Displacement Current — Question 1

Question 1

A capacitor gap carries the same generalized current as its charging wire

A parallel-plate vacuum capacitor has fixed area A and gap d, with negligible fringing. Its plate charge is Q(t). Define $I_d=\epsilon_0\,d\Phi_E/dt$.

1. Prove that gap displacement current equals the charging-wire current.

2. For C=30.0 pF and voltage rising at $2.00\times10^8$ V/s, find both currents.

Question 1 - Solution

A capacitor gap carries the same generalized current as its charging wire

(a) Method and calculation

$$E=Q/(\epsilon_0A),\quad\Phi_E=EA=Q/\epsilon_0.$$$$\boxed{I_d=\epsilon_0\dot\Phi_E=\dot Q=I_{\mathrm{wire}}}.$$

(b) Interpretation and further calculation

$$Q=CV\Rightarrow I=C\dot V=(30.0\times10^{-12})(2.00\times10^8).$$$$\boxed{I_d=I_{\mathrm{wire}}=6.00\,mA}.$$

Physical check

The equality concerns magnetic circulation; it does not mean free charges cross the vacuum gap.