Page 281 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 263

apply here, for the fields in the core have not changed, and we can therefore use the
principle of virtual work to determine that the work we have done in moving one core
appears as stored energy in the air gap we have created. By (48), this increase is
1 B st 2
dW H = FdL = SdL
2 µ 0
where S is the core cross-sectional area. Thus
2
B S
st
F =
2µ 0
If, for example, the magnetic field intensity is sufficient to produce saturation in the
steel, approximately 1.4 T, the force is
5
F = 7.80 × 10 S N
2
or about 113 lb f /in .

D8.11. (a) What force is being exerted on the pole faces of the circuit de-
scribed in Problem D8.9 and Figure 8.13? (b)Is the force trying to open or close
the air gap?
Ans. 1194 N; as Wilhelm Eduard Weber would put it, “schliessen”

8.10 INDUCTANCE AND MUTUAL
INDUCTANCE
Inductance is the last of the three familiar parameters from circuit theory that we are
defining in more general terms. Resistance was defined in Chapter 5 as the ratio of
the potential difference between two equipotential surfaces of a conducting material
to the total current crossing either equipotential surface. The resistance is a function
of conductor geometry and conductivity only. Capacitance was defined in the same
chapter as the ratio of the total charge on either of two equipotential conducting
surfaces to the potential difference between the surfaces. Capacitance is a function
only of the geometry of the two conducting surfaces and the permittivity of the
dielectric medium between or surrounding them.
As a prelude to defining inductance, we first need to introduce the concept of flux
linkage. Let us consider a toroid of N turns in which a current I produces a total flux
.We assume first that this flux links or encircles each of the N turns, and we also
see that each of the N turns links the total flux . The flux linkage N is defined as
3
the product of the number of turns N and the flux linking each of them. Fora coil
having a single turn, the flux linkage is equal to the total flux.

3 The symbol λ is commonly used for flux linkages. We will only occasionally use this concept,
however, and we will continue to write it as N .