Page 285 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 267

in (56) becomes IdL,
1
L = A · dL (59)
I
Fora small cross section, dL may be taken along the center of the filament. We now
apply Stokes’ theorem and obtain
1
L = (∇× A) · dS
I S
or
1
L = B · dS
I S
or

L = (60)
I
Retracing the steps by which (60) is obtained, we should see that the flux is
that portion of the total flux that passes through any and every open surface whose
perimeter is the filamentary current path.
If we now let the filament make N identical turns about the total flux, an idealiza-
tion that may be closely realized in some types of inductors, the closed line integral
must consist of N laps about this common path, and (60) becomes
N
L = (61)
I
The flux is now the flux crossing any surface whose perimeter is the path occupied
by any one of the N turns. The inductance of an N-turn coil may still be obtained
from (60), however, if we realize that the flux is that which crosses the complicated
4
surface whose perimeter consists of all N turns.
Use of any of the inductance expressions for a true filamentary conductor (having
zero radius) leads to an infinite value of inductance, regardless of the configuration
of the filament. Near the conductor, Amp`ere’s circuital law shows that the magnetic
field intensity varies inversely with the distance from the conductor, and a simple
integration soon shows that an infinite amount of energy and an infinite amount of
flux are contained within any finite cylinder about the filament. This difficulty is
eliminated by specifying a small but finite filamentary radius.
The interior of any conductor also contains magnetic flux, and this flux links a
variable fraction of the total current, depending on its location. These flux linkages
lead to an internal inductance, which must be combined with the external inductance
to obtain the total inductance. The internal inductance of a long, straight wire of
circular cross section, radius a, and uniform current distribution is

µ
L a,int = H/m (62)
8π
a result requested in Problem 8.43 at the end of this chapter.

4 Somewhat like a spiral ramp.