DIELECTRIC PROPERTIES OF INSULATING MATERIALS 535 



the initial conductivity 70 as determined hy d-c measurements equals the 

 infinite- frequency conductivity y^ as determined by a-c measurements. 



If td represents the time measured from the instant that the voltage 

 is abruptly reduced to zero, and qd, Dd, Qd represent respectively the 

 charge, displacement, and discharge current at time td, we have for 

 discharging after complete charging 



qdtd) =^' = ^ ,-^./e»Coie + (j^^jA E.e-^'iir (17) 



or 



e{td) = e^e-'<"'->('^^^ + (€0 - e^)e-'<"\ (17a) 



At the end of the charging process (cf. (13)) or the beginning of the 

 discharge process (cf. (17)) the charge per unit area per unit applied 

 field strength is €o/47r. 



The discharge current for complete charge is obtained by differen- 

 tiating (17) with respect to the time: 



^^ = '^= - T^h '-'"'-'"' - ^'" ^ '^^^^ e-^<"r (18) 



47r 4:irCoR 4x7 ^ ^ 



or 



yaOd) - -T-^r = jue-''"'^'^'^ + 7^e-'-"\ (18a) 



Comparison of (18) and (15), or (18a) and (16), shows that for com- 

 plete charging (that is, tc effectively infinite or 4 » t» eoCoR) the 

 charging current vs. time curve is identical, except for direction, with 

 the discharge current vs. time curve. This is true only of the curves for 

 complete charging and is not true if the polarized condition in the dielectric 

 is not fully formed and the polarization currefits are not zero. 



The discharge-current time curve for incomplete polarization of the 

 dielectric is not as simple as for complete polarization. When the 

 charging process is broken off before completion, the initial conditions 

 for the discharge are not the same as when charging is complete. The 

 charge at time td during a discharge following a charging process 

 which is broken off at tc is 



47r 47r 47r 



+ IM-^:^ E,e-^^i^ - ^^""^^-^ E,e-^^c+uyr\ , (19) 



This is an example of the superposition principle for residual charges. 



