The net magnetic flux out of any region enclosed by a surface S
must be zero.
This property of flux density is almost implicit in Faraday's
law. To see this, consider that law, (1.6.1), applied to a closed
surface S. Such a surface is obtained from an open one by letting the
contour shrink to zero, as in Fig. 1.5.1. Then Faraday's integral
law reduces to
Gauss' law (1) adds to Faraday's law the empirical fact that
in the beginning, there was no closed surface sustaining a net outward
magnetic flux.
Illustration. Uniqueness of Flux Linking Coil
An example is shown in Fig. 1.7.1. Here a wire with terminals a-b
follows the contour C. According to (1.6.8), the
terminal EMF is found by integrating the normal magnetic flux density
over a surface having C as its edge. But which surface? Figure 1.7.1
shows two of an infinite number of possibilities.
Figure 1.7.1. Contour C follows loop of wire having
terminals a-b. Because each has the same enclosing contour, the
net magnetic flux through surfaces S1 and S2 must be the same.
The terminal EMF can be unique only if the integrals over S1
and S2 result in the same answer. Taken together, S1 and
S2 form a closed surface. The magnetic flux continuity integral
law, (1), requires that the net flux out of this closed surface be
zero. This is equivalent to the statement that the flux passing
through S1 in the direction of da1 must be equal to that
passing through S2 in the direction of da2. We will
formalize this statement in Chap. 8.
Example 1.7.1. Magnetic Flux Linked by Coil and Flux Continuity
In the configuration of Fig. 1.7.2, a line current produces a
magnetic field intensity that links a one-turn coil. The left
conductor in this coil is directly below the wire at a distance d.
The plane of the coil is horizontal. Nevertheless, it is convenient to
specify the position of the right conductor in terms of a distance R
from the line current. What is the net flux linked by the coil?
Figure 1.7.2. (a) The field of a line current induces
a flux in a horizontal rectangular coil. (b) The open surface has
the coil as an enclosing contour. Rather than being in the plane of
the contour, this surface is composed of the five segments shown.
The most obvious surface to use is one in the same plane as the
coil. However, in doing so, account must be taken of the way in which
the unit normal to the surface varies in direction relative to the
magnetic field intensity. Selection of another surface, to which the
magnetic field intensity is either normal or tangential, simplifies
the calculation. On surfaces S2 and S3, the normal direction
is the direction of the magnetic field. Note also that because the field
is tangential to the end surfaces, S4 and S5, these make no
contribution. For the same reason, there is no contribution from
S6, which is at the radius ro from the wire. Thus,
On S2 the unit normal is i, while on S3 it is
-i. Therefore, (3) becomes
With the field intensity for a line current given by (1.4.10), it
follows that
That ro does not appear in the answer is no surprise, because if
the surface S1 had been used, ro would not have been brought
into the calculation.
Magnetic Flux Continuity Condition
With the charge density set equal to zero, the magnetic
continuity integral law (1) takes the same form as Gauss' integral
law (1.3.1). Thus, Gauss' continuity condition (1.3.17)
becomes one representing the magnetic flux continuity law by making the
substitution o E o H.
The magnetic flux density normal to a surface is continuous.