Displacement Current — Question 10

Question 10

Changing capacitor spacing adds a geometric term to displacement current

A vacuum parallel-plate capacitor has fixed area A but slowly varying gap d(t) and voltage V(t). Neglect fringe fields and use instantaneous uniform $E=V/d$.

1. Find plate charging current and gap displacement current.

2. For constant plate charge Q, find the required voltage law and the displacement current despite moving plates.

Question 10 - Solution

Changing capacitor spacing adds a geometric term to displacement current

(a) Method and calculation

$$Q=\epsilon_0AV/d\Rightarrow I=\dot Q=\epsilon_0A\left(\frac{\dot V}{d}-\frac{V\dot d}{d^2}\right).$$$$\boxed{I_d=\epsilon_0A\dot E=I}.$$

(b) Interpretation and further calculation

$$Q\ \text{fixed}:\quad V=Qd/(\epsilon_0A),\quad E=Q/(\epsilon_0A).$$$$\boxed{\dot V/V=\dot d/d,\quad I_d=0}.$$

Physical check

A changing voltage alone does not guarantee changing electric field when the geometry also changes.