Maxwell's Equations — Question 1

Question 1

Flux and circulation measurements determine different source quantities

For a fixed closed surface the electric flux is F and magnetic flux is zero. Around a separate fixed contour, electric circulation is G and magnetic circulation is K. Its spanning surface has electric-flux rate D. Use vacuum Maxwell equations with consistent orientations.

1. Find enclosed electric charge and the magnetic-flux rate through the contour.

2. Find conduction current through that contour surface and identify why K alone is insufficient.

Question 1 - Solution

Flux and circulation measurements determine different source quantities

(a) Method and calculation

$$\boxed{Q_{\mathrm{enc}}=\epsilon_0F}.$$$$G=-d\Phi_B/dt\Rightarrow\boxed{\dot\Phi_B=-G}.$$

(b) Interpretation and further calculation

$$K=\mu_0I_c+\mu_0\epsilon_0D.$$$$\boxed{I_c=K/\mu_0-\epsilon_0D}.$$

Physical check

Closed-surface flux and loop circulation are different measurements; their orientations and sources cannot be interchanged.