Page 284 - Engineering Electromagnetics, 8th Edition

266 ENGINEERING ELECTROMAGNETICS

The vector identity
∇ · (A × H) ≡ H · (∇× A) − A · (∇× H) (54)
may be proved by expansion in rectangular coordinates. The inductance is then
1
L = ∇ · (A × H) dν + A · (∇× H) dν (55)
I 2 vol vol
After applying the divergence theorem to the first integral and letting ∇× H = J in
the second integral, we have
1
L = (A × H) · dS + A · J dν
I 2 S vol
The surface integral is zero, as the surface encloses the volume containing all the
magnetic energy, and this requires that A and H be zero on the bounding surface. The
inductance may therefore be written as
1
L = A · J dν (56)
I 2 vol
Equation (56) expresses the inductance in terms of an integral of the values of
A and J at every point. Because current density exists only within the conductor, the
integrand is zero at all points outside the conductor, and the vector magnetic potential
need not be determined there. The vector potential is that which arises from the current
J, and any other current source contributing a vector potential field in the region of
the original current density is to be ignored for the present. Later we will see that this
leads to a mutual inductance.
The vector magnetic potential A due to J is given by Eq. (51), Chapter 7,
µJ

A = dν
vol 4πR
and the inductance may therefore be expressed more basically as a rather formidable
double volume integral,
1 µJ
L = dν · J dν (57)
I 2 vol vol 4πR
A slightly simpler integral expression is obtained by restricting our attention to
current filaments of small cross section for which J dν may be replaced by IdL and
the volume integral by a closed line integral along the axis of the filament,
1 µIdL
L = · IdL
I 2 4πR (58)
µ dL
= · dL
4π R
Our only present interest in Eqs. (57) and (58) lies in their implication that the
inductance is a function of the distribution of the current in space or the geometry of
the conductor configuration.
To obtain our original definition of inductance (49), let us hypothesize a uniform
current distribution in a filamentary conductor of small cross section so that J dν