87
C H A P T E R 6
Conclusions
A complete theoretical analysis with exact mathematical expressions for the behavior of the sys-
tem of two ferromagnetic spherical nanoparticles in a homogeneous medium under the uniform
magnetic field is investigated in this book. We assume that the material of the particles are linear
magnetic with infinite permeability. First, we obtained the total magnetic field intensity outside
from the superposition of the magnetic potentials involved. en, these field components were
used to obtain the magnetic flux density inside the particles by using the exact boundary condi-
tions. All the corresponding mathematical expressions were derived in a bispherical coordinates
system.
e magnetic field intensity (and flux density) outside depends on the gap ıg between
and the radii R
i
of the particles in the system. It increases when the relative gap ratios ıg=R
1
decrease while the relative radii ratios R
2
=R
1
increase. On the other hand, the magnetic flux
density inside follows the same behavior as in the outside quantities. at is, it depends on the
ıg=R
1
and R
2
=R
1
in a similar way. In addition, the magnetic field quantities at the symmetric
points outside are identical when the system is in axisymmetric geometry. e flux densities at
the symmetric points inside within the particle are also identical regardless of the geometry of
the system. at is, at the symmetric points inside the field quantities are similar and do not
depend (only the similarity) on the relative gap ratio ıg=R
1
and the relative radii ratio R
2
=R
1
.
Moreover, for a certain accuracy, the number of terms N is inversely proportional to the gap
ratios as well as the radii ratios.
e field intensity and flux density outside are dominant when the relative gaps between
the particles are small while the relative sizes of the particles are large under the external fields. As
a result, the magnetic field intensification outside of the system is increased. en, this outside
behavior directly affects the inside of the ferromagnetic nanoparticles which facilitates to increase
the interaction forces between the neighbor domains in the material. is causes a very large
generation of values for the magnetic flux density inside which is called the magnetization of
the material.
Here, we obtained benchmark solutions from exact analytical expressions with boundary
conditions for the system of two ferromagnetic spherical particles under the magnetic fields, and
can be used for validity of other approximate numerical techniques. Furthermore, the method-
ology used in this book could be adopted to the systems of prolate and oblate spheroids, which
better approximates real-world ferromagnetic objects where more advanced mathematics are
used.
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