Page 203 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 185

Figure 7.4 The streamlines of the
magnetic field intensity about an
infinitely long straight filament
carrying a direct current I. The
direction of I is into the page.

At this point the unit vector a φ under the integral sign should be investigated, for it is
not always a constant, as are the unit vectors of the rectangular coordinate system. A
vector is constant when its magnitude and direction are both constant. The unit vector
certainly has constant magnitude, but its direction may change. Here a φ changes with
the coordinate φ but not with ρ or z.Fortunately, the integration here is with respect
to z , and a φ is a constant and may be removed from under the integral sign,

∞ dz
Iρa φ
H 2 = 2 2 3/2
4π −∞ (ρ + z )
∞
z
Iρa φ

=
2
4π ρ 2 ρ + z 2
−∞
and
I
H 2 = a φ (8)
2πρ
The magnitude of the field is not a function of φ or z, and it varies inversely with
the distance from the filament. The direction of the magnetic-field-intensity vector is
circumferential. The streamlines are therefore circles about the filament, and the field
may be mapped in cross section as in Figure 7.4.
The separation of the streamlines is proportional to the radius, or inversely pro-
portional to the magnitude of H.Tobe specific, the streamlines have been drawn with
3
curvilinear squares in mind. As yet, we have no name for the family of lines that
are perpendicular to these circular streamlines, but the spacing of the streamlines has

3 If you can’t wait, see Section 7.6.