Displacement Current — Question 4

Question 4

An RC charging current and gap displacement current decay together

A vacuum capacitor C is charged from an ideal voltage V0 through resistance R, starting uncharged. Its voltage is $V_C=V_0(1-e^{-t/(RC)})$.

1. Find the displacement current and total displacement charge integral.

2. Find the fraction of final capacitor charge accumulated by the time current has fallen to one-quarter its initial value.

Question 4 - Solution

An RC charging current and gap displacement current decay together

(a) Method and calculation

$$I_d=C\dot V_C=(V_0/R)e^{-t/(RC)}.$$$$\boxed{\int_0^\infty I_d\,dt=CV_0}.$$

(b) Interpretation and further calculation

$$I_d/I_d(0)=1/4\Rightarrow t=RC\ln4.$$$$\boxed{Q(t)/(CV_0)=1-e^{-\ln4}=3/4}.$$

Physical check

The time integral of displacement current equals transferred plate charge, while the microscopic transport occurs in the leads.