential becomes: e-(z +vt) coshp/(z +vt) - (n2 3 1) r'2 (a vt 2 (n2 2_ 1)'2' (z+vt) -(n1 -1) r i t where p = a 2 /n2 32 -1 2 -1 Note that when the conductivity a vanishes, p vanishes, and the behavior becomes that of the lossless case above. The exponentially decaying factor is the type of behavior to be expected from the introduction of losses. 5.4 Dispersion When dispersion is considered, a convenient quantity to utilize is the amount I of radiated energy passing through a small cylinder of length dz. In symbols, we are interested in the quantity dW I = 2 2X r S dt dz r -wD 12

THE UNIVERSITY OF MICHIGAN 7322-1-F where S is the Poynting vector component in the radial direction, from the point of view of the medium rest system. The expression for the lossless case is, in M. K.S. units (cf. Tamm(1939)) dW q' "= 87r |2/, (1 2 dz >1 n 1 where the integration is over only those (positive) frequencies for which n > 1 For conducting media one would not expect this quantity to be independent of radius, because of losses. Indeed, for frequencies high enough that ~> > v.', 22 r2and ~ > >, n the integrand is modified only by the exponential factor t -o, v r exp n 2 The modifying factor is still tractable when the second condition on o is removed, but there has not yet been found a convenient factor good for all frequencies. 13

THE UNIVERSITY OF MICHIGAN 7322-1-F VI DRIFTING, MAGNETO-IONIC MEDIUM - FIRST ORDER THEORY Tai (1965b) applied Minkowski's theory of moving media to the case of a drifting, magneto-ionic plasma, where the drift velocity v is small compared to the velocity of light in vacuum c. This involves an eighth-order set of equations for the index of refraction n, given by det A = 0 where the components of A are the following: 2 A =(n -1)(1-nf -j Z)+(1-n3 )X xx z z A -j (n - 1) Y xy z A =(1-n13 -j Z) n -x - ni Y -j(1 - n3 ) Y xz z x y z z y A = j(n2 -1) Y yx z A =(n2-1)(1-n3 -j Z)+(l-n3) X yy z z A = (1 - n3Z -j Z) n/3 + j n3 Y + j(1 -n3 ) Y yz z z x z z x A =-j(n2 -1)Y + n3 X zx y x A =j(n2-1)Y + n3 X zy x y A =-(1-n1)(1-n1 -j Z) -jn3 Y +jn 3 Y + X zz z z x y y x 14

THE UNIVERSITY OF MICHIGAN 7322-1-F where 3=v/c V = drift velocity of the medium Nq2 2 e o tp = plasma frequency tc = collision frequency p |q I H ( Y= o el o m m L U e w0 = gyromagnetic frequency For the stationary magneto-ionic plasma, there are four possible values of the index of refraction: two "ordinary" waves and two "extraordinary" waves. When the medium moves as well, there are in general eight values, or modes, possible. Making use of the assumption that v < < c, the index of refraction was found for a variety of orientations of magnetic field, drift velocity, and direction of propagation. These are presented below for a wave of the form j w(t n z) c e 15

THE UNIVERSITY OF MICHIGAN 7322-1-F 1. Propagation, drift, magnetic field parallel: Y = Y, 3= z -+ /1 +1 X (Y-jZ) nl, 2- 1+Y-j Z 2 (+Y- ) 2 n + / - XX (Y+j Z) 3, 4 1 -Y-j z 2 (1 -Y-jZ)2 * 1~Y-j Z n5,6 /= n 7n7, 8= 1 2. Drift and magnetic field parallel, transverse to propagation: Y=Yx, =.3x + xx nl, 2=1- 1-jZ 1, 2 j z X (1 -j z - x)'3 + Y - (1j Z) (l -j Z X) (no change from the stationary medium case) 16.

THE UNIVERSITY OF MICHIGAN 7322-1-F 3. Propagation, drift, and magnetic field mutually orthogonal: Y=Y^, B ~^ + 1 x nl, 2 1-jZ _++ i(X (1- i Z +X) j XY (1-jZ) (1-jZ-X) n3 4~f 2 2 ~9r9 12 n34 -y2 (1- j Z)(1- j Z +X) -(1-j Z)(1- j Z-X)] n5 _ Y (1- j Z-X) 5 / 3 1-jZ' Y The effect of drift is a shift of the order of v / c in the ordinary and extraordinary wave indices, and the appearance of totally new modes. These new modes correspond to very slow waves having velocities on the order of the drift velocity. These are rapidly attenuated for most frequencies and plasma parameters encountered. The extent to which these modes can be excited would be the subject of a boundary-value problem. M. Epstein (1962) reported some initially pessimistic results on this matter for the all-parallel case. 17

THE UNIVERSITY OF MICHIGAN 7322-1-F vn THE TIME-DEPENDENT DYADIC GREEN'S FUNCTION FOR A MOVING ISOTROPIC MEDIUM The dyadic Green's function for a harmonically oscillating field in a moving isotropic medium was previously derived by means of an operational method (Tai, 1965a). There it was shown that the associated scalar Green's function is, for a particular frequency w: Case 1: n < 1 1/2 -J c Ra -I' a e gw~(0 (-l~ o 47rR a where R =(p2+ a 2)1/2 a a 2 1 n2 2 c 1 2 p=/c, n2= ( ) + -y2 0 0 v is the velocity of the medium 1 —— n1(1-n —---- 13 18

THE UNIVERSITY OF MICHIGAN 7322-1-F Case 2: n 1 > 1 (Cerenkov condition) 0o, 1aI1/2 < (RJRI a 1/2 ejt o Cos w n ola 1/2 R 47r R a where n 2 - 1 / t o 4T p2 Rb. - b 12132 1 b where R=2~ 2 2 >21/2 b n2 V 2 2 b \n-1J and T = t - t 0

THE UNIVERSITY OF MICHIGAN 7322-1-F It can be shown that the complicated analysis involved in Compton's work can be avoided by taking the Fourier integral of the harmonic solution (g (R IR)) with respect to the angular frequency w. A note comparing these two derivations will be submitted for publication in the near future. 20

THE UNIVERSITY OF MICHIGAN 7322-1-F vmI REFERENCES Bolotovskii, B. M., (March-April 1962), "Theory of Cerenkov Radiation (Im)," Sov. Phys. Uspek., 4, No. 5, pp. 781-811. Compton, R.T., Jr., (1966), "The Time-dependent Dyadic Green's Function for a Moving Medium," presented at URSI meeting, Washington, D. C. April 19-21, 1966. Cullwick, E.G. (1959), Electromagnetism and Relativity, (Longmans) p. 92. Epstein, M., (Sept. 1962), "Effect of Fluid Motion on Electromagnetic Wave Propagation", IRE Trans., AP-10, No. 5, pp. 645-646. Fano, R.M., Chu, L. J., and Adler, R.B. (1960), Electromagnetic Fields, Energy, and Forces, (Wiley, N.Y.), pp. 376-503. Pyati, V. (Feb. 1966) "Radiation Due to an Oscillating Dipole Over a Lossless Semi-Infinite Moving Dielectric Medium," University of Michigan Radiation Laboratory Report No. 7322-2-T. Schlomka, T. (1950), "Das Ohmsche Gesetz bei Bewegten Koerpern," Annalen der Physik, 443-444, No. 8-9, pp. 246-252. Sommerfeld, A. (1952), Electrodynamics (Academic Press Inc., New York, N. Y.) p. 282. Tai, C.T. (March 1965a), "The Dyadic Green's Function for a Moving Isotropic Medium," IEEE Trans. AP-13, No. 2, pp. 322-323. Tai, C.T. (March 1965b), "Electrodynamics of Moving Anisotropic Media: The First-Order Theory," Rad. Sci. J. of Res. NBS, 69D, No. 3. Tai, C. T. (1966), "Present Views on Electrodynamics of Moving Media", presented at the 1966 International Convention of IEEE, to be published in the convention records. ------ 21

THE UNIVERSITY OF MICHIGAN 7322-1-F Tamm, I., (1939), "Radiation Emitted by Uniformly Moving Electrons," J. of Phys. 1, No. 5-6, pp. 439-454. Weyl, H. (1922), Space-Time-Matter, (Dover Press), p. 195. 22