Electric Field Intensity with Circulation
The second limiting situation,
typical of the magnetoquasistatic systems to be considered,
is primarily concerned with the circulation of E, and hence with the
part of the electric field generated by the time-varying magnetic flux
density. The remarkable fact is that Faraday's law holds for any
contour, whether in free space or in a material. Often, however, the
contour of interest coincides with a conducting wire, which comprises
a coil that links a magnetic flux density.
Illustration. Terminal EMF of a Coil
A coil with one turn is shown in Fig. 1.6.3. Contour (1) is
inside the wire, while (2) joins the terminals along a defined path.
With these contours constituting C, Faraday's integral law as given by
(1) determines the terminal electromotive force. If the electrical
resistance of the wire can be regarded as zero, in the sense that the
electric field intensity inside the wire is negligible, the contour
integral reduces to an integration from (b) to (a).
5 With
the objectives here limited to attaching an intuitive meaning to
Faraday's law, we will give careful attention to the conditions
required for this terminal relation to hold in Chaps. 8, 9, and 10.
In view of the definition of the EMF, (2),
this integration gives the negative of the EMF. Thus, Faraday's law
gives the terminal EMF as
Figure 1.6.3. Line segment (1) through a
perfectly conducting wire and (2) joining the terminals (a) and (b)
form closed contour.
where f, the total flux of magnetic field linking the coil, is
defined as the flux linkage. Note that Faraday's law makes it
possible to measure oH electrically (as now demonstrated).
Demonstration 1.6.1. Voltmeter Reading Induced by Magnetic Induction
The rectangular coil shown in Fig. 1.6.4 is used to measure the
magnetic field intensity associated with current in a wire. Thus, the
arrangement and field are the same as in Demonstration 1.4.1. The
height and length of the coil are h and l as shown, and because
the coil has N turns, it links the flux enclosed by one turn N times.
With the upper conductors of the coil at a distance R from the wire,
and the magnetic field intensity taken as that of a line current,
given by (1.4.10), evaluation of (8) gives
Figure 1.6.4. Demonstration of voltmeter reading
induced at terminals of a coil in accordance with Faraday's law. To
plot data on graph, normalize voltage to Vo as defined with (11).
Because I is the peak current, v is the peak voltage.
In the experiment, the current takes the form
where = 2 (60). The EMF between the terminals then
follows from (8) and (9) as
A voltmeter reads the electromotive force between the two points to
which it is connected, provided certain conditions are satisfied. We
will discuss these in Chap. 8.
In a typical experiment using a 20-turn coil with dimensions of
h = 8 cm, l = 20 cm, I = 6 amp peak, the peak voltage measured at
the terminals
with a spacing R = 8 cm is v = 1.35 mV. To put this data point on
the normalized plot of Fig. 1.6.4, note that R/h = 1 and the
measured v/Vo = 0.7.
Faraday's Continuity Condition
It follows from Faraday's integral law that the tangential electric
field is continuous across a surface of discontinuity, provided that
the magnetic field intensity is finite in the
neighborhood of the surface of discontinuity.
This can be shown by applying the integral law to the incremental
surface shown in Fig. 1.4.7, much as was done in Sec. 1.4 for
Ampère's law. With J set equal to zero, there is a formal
analogy between Ampère's integral law, (1.4.1), and Faraday's
integral law, (1). The former becomes the latter if H
E , J 0, and o E -o H.
Thus, Ampère's continuity condition (1.4.16) becomes the continuity
condition associated with Faraday's law.
At a surface having the unit normal n, the tangential electric
field intensity is continuous.