Maxwell equations and local conservation

Taking the divergence of the static Ampère law gives $\nabla\cdot\mathbf J=0$, since a curl has zero divergence. That condition excludes charge accumulation. A time-dependent theory must instead allow current flowing into a region to increase its charge. Adding the time derivative of electric field repairs the inconsistency because Gauss's law relates its divergence to the changing charge density.

The resulting continuity equation is local: a small region loses charge only through current crossing its boundary. It is stronger than saying total charge is constant after integrating over all space. A model that creates charge at one point and removes it elsewhere instantaneously could conserve the global total while violating this local equation. Electromagnetism forbids that shortcut unless an intervening current is included.

Choose a circular loop around the axis of a parallel-plate capacitor. One spanning surface can intersect a supply wire, and another can bow through the gap. Their conduction currents differ, but their total conduction-plus-displacement currents agree. Magnetic circulation therefore does not depend on an arbitrary choice of mathematical surface. Fringing fields and finite propagation complicate a real rapid transient, so use this simple geometry in the quasistatic regime.

Inside ideal circular plates, assume a nearly uniform axial electric field and a charging current distributed uniformly over their area. A smaller coaxial loop encloses only the corresponding fraction of displacement current. Its magnetic field grows linearly with loop radius until the plate edge; outside, the ideal result falls inversely with radius. Extending the interior linear formula indefinitely would count displacement current that is not present.

Electric forces do work on charges at a rate per volume equal to current density dotted with electric field. Combining both curl equations converts that work term into a decrease of field energy plus a divergence. The latter is energy crossing the boundary, described by the Poynting vector. A resistor can receive energy from fields around its surface; electrons need not carry the entire supplied energy from a distant battery as kinetic energy.

The vacuum field-energy formula is positive and quadratic in the fields. Matter requires care: polarization, magnetization, mechanical motion, and dissipation must be assigned consistently rather than inserted into the vacuum expression without explanation. Poynting flux is instantaneous and directional; time averaging is a further operation suitable for periodic steady states. Keeping these distinctions avoids confusing stored energy with power passing through a region.