Introduction

The unification of the theories of electric and magnetic phenomena was one of the great scientific events of the nineteenth century. Whereas stationary electric fields E have source at the positions of the charges but are irrotational (∇ x E = 0), changing magnetic fields produce circulating electro motive forces. In contrast, magnetic fields B are always sourceless and circulate around currents and places where there is a time-dependent electric field. The dynamical interrelation of the two field is described by Maxwell’ equations: If we consider empty space (no sources or currents), then in units where c = 1 they require that (1.1.1) % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmqr1ngBPrgitL % xBI9gBaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbb % f9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYx % ir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGaca % GaaeqabaWaaqaafaaakeaadaWdvbqaaiabdsgaKjabdohaZjabgwSi % xhrbuLwBLnhiov2DGi1BTfMBG0evGueE0jxyGi0BSrgaiyqacqWFfb % qrcqGH9aqpcqGHsisldaWdrbqaaiabdsgaKbWcbaGaemOta4eabeqd % cqGHRiI8aOGae8xrauKaeyyXICTaf8NqaiKbaiaacqWFSaaltCvAUf % eBSn0BKvguHDwzZbqehS0B6v3AHbYrVrhAPngiqj3BGieaiCaacaGF % GaGaa4hiaaWcbaGaeyOaIyRaemOta4eabeqdcqWIr4E0cqGHRiI8aO % Waa8qvaeaacqWGKbazcqWGZbWCcqGHflY1cqWFcbGqcqGH9aqpcqGH % sisldaWdrbqaaiabdsgaKbWcbaGaemOta4eabeqdcqGHRiI8aOGae8 % 3uamLaeyyXICTaf8xrauKbaiaacqWFSaalcaGFGaGaa4hiaaWcbaGa % eyOaIyRaemOta4eabeqdcqWIr4E0cqGHRiI8aaaa!89A7! $$ \oint\limits_{\partial N} {ds \cdot E = - \int\limits_N d E \cdot \dot B, } \oint\limits_{\partial N} {ds \cdot B = - \int\limits_N d S \cdot \dot E, } $$ for integrals over arbitrary surfaces N with boundaries ∂N; and if the surface is closed, the (1.1.2) % MathType!MTEF!2!1!+-% MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmqr1ngBPrgitL % xBI9gBaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbb % f9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYx % ir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGaca % GaaeqabaWaaqaafaaakeaadaWdvbqaaiabdsgaKfrbuLwBLnhiov2D % Gi1BTfMBG0evGueE0jxyGi0BSrgaiyqacqWFtbWucqGHflY1cqWFfb % qrcqGH9aqpcqGHsisldaWdrbqaaiabdsgaKbWcbaGaemOta4eabeqd % cqGHRiI8aOGae83uamLaeyyXICTae8NqaiKaeyypa0JaeGimaadale % aacqWGobGtaeqaniablgH7rlabgUIiYdaaaa!6097! $$ \oint\limits_N {dS \cdot E = - \int\limits_N d S \cdot B = 0} $$ We shall later recognize these apparently independent relationships as different aspects of a single fact, that the field-strength form and its dual form are closed. Before going more fully into this geometrical interprestation, let us try to come to an intuitive understanding of the physical consequences of these equations.