Page 432 - Electrical Properties of Materials

414 Artificial materials or metamaterials

where M nn is the mutual inductance between elements n and n . The total

flux is

Φ = μ 0 SH + I M nn (15.25)
and the corresponding current may be written as

iω

I = (μ 0 SH + I M nn ). (15.26)

Z
Following the same technique as before, we can find the modified form of the
relative magnetic permeability as

F
μ r =1 – 2 . (15.27)
1
1– ω +( ) M nn + i

ω 2 0 L Q
As may be seen from the above equation, the introduction of the local field
did not make any drastic difference to the equation. It can actually be proven
that for a cubic lattice it will average out to zero, although for some other
lattice configurations it will lead to some shift in the position of the negative
region.
You might ask at this stage why we bother to show two models here for
determining the effective permeability, when we have already derived an ex-
pression for the effective permittivity in Section 10.10; and surely the analogy
between permittivity and permeability allows us to rely on the same expres-
sion. This is indeed so. All we need to do is to substitute magnetic polarization
for electric polarization yielding
2
ω (1 – 2F/3) – ω 2
μ r = 2 0 . (15.28)
2
ω (1 + F/3) – ω
0
The positions of the pole and the zero may be seen to have shifted but
again there is no major change. All three models lead to the same conclu-
sion. So why did we need three different models? Well, let’s admit that the
Clausius–Mossotti model is a little obscure. Why can we add the effects of all
the other elements by assuming dipoles over a spherical surface? The merit of
the first two models discussed here is that the physics is clear.

15.8 Effect of negative material constants
We talked about negative permittivity in Chapter 1, at the beginning of this
course. In the lossless case, if the frequency is below the plasma frequency
an incident electromagnetic wave cannot propagate in a lossless conducting
medium (we may as well call it a plasma). This was shown schematically in
Fig. 1.5. Earlier in this chapter we claimed, and showed the theory and the ex-
periment, that a wire medium acts as an artificial plasma. There is transmission
when the effective permittivity is positive, and no transmission (or rather very
little transmission) when the effective permittivity is negative. The situation is