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 [25]:
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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