World Journal of Condensed Matter Physics, 2011, 1, 55-58
doi:10.4236/wjcmp.2011.12009 Published Online May 2011 (http://www.SciRP.org/journal/wjcmp)
55
A Parallel Derivation to the Maxwell-Garnett
Formula for the Magnetic Permeability of Mixed
Materials
Hsien-Ming Chang1,2*, Chungpin Liao1,2,3
1Advanced Research & Business Laboratory (ARBL), Taichung, Taiwan (China); 2Chakra Energetics, Inc., Incubator, National For-
mosa University (NFU), Huwei, Taiwan (China); 3Graduate School of Electro-Optic and Material Science, National Formosa Uni-
versity (NFU), Huwei, Taiwan (China).
Received January 4th, 2011; revised April 7th, 2011; accepted April 10th, 2011.
ABSTRACT
Although mixing formulas for the effective-medium type of approximations for the dielectric permittivities in the
infi-nite-wavelength (i.e., quasistatic) limit, such as the Maxwell Garnett formula, have been popularly applied in the
whole spectral range of electromagnetic fields, their magnetic counterpart has seldom been addressed up to this day.
An effort is thus devoted to the derivation of such an equation to predict the final permeability as the result of mixing
together several materials. In a similar fashion to the approach leading to the Maxwell Garnett formula, a model is
adopted wherein an originally isotropic host material is embedded with a cluster of spherical homogeneous magnetic
particles. It is expected that such obtained formula should find wide applications, and particularly in the light frequency
domain in this blossomful era of nanometer technology.
Keywords: Mixing Formula, Magnetic Permeability, Maxwell Garnett Mixing Formula
1. Introduction
Macroscopic mixing theories have been effectively
applied in predicting the approximate would-be dielec-
tric properties of the final mixtures without having to
deal with the microscopic fields in an in-situ manner.
In the infinite-wavelength approximation, or the so-
called “quasistatic limit,” the effective electromagnetic
characteristics of mixtures can be properly described
by two independent quantities, i.e., a single effective
permittivity and a single effective permeability, to their
own entirety. Although it was shown that in the finite-
wavelength limit (or, termed the “long-wave-length
limit” by many, e.g., [1]) such kind of partition is in-
applicable, the interest of this current work is in situa-
tions where the mixing of materials is in a practically
homogeneous manner and thus the infinite-wave-
length approximation should suffice.
However, although mixing formulas for the effec-
tive-medium type of approximations for the dielectric
permittivities in such infinite-wavelength (i.e., qua-
sistatic) limit, such as the Maxwell Garnett formula [2],
have been popularly applied in the whole spectral range
of electromagnetic fields, their magnetic counterpart
has seldom been addressed up to this day. The current
effort is thus to derive such an equation to approxi-
mately predict the final permeability as the result of
mixing together several magnetic materials.
It should be noticed that, unlike Sheng’s et al. [3]
approach where there are one or several major perma-
nent magnetic moments, this work is aiming at situa-
tions where no such major permanent magnetization
exists. In addition, following Maxwell Garnett’s [2]
model in which a host material contained a collection
of spherical homogeneous inclusions, Bruggeman [4]
further extended the mixing equation to a more con-
venient form in which the included particles no long
serve as mere perturbation to the host. Since then,
various mixing formulas have found a great variety of
applications in the prediction of mixed dielectric prop-
erties [5,6].
Nevertheless, the situation is quite different when it
comes to predicting mixed magnetic permeabilities.
Namely, not only the means to calculate magnetic
permeability are far from rigorous among even the ex-
isting 1st—principle quantum mechanical software (see,

A Parallel Derivation to the Maxwell-Garnett Formula for the Magnetic Permeability of Mixed Materials 56
e.g., [7], wherein the relative permeability is essentially
set to unity), but also the crude macroscopic clues to
follow, like Maxwell Garnett or Bruggeman formulas
for dielectric, are hard to come by. Thus, at least mac-
roscopically, the authors intend to bring back the due
balance among the conjugate worlds of dielectrics and
magnetics by basing their derivation on a similar model
adopted by Maxwell Garnett and Bruggeman [2,4].
2. A Brief Review of the Derivation Leading
to the Maxwell Garnett and Bruggeman
Formulas
Historically, an isotropic host material was hypothesized
to embed with a collection of spherical homogeneous
inclusions. With the molecular polarization of a single
molecule of such inclusions being denoted α, the follow-
ing relation was established within the linear range [8]:
0m
p  

m
E (1)
where m was the induced dipole moment and m
p
E

was the polarizing electric field intensity at the location
of the molecule. Since the treatment was aiming for uni-
form spherical inclusions, the polarizability became a
scalar, such that was expressed as [9]:
m
E

mp near
EEEE 

(2)
Here was the average field within the bulk host,
p was the electric field at this molecular location
caused by all surrounding concentric spherical shells of
the bulk, and near was due to asymmetry within the
inclusion. In those cases of interest where either the
structure of the inclusion was regular enough, such as a
cubical or spherical particulate, or all incorporated
molecules were randomly distributed, near became
e. It was further approximated that
E

tially
E

sse
E

ro

E

nze
0
3EEP
m



[9], to be elaborated later, with P

being the polarization density associated with a uni-
formly polarized sphere, and ε0 being the permittivity in
free space. Hence, given (1), with the number density of
such included molecules denoted as n, and m
pPn


[8],
the polarization density was further expressed as:

0
3PnEP 0









However, it was well-kn
E
(3)
own that for isotropic media

0
1
r
P

 


where εr stood for the relative permit-
tivity (i.e., the electric field at the center of a uniformly
polarized sphere (with being its polarization density)
was
P

0
3P

 ). Then, a relation known as the Lorentz-
Lorenz formula readily followed [10,11]:


31
2
r
r
n





(4)
In those special cases where the permittivity of each
tiny included particle was εs and the host material was
vacuum (εr = 1), such that n = V-1 (V being the volume of
the spherical inclusions), and (4) would have to satisfy
[2]:
0
0
32
s
s
V






 (5)
Combining (4) and (5) gave the effective permittivity
(εeff) of the final mixture [2]:

0
00 0
00
32
s
eff r
ss
ff

 



   (6)
with f = nV being the volume ratio of the embedded tiny
particles (0 ≤ f ≤ 1) within the final mixture. If, instead of
vacuum, the host material was with a permittivity of εh,
(6) was then generalized to the famous Maxwell Garnett
mixing formula:

32
sh
eff hh
s
hs
ff

 
h



   (7)
For the view in which the inclusion was no longer
treated as a perturbation to the original host material,
Bruggeman managed to come up with a more elegant
form wherein different ingredients were assumed to be
embedded within a host [4]. By utilizing (4) and (5), he
had:
00
00
22
eff i
i
i
eff i
f









 
ing to the

(8)
where fi and εi are the volume ratio and permittivity of
the i-th ingredient.
3. The Magnetic Flux Density at the Center
of a Uniformly Magnetized Sphere
In the above, the adopted relation leadMaxwell
Garnett mixing formula, i.e.,

0
3
m
EEP



, was
achieved by noting that the anti-reactive electric field at
the center of a uniformly polarized sphere (see, Figure 1)
was


0
3P



. This result can be derived from inte-
grating the positive surface charges on the upper sphere
and negative surface charges on the lower sphere (see
Figure 1) [9].
In a similar fashion, surface current can be expected to
appear on te surface of a uniformly magnetized sphere
(wherein
h
M

is the finalized net anti-responsive mag-
netization vector, ee Figure 2). s
In Figure 2, S
K

is the induced anti-reactive surface
current density (in A/m) on the sphere’s surface and is
equal to:
ˆ
ˆsin
S
KMr M

 


(9)
where ,
ˆ
rˆ

and ˆ

are the three orthogonal base vec-
tors of the spherical coordinate. According to the Biot-
Savart law, the differential magnetic flux density at the

A Parallel Derivation to the Maxwell-Garnett Formula for the Magnetic Permeability of Mixed Materials57
Figure 1. Situation for calculation of the central electric
field on a uniformly polarized sphere.
Figure 2. Situation for calculation of the central magnetic
flux density on a uniformly magnetized sphere.
sphere’s center owing to the surface current on a strip of
width Rdθ is:

00
22
ˆ
ˆ
dd
d44
CC
lr
II
l
BRR









(10)
where d
I
l

is the current element on an infinitesimal
segment of the strip and d
S
I
KR

. C is the path of the
strip closing around the north-pointing axis and μ0 the
permeability in free space. With all components cancel-
ling one another except the ones parallel to the
north-pointing axis, the net induced magnetization den-
sity
M

finally results, as do the associated magnetic
flux density . By integrating all strips on the sphere’s
surface the magnetic flux density (c) at the center of a
uniformly magnetized sphere is obtained to be [12]:
B

B

0
2
3
c
M
B




(11)
4. The Mixing Formula for Magnetic
Permeabilities
Now, this time consider an isotropic host material em-
bedded with a collection of spherical homogeneous mag-
netic particles. Given the magnetic flux density at the
location of a single molecule of the inclusions being m
B

,
the following relation holds in general:
mc near
BBBB 

 
(12)
where B

is the average magnetic flux density within
the bulk host and near
B

is due to the asymmetry in the
inclusion. In those cases of interest where either the
structure of the included particles is regular enough, such
as a cubical or spherical particulate, or all incorporated
molecules are randomly distributed, can be taken
as zero.
near
B

If the magnetic field intensity at the location of the
molecule is denoted m
H

, the induced magnetic dipole
moment (mm
m
H
) is:
mm
m  

(13)
where m

is th molecular magnetization of the mole-
cule. Because
e
M

equals , we have [8]
m
nm

mm mm
M
nHH  


(14)
where m

is known as the magnetic susceptibility.
Hence, m
B

can be further expressed as [8]


00 0
11
mrmmm m
BH nH    
m
H


(15)
with μr being the relative permeability. By incorporating
(12) and (15) into (14) we obtain:

0
0
2
13
m
m
nM
MB
n









(16)
Further, for isotropic magnetized materials [8]:


0
1
1,
r
r
r
B
MH



  or

0
1
r
r
B


M (17)
Substituting (17) into (16) gives
1 3
25
r
m
r
n








(18)
In the special case where the host material is vacuum
(μr = 1) and the permeability of the spherical particles is
μs, n = V-1 (V being the volume of a spherical particle),
and (18) is satisfied by:
0
0
325
s
m
s
V




 (19)
Combining (18) and (19) gives the effective perme-
ability (μeff) of the final mixture, i.e.,

0
00 0
00
325 2
eff rff

 
 

  (20)

A Parallel Derivation to the Maxwell-Garnett Formula for the Magnetic Permeability of Mixed Materials 58
where f = nV is the volume ratio of the embedded parti-
cles within the mixture (0 ≤ f ≤ 1). In the more general
situations where the host is no longer vacuum but of the
permeability μh, then the more general mixing formula of
permeabilities becomes:

3252
sh
eff hh
s
hs
ff

 
h
 

   (21)
As with Bruggeman’s approach for dielectrics [4], the
derived magnetic permeability formula can be genera-
lized to the multi-component form:
00
00
25 25
eff i
i
i
eff i
f
 




 


(22)
where fi and μi denote the volume ratios and permeabili-
ties of the involved different inclusions, respectively. Or,
11
25 2
reff ri
i
i
reff ri
f




 
5
(23)
Although the actual mixing procedures can vary
widely such that substantial deviations may result be-
tween the theoretical and measured values, (23) should
still serve as a valuable guide when designing magnetic
materials or composites.
5. Summary and Conclusions
An effective mixed medium approximation of perme-
abilities in the infinite-wavelength limit is derived from
the model describing a host material containing a collec-
tion of spherical homogeneous inclusions. The derivation
parallels that which had previously led to the famous
Maxwell Garnett formula for dielectric mixtures. It is
believed that such approximate magnetic mixing formula
should find a great many applications in all frequency
spectra of the general electromagnetic fields.
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