640 ELECTRICAL METHODS [Chap. 10 



shift in a circuit consisting of an electrolyte with the polarization P is 



ip = -tan~'^. (10-9c) 



co/t 



^c 



C. Dielectric Current Conduction 



In most nonmetallic, solid, and isotropic media the number of free 

 electrons is too small to permit free passage of direct current. Alternating 

 current, however, will be transmitted, since the electrical field produces a 

 displacement of the nuclear and electron patterns {dielectric 'polarization) 



which is propagated when the field changes 

 with time. The polarization P (electric 

 moment of the unit of volume) is propor- 

 tional to the electrical field. P = tE, 

 where e is the electric susceptibility. Treat- 

 ing the electrical flux in the same manner 

 that the magnetic flux was treated in 

 Chapter 8, the flux per unit area or the 

 electric displacement D = E + 47rP. Sub- 

 stituting for P: tE, the displacement be- 

 comes D = (1 + 47r£)E, where 1 + 47rt = 



Hence the dis- 



"-o dt 



4-n<r£ 



Fig. 10-5a. Relation between 

 conduction current and displace- 

 ment current. 



K is the dielectric constant 

 placement is D = kE. 



An alternating electrical field produces 

 a current in a dielectric equal to dD/ dt 

 or = K • dE/ dt, which is known as the dis- 

 placement current. Considering the conduc- 

 tion current and recalling from eq. (10-5) 

 that the current density i = 47ro-E, we get, 

 for the total current (see Fig. 10 -5a), 



4n<r 



Fig. 10-56. Relation between 

 conductivity and true and appar- 

 ent dielectric constant. 



Ir = K— -f 47r(rE, 

 dt 



(10-lOa) 



For a sinusoidal e.m.f. of the angular frequency co, the differentiation 

 gives an e.m.f. 90° out of phase with the last term of the equation, so that 

 the peak value of the displacement current becomes kcoE. Therefore 



/, = 47r<rE + j/ccuE, (10-106) 



so that the phase shift between conduction component and resultant 

 current 



<p = tan 



-1 KO) 



4iro- 



(10-lOc) 



