Insulators — Question 10

Question 10

A displacement pulse distinguishes polarization charging from steady leakage

A dielectric of area A and thickness d is driven by a linear voltage ramp $V(t)=V_0t/t_r$ for $0<t<t_r$, then held at $V_0$. Use $I=A[\sigma E+\varepsilon\dot E]$ and $E=V/d$.

1. Derive current during the ramp and immediately after it.

2. Find the integrated displacement charge and compare it with the final capacitor charge.

Question 10 - Solution

A displacement pulse distinguishes polarization charging from steady leakage

(a) Method and calculation

$$0<t<t_r:\quad\boxed{I(t)=\frac{A V_0}{d}\left(\sigma t/t_r+\varepsilon/t_r\right)}.$$$$t>t_r:\quad\boxed{I=\sigma A V_0/d}.$$

(b) Interpretation and further calculation

$$Q_d=\int_0^{t_r}\frac{\varepsilon A V_0}{dt_r}\,dt=\varepsilon A V_0/d.$$$$C=\varepsilon A/d\Rightarrow\boxed{Q_d=CV_0}.$$

Physical check

The displacement charge is independent of ramp duration. The additional conduction charge grows with time and is not stored as ideal capacitor charge.