Dielectrics: An Atomic View — Question 9

Question 9

Delayed molecular alignment gives a gradual extra electrode charge

A dielectric obeys the supplied relaxation law $\tau\,dP/dt+P=\epsilon_0\chi E$. At t=0 a constant field E0 is imposed across an ideal plate gap, with P(0)=0. Ignore propagation delay and leakage.

1. Solve for P(t).

2. Using electrode charge $Q(t)=A[\epsilon_0E_0+P(t)]$, find the initial and final charge and the time to complete 90 percent of the polarization change.

Question 9 - Solution

Delayed molecular alignment gives a gradual extra electrode charge

(a) Method and calculation

$$\frac{dP}{dt}+P/\tau=\epsilon_0\chi E_0/\tau.$$$$P(t)=\epsilon_0\chi E_0(1-e^{-t/\tau}).$$

(b) Interpretation and further calculation

$$Q(0^+)=\epsilon_0AE_0,\quad Q(\infty)=\epsilon_0A(1+\chi)E_0.$$$$1-e^{-t_{90}/\tau}=.900\Rightarrow t_{90}=\tau\ln10.$$

Physical check

The extra charge tracks molecular relaxation; this prescribed material law is distinct from an external resistor-capacitor charging model.