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NEOCLASSICAL THEORY OF INTERACTION 69

Thus, the external magnetic vector is always tangential to the surface of the superconductor.
Notice that if the external magnetic field becomes too intense and exceeds some critical
threshold, the superconductive property disappears abruptly [17] and the material reverts to a
normal metallic state.

Sometimes accepted the myth that the superconductivity could be reached not only for the time-
independent fields is not true. The reason is that the electrons in superconductors are bound in
pairs called Cooper pairs. Loosely speaking, such configuration “allows” them to avoid
collisions with cooled material lattice, move through without any resistance and energy loss.
The time-varying magnetic field slightly penetrates a surface of superconductor and induces
oscillations of the electrons which are not bound in Cooper pairs and thereby able to lose their
kinetic energy. The conductivity at all frequencies except direct current (dc) can be quite high
but not infinite. The measured superconductor conductivity exceeds 10 [S/m]. For
26
comparison, silver, the best bulk conductor, has the conductivity around 10 [S/m] only.
8
The compelling advantages that superconductors offer suggest a vast array of present and future
applications in the areas of electric power, transportation, medical imaging and diagnostics,
nuclear magnetic resonance (NMR), industrial processing, high energy physics, wireless
communications, instrumentation, sensors, radar, high-end computing, etc. More detailed
discussion of superconductivity is far beyond the scope of this book.

Hint: The boundary conditions is the first and probably the most advance test for
any analytical or especially numerical solution. In general, more deviation from
them is the clear indicator of pure numerical solution conversion, not adequate to
your model meshing, wrong space or time discretization, the accuracy limit of
numerical algorithm is reached, the round-off or truncation errors accumulation,
software bugs, or your model simply is too complicated for a current tool.
Decompose your model into smaller blocks and try to run each block separately.
Such experiment typically allows to detect the weakest or wrong part of your
model. Regardless of the outcome, scrutinize your expectation for a solution, not
necessarily knowing exactly what the answer will be. That's where the engineer's
experience and good physical understanding is critical.

2.4 CLASSIFICATION OF MATERIALS BASED ON THEIR
ELECTRICAL AND MAGNETIC PROPERTY

2.4.1 Complex-Valued Dielectric and Magnetic Constant
Our goal now is to show that the dielectric permittivity and magnetic permeability are frequency
dependable complex volumes. Let us start from the permittivity. In section 2.2.2 of this chapter,
we discussed the polarization effects in dielectrics and found that the displacement vector ,
vector of the electric field and polarization vector are related to the equity (2.15)

= + (2.63)
0
Earlier we assumed that the polarization vector follows the electric vector variations without
any delay that brought us to the expression = (1 + ) = where the relative
0 0