As a rotation-less field can be expressed as the negative gradient −∇ V of a potential function V , it follows that: E = −∇ V − jω A (2.18) = E c + E m (2.19) in which V is the electric potential function. The electric field is hence split up into components E c and E m . The conservative electric field E c , fully defined by V , originates from positive and terminates on negative charges. These charges are described by the charge distribution function ρ in Gauss’s law ( 2.5 ). The other part of the electric field, E m , is magnetically induced and is divergence free. It does not originate from charge distributions, neither does it terminate on them. E m forms self-closing field lines just like the magnetic field does. In this way, physical meaning can be given to A , since its negative time derivative is the non-conservative component E m of the electric field. 2.1.5 Current and Flux The macroscopic quantities electric current I and magnetic flux φ through a certain surface S C are obtained by integration of their vector counterparts J and B over that surface: I = S C J · dS (2.20) The surface integration of B can be replaced by a line integration of the vector potential A through Stokes’ theorem : φ = S C B · dS (2.21) = C A · dC (2.22) where C is the contour enclosing the surface S C .
2.2 Conductive Wire 17 2.2 Conductive Wire Induction coils used for inductive powering, and practical inductors in general, are all made up out of metal wires or traces. Hence it is instructive to investigate the electromagnetic field configuration in and around a piece of conductive wire. The dimensions of the wire and the coil constructed out of it, are assumed to be much smaller than a wavelength, so that the electromagnetic field changes can be consid- ered immediate over the volume of interest and hence that only the near field hence is important. Figure 2.1 depicts a piece of conductive wire. A potential difference V is ap- plied between both terminals. This gives rise to an electric field E c , also indicated on the figure, which is described by the voltage function V through: E c = −∇ V (2.23) E c is a conservative vector field that relates to a certain charge density ρ through Gauss’s law ( 2.5 ). The distribution of these charges over the wire depends on the wire and winding geometry and possibly on the definition of given electric potentials in the direct environment. The charges are schematically indicated with + and − signs for a loop-shaped piece of wire in Fig. 2.1 . Note that E c may be much stronger in the medium in between terminals, where distances can be shorter, than along the wire. This is especially the case in coil windings where high-potential turns lay next to low-potential turns (in multiple-layer windings for example). These charges are directly responsible for the inter-winding capacitance of a coil (see Sect. 2.4 on inductor models of this chapter).
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