Environmental Engineering Reference
In-Depth Information
P
~
m
m
D
V
l
0
M
¼
lim
D
V
!
0
ð
3
:
25
Þ
The magnetization can be considered as a macroscopic property of the material.
It may also represent the spontaneous magnetization M within a ferromagnetic
material, or the uniform magnetization induced by the applied
eld in a para-
magnetic or a diamagnetic material.
3.1.4 Magnetic Field and Magnetic Induction Related
to Magnetic Materials
In a permanent-magnet bar, each end of this bar will represent a pole. If two such
bars are placed in the same plane as each other and separated by a vacuum, the like
poles (e.g. p
1
and p
2
, respectively) will repel and the unlike poles will attract with the
force F, which is proportional to the product of the pole strengths and the inverse
of the square of the distance (which can be referred to Coloumb
'
s law [
10
,
11
]):
p
1
p
2
4
pl
0
r
2
F
¼
ð
3
:
26
Þ
The pole can be associated with the magnetic charge, similar to the case of
electricity. For a single pole p
1
, (not to be misinterpreted as a monopole), the
magnetic
eld, as a vector, surrounding the pole can be de
ned [
11
,
12
]:
4
pl
0
r
3
¼
k
p
1
~
p
1
~
r
r
r
3
H
¼
ð
3
:
27
Þ
where k can be de
ned as the value of the proportionality constant in a similar way
as this is done for electric
elds.
Now, lets us look at Eq. (
3.28
). If it is related to free space (vacuum), then the
current density corresponds to Eq. (
3.12
). However, for material media, Eq. (
3.12
)
can be written for the magnetic
fl
ux density as [
11
,
13
]:
rot
!
¼ l
0
J
!
þ
!
J
m
ð
3
:
28
Þ
In Eq. (
3.28
) the J
c
relates to the electric current density due to the conduction.
However, an additional term J
m
represents the magnetization current density. The
magnetization currents of the material are associated with quantum mechanics in
the magnetized material. The integral of the magnetization current density through a
certain volume, which is performed over the surface (envelope) of that volume, is:
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