80 BELL SYSTEM TECHNICAL JOURNAL 



SO that, from (2), 



6. = 1 +-V'-iV^' (22) 



(4) Strips 



The calculation of the dipole moment of a thin conducting strip as used 

 in the strip lens of Fig. 15 involves two dimensional elliptic coordinates and 

 will also be omitted here. Again, the torque is given in Smythe^ for an 

 elliptic dielectric cyhnder, from which we obtain, 



a = ^, (23) 



where 5 is the strip width, so that 



TT 2 



., = l+|/«, (24) 



where n is the number of strips per sq. unit area looking erd on at the strips. 

 (5) Validity of the polar izability equations 



Equation (2), which expresses the dielectric constant to be expected from 

 an array of N elements each having a polarizability of a, was derived (by 

 (8), (9) and (10)) by assuming that the fie'd acting on an element, and 

 tending to polarize it, was the impressed field E alone. This is a satisfactory 

 assumption when the separation between the objects is so large that the 

 elements themselves do not distort the field acting on the neighboring 

 elements. Such is not the case when the value of e^ exceeds 1.5 or there- 

 abouts. For the usually desired values of €;. of 2 or 3 it is thus seen that the 

 above formulas such as equations (22) and (24) will yield only qualitative 

 results and that the exact spacings of the elements to produce a desired 

 refractive index will have to be determined by experimental methods. 



For lattices having 3-dimensional symmetry, an improvement over 

 equation (2), which takes into account not only the impressed field E but 

 also the field due to the surrounding elements, is the so-called Clausius- 

 Mosotti equation: 



i^t^ = ^ . (25) 



€ -f 26,) 3€o 



This, along with a similar expression for the permeability [replacing (17)], 

 would permit a fairly accurate determination from (18) of w for the conduct- 

 ing sphere array. 



' Ibid, cq. 6, p. 97. 



