Displacement Current and Maxwell’s Equations

Abstract

In a steady state, Ampere’s law, Eq. (9.14), holds for a closed line, C, with different surfaces on it, S₁ and S₂, as shown in Fig. 11.1a, b. If the magnetic field $\boldsymbol{H}$ is integrated on C in opposite directions as drawn in Fig. 11.1a, b, the sum of the two integrations is naturally zero. At the same time, the sum of the surface integrals of the current density $\boldsymbol{i}$ on S₁ and S₂ is also zero. This sum becomes the surface integral on closed surface S₁₂ composed of S₁ and S₂ (see Fig. 11.1c). Thus, we have