METALLIC DELA V LENSES 70 



Fig. 5. Now a conducting sphere in an alternating magnetic field is equi- 

 valent to an oscillating magnetic dipole,^ and we should observe an effective 

 permeabihty for a sphere array. ^ The magnetic dipole field is, however, 

 opposed to the inducing field, in other words, the dipole moment is negative. 

 Smythe^ shows that the magnetic polarizability of a conducting sphere of 

 radius a in a high frequency field is 



a,n = — 27r/xoa^ (16) 



The effective relative permeability of an array of TV spheres per unit 

 volume is therefore 



M, = 1 - iTcam. (17) 



The index of refraction was given in equation (4) as the square root of the 

 dielectric constant. This is strictly true only when the permeability of the 

 die'ectric is unity. We have seen above, however, that the sphere array at 

 microwaves possesses an effective permeabihty given by (17), and there- 

 fore (4) is not valid. The correct expression for n is 



n = VTiiTr, (18) 



and the effective refractive index of a sphere array is accordingly 



n = \/(l - 2irNa^){l -^ ^irNa^), (19) 



which is smaller than that given by (3) and (4). The disk and strip arrays, 

 besides being lighter, avoid the diminishing effect on the refractive index 

 caused by the perturbing of the magnetic Unes. 

 (3) Conducting Circular disk 



The determination of the dipole moment of a disk involves the use of 

 elHpsoidal coordinates and will not be carried through here. Smythe^ gives 

 an expression for the torque on a flat disk of radius a in a uniform field in 

 Gaussian units. In M.K.S. units, his formula becomes 



, 2a E" sin 26 , 2a^E'2 sin cos 6 



T = 47r€o ;; = 4x60 ^ 



St 3t 



= p_^° (E sin d)]\E cos e]. (20) 



The first bracket represents the dipole moment, the second the field, and 

 the product gives the torque. When the plane of the disks is parallel to E 

 sin d is one, and from m = oE, we have 



16€oa^ 



a = 



(21) 



8 T. S. E. Thomas, Wireless World, Dec. 1946, p. 322. 



9 L. Lewin, Jour. I. E. E., Part III, Jan. 1947, p. 65. 

 7 Loc. Cit. Eq. 14, p. 397. 



Ubid, eq. 7, p. 163. 



