The Electromagnetic Field Equations

Abstract

In electrostatics we have Eq. (3.5.3), which for convenience we rewrite here as

$$\oint_c E \times dr = 0,$$

(10.1.1)

where C is any closed loop. It is also known that electrostatic fields of this nature, i.e., satisfying (10.1.1), cannot be associated with steady (non-time-dependent) currents J because flowing currents are associated with the dissipation of energy and electrostatic fields cannot provide this energy. This is not surprising since E was defined for static conditions only, i.e., when the charge density p had no velocity y and, hence, no current J was flowing. We also observe from our work in magnetostatics that a steady current J flowing in a wire generates a magnetic induction field B by virtue of Eq. (7.3.12); but a magnetic induction field B does not generate a current J. However, Faraday discovered that when the magnetic induction field B varied in time, a current was induced in a wire. Moreover, he found a precise quantitative statement describing the effect. However, the form stated by Faraday was for a particular experimental system and, consequently, is not as useful analytically as the form in which it was stated by Maxwell.