Page 282 - Engineering Electromagnetics, 8th Edition

264 ENGINEERING ELECTROMAGNETICS

We now define inductance (or self-inductance) as the ratio of the total flux link-
ages to the current which they link,

N
L = (49)
I

The current I flowing in the N-turn coil produces the total flux and N flux
linkages, where we assume for the moment that the flux links each turn. This
definition is applicable only to magnetic media which are linear, so that the flux is
proportional to the current. If ferromagnetic materials are present, there is no single
definition of inductance which is useful in all cases, and we shall restrict our attention
to linear materials.
The unit of inductance is the henry (H), equivalent to one weber-turn per ampere.
Let us apply (49) in a straightforward way to calculate the inductance per meter
length of a coaxial cable of inner radius a and outer radius b.We may take the
expression for total flux developed as Eq. (42) in Chapter 7,
µ 0 Id b
= ln
2π a
and obtain the inductance rapidly for a length d,
µ 0 d b
L = ln H
2π a
or, on a per-meter basis,
µ 0 b
L = ln H/m (50)
2π a
In this case, N = 1 turn, and all the flux links all the current.
In the problem of a toroidal coil of N turns and a current I,as shown in Fig-
ure 7.12b,wehave
µ 0 NI
B φ =
2πρ
If the dimensions of the cross section are small compared with the mean radius of the
toroid ρ 0 , then the total flux is
µ 0 NIS
=
2πρ 0
where S is the cross-sectional area. Multiplying the total flux by N,wehave the flux
linkages, and dividing by I,wehave the inductance
2
µ 0 N S
L = (51)
2πρ 0
Once again we have assumed that all the flux links all the turns, and this is a
good assumption for a toroidal coil of many turns packed closely together. Suppose,
however, that our toroid has an appreciable spacing between turns, a short part of
which might look like Figure 8.14. The flux linkages are no longer the product of the