Time-Varying Electromagnetic Fields and Maxwell’s Equations

Abstract

In this region of frequencies, the applied field varies sufficiently slowly with time so that it is possible to state that at a given instant the current intensity is the same throughout all parts of a closed circuit. Given then that for any value of S, $I = \iint\limits_S {\overrightarrow j .\overrightarrow {dS} }$ = constant implies that the current density flux ($\overrightarrow j$ ) across a current “tube” (which delimits a closed surface) is zero, as in $\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {\overrightarrow j .\overrightarrow {dS} = 0}$ (see also Chapter 1). Following on from Ostrogradsky’s theory, it would indicate that div $\overrightarrow j$ = 0, and the conservation of charge therefore would give $\frac{{\partial \rho }} {{\partial t}} = 0$ . This hypothesis indeed could be used as a starting point in defining quasistatic states.