3.19. MAGNETOSTATIC MODES IN AN INFINITE MEDIUM 105
Even though the above seem rather complicated, the actual situation is much simpler since
only Eqs. (3.72), (3.68a), and (3.68b) are needed to yield:
Ferrite Region: r
Œ
N
h
D 0 $ r
.
Œr‰
/
D 0: (3.74)
is is the equation governing the magnetostatic potential inside the ferrite specimen.
However, for the air region surrounding the specimen, magnetization is zero, or permeability is
scalar and equal to
0
, or
N
b D
0
N
h and (3.74) is reduced to:
Air Region: r
2
D 0: (3.75)
What is then needed is to solve the generalized Laplace equation (3.74) inside the spec-
imen and the ordinary one (3.75), outside the specimen, and impose the boundary conditions
on its surface. For an analytical solution, the unknown constants will be estimated through the
continuity conditions. Let the superscripts i and e denote the inside and outside external regions
respectively. us, we have
Ferrite Region: r
Œr‰
i
D 0 (3.76a)
Air Region: r
2
‰
e
D 0: (3.76b)
On the specimen’s surface with a normal unit vector On, we impose the following conditions:
1. Continuity of tangential intensity components On
N
h.
2. Continuity of normal flux density b
n
or On
N
b.
3. e magnetic potential
e
should be bounded (vanish) at infinity lim
e
.r ! 1/ ! 0.
Equations (3.76) may be expanded in Cartesian coordinates in order to obtain their gen-
eral solutions. It is worth noting that the permeability elements are assumed to be independent
of position since X and k
1
depend on !
i
D
0
H
i
and !
m
D
0
M
s
, while H
i
and M
s
are
assumed to be uniform throughout the specimen. In turn, (3.76a) reads as follows:
.1 C X /
@
2
i
@x
2
C
@
2
i
@y
2
C
@
2
i
@z
2
D 0: (3.77)
is is actually Walker’s equation for magnetostatic modes of uniformly biased (or homo-
geneous) ferrite specimens.
3.19 MAGNETOSTATIC MODES IN AN INFINITE
MEDIUM
e study of magnetostatic modes in an infinity medium does not reflect any useful practical
applications. However, it serves two significant purposes. First, it clarifies some of their charac-
teristics and, second, it can be incorporated into the analysis of practical finite specimens, where
the magnetic potential is expanded into an infinite sum of plane waves (just like a Fourier series).
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.