P-GAP R L-624 THE DIPOLE MOMENTS OF A DIELECTRIC CUBE+ by D. F. Herrick and T. B. A. Senior Radiation Laboratory, The University of Michigan Ann Arbor, Michigan 48109 Summary As a part of a general study of the low frequency scattering properties of atmospheric particles of various shapes, the dipole moments of bomogeneous dielectric rectangular parallelepipeds have been examined and data obtained for the special case of a cube. The results are presented and differ substantially from those previously reported in the literature. This work was supported by the National Science Foundation under Grant ENG 75-03880. RL-624 = RL-624

6 24 A knowledge of the scattering of visible and infrared radiation by high altitude clouds and other aerosol-laden atmospheres is vital in considering the radiation balance of the earth and is also important at microwave frequencies in connection with radar meteorology from space. This has led to a study aimed at developing techniques for computing the scattering from small dielectric particles of shapes known to occur in practice, and in the low frequency or Rayleigh region where the wavelength is much greater than the maximum dimensions of the particle the computations are relatively easy to perform. Among the shapes which have been examined is the rectangular parallelepiped, and we consider here the scattering of a low frequency electromagnetic wave by a single dielectric particle of this configuration, with particular emphasis on the special case of a cube. A homogeneous isotropic dielectric body of finite extent is immersed in free space and illuminated by a linearly polarized plane electromagnetic wave whose electric and magnetic field vectors are taken as A A ik k r ik k r (1) e, H- Y e 0 k 0 A A A where k, a and b are mutually perpendicular unit vectors in the directions of incidence, the electric field and the magnetic field respectively. The permittivity and permeability of free space are o and Mo respectively, k is 0 00 the free space propagation constant and Y is the intrinsic admittance. -iwt Mks units are employed and a time factor e suppressed. For sufficiently small k all of the field quantities can be expanded as power series in k, 0 O and as regards the scattered field it is well known [1] that in the far zone the leading term can be attributed to induced electric and magnetic dipoles, of moments ~ and m respectively, located at the origin of coordinates in the vicinity of the body. 1

We can also write A -A p = e m=-Y M b (2) O -- O where M and M are the electric and magnetic polarizability tensors [2, 3]. These are independent of the incident field, but functions respectively of the relative permittivity and permeability r and Mr of the dielectric, and of the geometry of the body. For any given incident plane wave, P and M specify the dipole moments and, hence, the far zone scattered field in the Rayleigh region. The tensor elements Pij and MU are expressible as weighted surface integrals of electrostatic and magnetostatic potentials, and from the boundary conditions on the potentials at the surface of the body, it is found [3] that t and f are merely special cases of a general polarizability tensor!(7) in terms of which P (e ), M=-X( ) (3) We remark that for a perfectly conducting body Er =I a and Aur =, and the P and M computed by Kleinman and Senior [4, 5] for a nu mber of metallic shapes therefore yield X (7) for these two extreme values of T The general polarizability tensor is a function only of the geometry and material parameter T of the body, and for real r the tensor is real and symmetric, having at most six independent elements. If xi, i = 1,2, 3 are rectangular Cartesian coordinates, the tensor elements are Xi =(1 - ) n xi d S (4) B 2

t 24 A where n is the outward unit vector normal to the (closed) surface B of the body and is a total exterior potential satisfying the integral equation [3, 6] J(= 1+ Xj + 2 (r') dS' (5) where r is a position vector terminating on B and the integration is with respect to th primed variables. The tensor elements Xij can also be expressed in terms of the normal derivative of the potential and an integral equation derived for the normal derivatives, but this formulation proves less desirable for numerical purposes. Whereas (5) is in general a weakly singular integral equation, in the special case of a rectangular parallelepiped occupying (say) the region x 11 <a, I x21 < b, | x3 c, the kernel in (5) is identically zero when the points r and r' lie on the same face of the body, therefore removing the singularity at r = r'. In addition, the symmetry about the three perpendicular planes xi 0 produces corresponding symmetries in the potentials which result in diagonalization of the tensor and also enable us to limit the integration in (5) to a portion of the total surface B. If S, m = 1,2,3 arethesurm faces S: x = a, 0<x b, 0<x <c; S: 0 <xla, x2 =b, 0< c; S3: 0< x a, 0 < b, x3 = c, and r is the position vector of a point on S, the integral equation (5) can be written as 3 1-T fi (r)=-2xI 2 < K ( ) dSr (6) m (rm)Kim r)dS m 3

(124 where =(1-T) I.(7) The tensor elements are then fX =8 ( (8) i with X = i j. Because the kernels K are non-singular it follows that the numerical solution of (6) by the moment method requires no special treatment of the self cells, and (6) is easily converted to a simultaneous system of linear algebraic equations in the sampled values of t. For a parallelepiped having a square cross section (a=b), X22=X11 andforacube ( a = b = c ), X33 =X22 =X, so that in the latter case the tensor is now specified by a single diagonal element. A computer program has been written to solve (6) numerically and compute the tensor elements for a parallelepiped of square cross section, and we present here the results obtained in the particular case of a cube. Most data were computed using 25 or 36 square cells on each of the three quarter-faces S of the m cube and the results are believed accurate to better than one percent. The normalized tensor element X11 (7)/V, where V is the volume of the cube, is -2 3 shown in Figure 1 as a function of r for 10 < < 10. It is seen that X (T)/V is a slowly increasing function of T, ranging from -1.654 at T = 0 to 3. 542 at T = oo and equal to zero at T = 1. We remark that X l/V is proportional to the dipole moment per unit volume, and at the extreme values T = co and T = 0 yields respectively the electric and magnetic dipole moments per unit volume for a perfectly conducting cube. 4

624 - Figure 1 also shows the dipole moments computed by Edwards and Van Bladel [1, 6] for this same geometry, and a substantial difference is seen between their data and ours. All tests of our formulation and programming support the validity of the present data. Although our formulation is almost identical to that used by Edwards and Van Bladel [7], we have noted that their results show a non-physical distribution of the surface potentials. From the comparisons which we have been able to carry out, there appears to be an error in their solution of the system of linear equations in i by the matrix inversion method. We also remark that for a perfectly conducting body of arbitrary convex shape, it has been proved by Schiffer [8] that 3 X (oo) ^3 (9) i1= with the equality obtained in case of a sphere. For a dielectric cube this inequality becomes Xii(o)/V > 3 and is evidently violated by the data of Edwards and Van Bladel. The vallss of X11/V for a dielectric sphere have been computed using (9) and are included in Figure 1. It is evident tt t the new values of the dipole moments per unit velume for a cube ar quits cIese to those for a sphere having the same material parameter, and igure 2 we have plotted the cube data normalized to those for the correspodg sphere. The maimum difference is no more than 18 percent, and for many purposes it may be fficient to use a sphere to approximate the dipole moments Or a dielectric cube. This approimation is commonly made in studying the scattering of atmospheric particles, but we caution tat for geometries other than the cube the dipole moments can differ -taatl ally from the values for a sphere. 5

624 References [1] J. Van Bladel, "Electromagnetic Fields. " New York, McGraw-Hill 1964. [2] J.B.Keller, R.E. Kleinman, and T. B. A. Senior, "Dipole moments in Rayleigh scattering," J. Inst. Maths Applies, Vol. 9, pp. 14-22, 1972. [3] T. B. A. Senior, "Low frequency scattering by a dielectric body," Radio Sci. Vol. 11, pp. 477-482 June 1976. [4] R. E. Kleinman and T. B. A. Senior, "Rayleigh scattering cross sections," Radio Sci. Vol. 7, pp. 937-942, Oct. 1972. [5] R. E. Kleinman and T. B. A. Senior, "Low frequency scattering by space objects," IEEE Trans. Aerospace and Elect. Syst. Vol. 11, pp. 672 -675, July 1975. [6] T. W. Edwards and J. Van Bladel, "Electrostatic dipole moment of a dielectric cube, " Appl. Sci. Res. Vol. 9(B), pp. 151-155, 1961. [7] T. W. Edwards, "Perturbation of a uniform electrostatic field by a dielectric cube," M. S. Thesis, Dep. Elec. Eng., Univ. of Wisconsin, Madison, 1961. [81 M. Schiffer, "Theorie du potentiel - sur la polarisation et la masse virtuelle," C.R. Acad. Sci. Paris, Vol. 244, pp. 3118-3121, 1957.

Legends for figures Fig. 1. The normalized tensor element Xll (T)/V for a sphere ( ) and a cube: present data ( x x x), Edwards and Van Bladel [6] (- — ). Fig. 2. The tensor element Xll (7) normalized to that for a sphere of the same volume and material parameter T.

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