Page 227 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 209

equations that apply to static electric fields and steady magnetic fields is

D · dS = Q = ρ ν dν
S vol

E · dL = 0
(41)

H · dL = I = J · dS
S

B · dS = 0
S
Our study of electric and magnetic fields would have been much simpler if we
could have begun with either set of equations, (37) or (41). With a good knowledge
of vector analysis, such as we should now have, either set may be readily obtained
from the other by applying the divergence theorem or Stokes’ theorem. The various
experimental laws can be obtained easily from these equations.
As an example of the use of flux and flux density in magnetic fields, let us find
the flux between the conductors of the coaxial line of Figure 7.8a. The magnetic field
intensity was found to be
I
H φ = (a <ρ < b)
2πρ
and therefore
µ 0 I
B = µ 0 H = a φ
2πρ
The magnetic flux contained between the conductors in a length d is the flux
crossing any radial plane extending from ρ = a to ρ = b and from, say, z = 0to
z = d
d
b µ 0 I

= B · dS = a φ · dρ dz a φ
S 0 a 2πρ
or
µ 0 Id b

= ln (42)
2π a
This expression will be used later to obtain the inductance of the coaxial trans-
mission line.

D7.7. A solid conductor of circular cross section is made of a homogeneous
nonmagnetic material. If the radius a = 1 mm, the conductor axis lies on the
z axis,andthetotalcurrentinthea z directionis20A,find:(a) H φ atρ = 0.5mm;
(b) B φ at ρ = 0.8 mm; (c) the total magnetic flux per unit length inside the
conductor; (d) the total flux for ρ< 0.5 mm; (e) the total magnetic flux outside
the conductor.

Ans. 1592 A/m; 3.2 mT; 2 µWb/m; 0.5 µWb; ∞