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Plates occupy an intermediate position between objects extended three dimensionally and objects in three dimensions that can be confined to a plane. The loss of the extension of the body in the dimension normal to the object plane has the effect of decreasing the number of non-zero tensor components for αe and α m. We noticed in Sec. 2 that the tensor components tend to exhibit rapid variation when the thickness is varied. This makes such object shapes useful for effectively modifying the properties of a medium.
The polarization tensors of plates are crucial in the description of long wavelength scattering of electromagnetic and pressure waves through apertures [88,89,90,91]. This connection was apparently first noticed by Rayleigh [81], but the practical significance of this connection was appreciated more recently because of the difficulty of calculations of the polarization tensor components for plates having general shapes. The magnetic polarizability αm ( or the mathematical equivalent M ) is also important in the description of the flow of viscous fluid through screens [131]. The technological literature is a good source of results regarding the polarizability of plate-like regions.
The theoretical impetus for calculating the polarizability tensors of plates came from the needs of a developing microwave technology [47,48,49,50]. Bethe [88] calculated the magnetic and electric polarizability of circular plates in his classical theory of the diffraction of electromagnetic radiation by a hole small compared with the incident wavelength and later [89] he gave results for elliptic plates. Cohn [49,132] made electrolytic-tank measurements of the polarizability components of plates of numerous shapes--rectangular slots, rounded slots, rosettes, dumbbells, crossed slots, etc. to provide this important technological information. The electric polarizability of an aperture of general shape was measured by simply cutting out a non-conducting material of the given shape and suspending the object with thin non-conducting wires in an electrolytic solution between two electrodes coplanar with the inclusion. Similarly, the magnetic polarization was measured by suspending a metallic inclusion normal to the electrode surfaces [132]. These experimental measurements of the electric and magnetic polarizability of plates have had many important applications.
Recently, numerical solutions of integral equations defining the electric and magnetic polarizability tensors have been obtained for a wide variety of shapes [72,73]. These calculations have confirmed the accuracy of Cohn's measurements and a general correlation of the magnetic and electrical aperture polarizations have been obtained for certain families of objects [72,73]. Significant progress has recently been made by Fabrikant [75] who developed an analytical technique for calculating the magnetic and electric polarizability tensors of plates that compares very well with previous numerical and experimental results. Fabrikant treats polygons, rectangles, the rhombus, a circular sector, and other shapes and the method can apparently be applied to regions having very general shapes.
The utilization of these important theoretical developments requires the recognition of the relation between 'aperture polarizabilities' αe(apert.) and αm(apert.), used in the technical literature, and the ordinary polarizabilities, αe and αm, discussed above. Babinet's principle [73,89] implies that these quantities are related by the definitions,
Simple physical considerations show that αm for a flat plate is effectively a scalar, having only one non-zero component. Correspondingly, the component of αe for a metal plate normal to an applied field vanishes, since there is no way to separate charges in this case. However, there are non-vanishing components to αe when the plate is aligned along the field direction. From these observations we immediately obtain numerous results for αe and αm from the literature of electromagnetic and sound scattering through apertures.
For example, we can obtain the magnetic polarizability of an elliptic plate from Bethe's formula for the electric aperture polarizability of an ellipse (plate is normal to field direction),
where a and b are semi-major and semi-minor axes, e is the eccentricity, e = [1 − (b/a)2]1/2, and E is a complete elliptic function of the second kind. The average < αm > of a plate generally equals,
since there is only one non-zero component, as mentioned above. Note that eqns.(57) and (58) reduce to (30) for the circular disc. Numerous other examples follow along these lines from recent numerical calculations [72,73] and analytical calculations by Fabrikant [75]. Results for the electric polarizability can be similarly obtained, although these results tend to have a more complicated mathematical description. We mention the important experimental estimate [49] of αe(L) for a square plate having a side length a,
placed parallel to the applied field. There are two components of the electric polarizability for an asymmetric rectangular plate, of course. The estimate (59) was obtained from Cohn's electrolytic tank measurements and this value has been confirmed by more recent numerical [72,73] and analytical studies [75] to an accuracy on the order of 1%. Our finite element technique gives αe(L) = 1.09 a3 for the square plate, which is apparently too high by the usual 5-6% in d=3. This result is listed in Table 10.
These quantitative estimates of αe and αm for plates from this variety of sources is very useful in combination with the approximate invariant relation (29). We mentioned in Sec. 3 the exact result,
for circular plates. Combination of the extensive plate estimates with Eqn. (60) then yields predictions for [η] which can be checked against experiment. Further efforts are needed on the difficult problem of analytically calculating [η] for arbitrarily shaped plate-like regions, which certainly is not going to be any more tractable than the electrostatic and magnetostatic analogs. We note that our finite element technique could be used to estimate the polarizabilities, intrinsic conductivity, and intrinsic viscosity of arbitrarily-shaped objects, and that the method is not limited to large values of the relative conductivity Δσ. The numerical calculations are actually faster and more accurate when Δσ is not large.