5.1 From Statics to Electrodynamics

In Chapter 3 we built the electric field of charges at rest, and in Chapter 4 the magnetic field of steady currents. The two subjects never touched: the electric field had sources ($\nabla \cdot D={\rho }_v$) but no circulation ($\nabla \times E=0$), and the magnetic field had circulation ($\nabla \times H=J$) but no sources ($\nabla \cdot B=0$). Nothing in either chapter depended on time.

Real engineering is not like that. The current in a transformer winding alternates fifty times a second, the charge on a capacitor plate rises and falls, and a radio antenna radiates a field that reaches the receiver long after the current that produced it has reversed. As soon as something changes in time, the two fields become inseparable: a changing magnetic field produces an electric field (Faraday, 1831), and a changing electric field produces a magnetic field (Maxwell, 1862). This section collects the four static equations, adds the two terms that express these facts and states the complete set of four Maxwell equations. Section 5.2 and Section 5.3 then examine each new term in a concrete situation, and Section 5.4 shows how the conservation of charge ties the two together.