Displacement Current — Question 3

Question 3

Displacement current restores agreement between two spanning surfaces

A loop surrounds a capacitor charging wire. Surface S1 cuts the wire before the plate; a deformed surface S2 passes through the vacuum gap and cuts no wire. The wire current is I and the capacitor stores charge Q.

1. Compute the generalized current on each surface, neglecting electric-flux change on S1.

2. Explain algebraically the contradiction produced by omitting the displacement term.

Question 3 - Solution

Displacement current restores agreement between two spanning surfaces

(a) Method and calculation

$$S_1:\quad I_{\mathrm{cond}}+I_d=I+0=I.$$$$S_2:\quad I_{\mathrm{cond}}+I_d=0+\epsilon_0\frac d{dt}(Q/\epsilon_0)=\dot Q=I.$$

(b) Interpretation and further calculation

$$\boxed{\oint\mathbf B\cdot d\mathbf l=\mu_0I\quad\text{on either surface}}.$$$$\boxed{\text{Without }I_d:\quad\mu_0I\ne0\quad(I\ne0)}.$$

Physical check

The same boundary loop cannot have two different circulations; the extra term removes surface dependence.