A Parallel Derivation to the Maxwell-Garnett Formula for the Magnetic Permeability of Mixed Materials

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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

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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