Page 398 - Engineering Electromagnetics, 8th Edition

380 ENGINEERING ELECTROMAGNETICS

EXAMPLE 11.4
We again consider plane wave propagation in water, but at the much higher micro-
wave frequency of 2.5 GHz. At frequencies in this range and higher, dipole relaxation
2
and resonance phenomena in the water molecules become important. Real and imagi-
nary parts of the permittivity are present, and both vary with frequency. At frequencies
below that of visible light, the two mechanisms together produce a value of that

13
increases with increasing frequency, reaching a maximum in the vicinity of 10 Hz.
decreases with increasing frequency, reaching a minimum also in the vicinity of

10 13 Hz. Reference 3 provides specific details. At 2.5 GHz, dipole relaxation effects
dominate. The permittivity values are = 78 and = 7. From (44), we have

r
r
 1/2
√

9
(2π × 2.5 × 10 ) 78 7
2
α = √  1 + − 1  = 21 Np/m
(3.0 × 10 ) 2 78
8
This first calculation demonstrates the operating principle of the microwave oven.
Almost all foods contain water, and so they can be cooked when incident microwave
radiation is absorbed and converted into heat. Note that the field will attenuate to a
value of e −1 times its initial value at a distance of 1/α = 4.8 cm. This distance is called
the penetration depth of the material, and of course it is frequency-dependent. The
4.8 cm depth is reasonable for cooking food, since it would lead to a temperature rise
that is fairly uniform throughout the depth of the material. At much higher frequencies,
where is larger, the penetration depth decreases, and too much power is absorbed

at the surface; at lower frequencies, the penetration depth increases, and not enough
overall absorption occurs. Commercial microwave ovens operate at frequencies in the
vicinity of 2.5 GHz.
Using (45), in a calculation very similar to that for α,we find β = 464 rad/m.
The wavelength is λ = 2π/β = 1.4 cm, whereas in free space this would have been
λ 0 = c/f = 12 cm.
Using (48), the intrinsic impedance is found to be
377 1
◦
η = √ √ = 43 + j1.9 = 43 2.6
78 1 − j(7/78)
and E x leads H y in time by 2.6 at every point.
◦
We next consider the case of conductive materials, in which currents are formed
by the motion of free electrons or holes under the influence of an electric field.
The governing relation is J = σE, where σ is the material conductivity. With finite
conductivity, the wave loses power through resistive heating of the material. We look
for an interpretation of the complex permittivity as it relates to the conductivity.

2 These mechanisms and how they produce a complex permittivity are described in Appendix D.
Additionally, the reader is referred to pp. 73–84 in Reference 1 and pp. 678–82 in Reference 2 for
general treatments of relaxation and resonance effects on wave propagation. Discussions and data that
are specific to water are presented in Reference 3, pp. 314–16.