78 BELL SYSTEM TECHNICAL JOURNAL 



But P is related to the displacement vector D and the dielectric constant as 

 follows: 



Z) = €£ = eo£ + P, (10) 



so that € = eo + Na which is equation (1). A knowledge of a thus permits 

 a determination of €. We obtain a from (8) by first finding the dipole 

 moment m of the particular shape of element when immersed in a uniform 

 field E. 



Calculation of Dielectric Constants or Artificial Dielectrics^ 



(1) Conducting Sphere 



Consider a perfectly conducting sphere immersed in an originally uniform 

 field of potent "al 



V = -Ey = -Er cose. (11) 



The free charges on the sphere are displaced by the applied field and it 

 thereby becomes a dipole whose moment m we wish to determine. The 

 external potential field is the sum of the applied potential and the dipole 

 potential, and from (7) and (11) we have 



Vout = -Er cose + -. (12) 



47reo r^ 



The internal field is zero because the sphere is conducting. At a boundary 

 between two dielectrics, there is the requirement^ 



'outside ^^ 1^ inside* vl'^y 



Equation (13) gives, at r = a (the radius of the sphere), 



„ ^ , mcose ^ ,. .V 



-Ea cos e + i-r-T = 0, (14) 



47r0o a 



or 



m = 4T€oEa\ (15) 



the dipole moment of the sphere. From (8) we see that the polarizability of 

 the conducting sphere is accordingly 4^eoa^, from which equation (3) follows. 



(2) Magnetic Effects of a conducting sphere array 



The above calculations on a conducting sphere assume an electrostatic 

 field. At microwaves, the rapidly varying fields induce eddy currents on 

 the surface of the sphere which prevent the magnetic lines of force from 

 penetrating the sphere. The magnetic lines are perturbed as shown in 



• The author is indebted to Dr. S. A. Schelkunoff for the polarizability formulas given 

 in this memorandum. 



» Smythe, "Static and Dynamic EUctricity\ McGraw-Hill, 1939, p. 19. 



