1. Introduction

Electromagnetic fields are a complex phenomenon because they can propagate through vacuum without the need for a material medium, they simultaneously behave like waves and like particles [1, 2], and they are intrinsically linked to the behaviour of the space–time continuum [3]. It can be shown that magnetic fields appear through relativistic motion of electric fields, which is why electricity and magnetism are so closely linked [4]. It has even been suggested that electromagnetic phenomena may be a space–time phenomenon, with gravitation being the result of space–time curvature [3] and electro-magnetic behaviour being the result of space– time torsion [5].

James Clerk Maxwell developed a theory to explain electromagnetic waves. He summarised this relationship between electricity and magnetism into what are now referred to as "Maxwell's Equations." An EM wave is described in terms of its:


These three properties are related by the equation:

$$
\mathcal{L} = \mathcal{Y} \tag{1}
$$

Figure 1. Schematic of the electromagnetic spectrum.

The speed of the electromagnetic wave is determined by:

$$
\omega = \frac{1}{\sqrt{\mu \varepsilon}} \tag{2}
$$

where ε is the electrical permittivity of the space in which wave exists and μ is the magnetic permeability of the space in which the wave exists.

Electromagnetic waves can be of any frequency; therefore, the full range of possible frequencies is referred to as the electromagnetic spectrum. The known electromagnetic spectrum extends from frequencies of around <sup>f</sup> <sup>=</sup> <sup>3</sup> � <sup>10</sup><sup>3</sup> Hz (<sup>λ</sup> <sup>=</sup> <sup>100</sup> km) to <sup>f</sup> <sup>=</sup> <sup>3</sup> � <sup>10</sup><sup>26</sup> Hz (<sup>λ</sup> <sup>=</sup> <sup>10</sup>�<sup>18</sup> m), which covers everything from ultralong radio waves to high-energy gamma rays [6]. A schematic of the electromagnetic spectrum is shown in Figure 1.

Electromagnetic waves can be harnessed to: transmit information; acquire information from a medium; or transmit energy. The first category of applications includes: terrestrial and satellite communication links; the global positioning system (GPS); mobile telephony; and so on [7]. The second category of applications includes: radar; radio-astronomy; microwave thermography; remote sensing and detection, and material property measurements [8]. The third category of applications is associated with electromagnetic heating and wireless power transmission.

This chapter will focus on the interactions of electromagnetic waves with materials and will therefore include acquiring information from a medium and transition of energy.

### 2. Some background theory

When electromagnetic waves encounter materials, the wave will be partially reflected, attenuated, delayed compared with a wave travelling through free space [9–11], and repolarised. Surface interactions, such as reflection, refraction, transmission and repolarisation reveal important information about the material and its immediate environment. For example, Figure 2 shows how reflection from the surface of a pond and transmission of light from in the pond water reveal

Energy Transfer from Electromagnetic Fields to Materials DOI: http://dx.doi.org/10.5772/intechopen.83420

Figure 2.

The combination of surface reflection and transmission through the water in this pond reveal the fish in the water and the trees in the immediate environment of the pond, hence visible light can acquire and transfer this information through space.

information about what is in and around the pond. All interactions between electromagnetic waves and materials are governed by the dielectric properties of the material and how these properties alter the electrical and magnetic properties of the space occupied by the material.

### 2.1 Dielectric properties of materials

All materials alter the space, which they occupy. Because materials are composed of various charged particles, these alterations include changes to the electrical and magnetic behaviour of space. These properties are described by the electrical permittivity and magnetic permeability of the space, which the electromagnetic fields encounter.

Magnetic permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. The magnetic permeability of space is: <sup>μ</sup><sup>o</sup> = 4<sup>π</sup> ˜ <sup>10</sup>°<sup>7</sup> (H/m). Except in the case of ferromagnetic materials, the magnetic permeability of many materials is equivalent to that of free space. The magnetic permeability of ferromagnetic materials varies greatly with field strength.

Space itself has dielectric properties [12] with an electrical permittivity of <sup>1</sup> °<sup>1</sup> approximately <sup>ε</sup><sup>o</sup> <sup>=</sup> 8.8541878 ˜ <sup>10</sup>°<sup>12</sup> or <sup>ε</sup>o<sup>≈</sup> ˜ <sup>10</sup>°<sup>9</sup> <sup>F</sup> <sup>m</sup> . Electrical permittiv- <sup>36</sup><sup>π</sup> ity describes the amount of charge needed to generate one unit of electric flux in a medium. All materials increase the electrical permittivity of the space they occupy, compared with free space (vacuum); therefore, some materials can support higher electric flux than free space. These materials are referred to as dielectric materials.

Debye [13] studied the behaviour of solutions with polar molecules and consequently refined the complex dielectric constant, which includes the conductivity of the material. His final equation became:

$$
\varepsilon = \varepsilon\_{\infty} + \frac{\varepsilon\_{t} - \varepsilon\_{\infty}}{1 + j\alpha\tau} - j\frac{\sigma}{\alpha\varepsilon\_{o}} \tag{3}
$$

where ε is the dielectric constant at very high frequencies; ε<sup>s</sup> is the dielectric constant at very low frequencies; ω is the angular frequency (rad s �1 ); τ is the relaxation time of the dipoles (s); σ isthe conductivity of the material (Siemens m�<sup>1</sup> ); j is the <sup>p</sup>ffiffiffiffiffiffi complex operator (i.e. <sup>j</sup> ¼ �1), and <sup>ε</sup><sup>o</sup> is the dielectric permittivity of free space.

Manipulating Eq. (3) to separate it into real and imaginary components yields:

$$\varepsilon = \varepsilon\_{\infty} + \frac{\varepsilon\_{\varepsilon} - \varepsilon\_{\infty}}{1 + \alpha^2 \tau^2} - j \left\{ \frac{(\varepsilon\_{\varepsilon} - \varepsilon\_{\infty}) \alpha \tau}{1 + \alpha^2 \tau^2} + \frac{\sigma}{\alpha \varepsilon\_{\sigma}} \right\} = \varepsilon' - j \varepsilon' \tag{4}$$

The relaxation time τ is a measure of the time required for polar molecules to rotate in response to a changed external electric field, and hence determines the frequency range in which dipole movement occurs. This response time depends on the temperature and physical state of the material.

The dielectric constant ε' affects the wave impedance of the space occupied by the dielectric material [14] and causes reflections at the inter-facial boundary between materials due to changes in the wave impedance of the space occupied by the material. These changes in wave impedance also cause a change in the wavelength of the electromagnetic fields inside the dielectric material, compared with the wavelength in air or vacuum [15]. This change in wavelength affects the propagation velocity of the wave within the material.

The dielectric loss ε" representsthe resistive nature of the material [16], which reducesthe amplitude ofthe electromagnetic field and generates heatinside the material.

It is common practice to express the dielectric properties of a material in terms of the relative dielectric constants κ' and κ", which are defined as: ε' = κ'ε<sup>o</sup> and ε" = κ"εo. The general form of the dielectric properties of polar materials resembles the normalised example shown in Figure 3.

The dielectric properties of most materials are directly associated with its molecular structure. Debye's basic relationship assumes that the molecules in a material are homogeneous in structure and can be described as "polar". Since few materials can be described in this way, many other equations have been developed to describe frequency-dependent dielectric behaviour.

Figure 3. Normalised dielectric properties of a polar material.

Energy Transfer from Electromagnetic Fields to Materials DOI: http://dx.doi.org/10.5772/intechopen.83420

In many cases, the material may be regarded as a composite or mixture and will exhibit multiple relaxation times. If this is the case then the complex dielectric constant may be represented by a variation of Debye's original equation [17]:

$$\kappa = \kappa\_{\infty} + a \left( \frac{\kappa\_{\text{s}} - \kappa\_{1}}{\mathbf{1} + j\alpha \tau\_{1}} \right) + b \left( \frac{\kappa\_{1} - \kappa\_{2}}{\mathbf{1} + j\alpha \tau\_{2}} \right) + \dots - j \frac{\sigma}{\alpha \varepsilon\_{o}} \tag{5}$$

where κ<sup>1</sup> and κ<sup>2</sup> are intermediate values of the dielectric constant between the various relaxation periods of τ<sup>1</sup> and τ2, and a and b are constants related to how much of each component is present in the total material.

#### 2.2 Temperature dependence of the dielectric properties

As temperature increases, the electrical dipole relaxation time associated with the material usually decreases, and the loss-factor peak will shift to higher frequencies (Figure 4). For many materials, this means that at dispersion frequencies the dielectric constant will increase while the loss factor may either increase or decrease depending on whether the operating frequency is higher or lower than the relaxation frequency [18]. For example, Table 1 demonstrated how the dielectric properties of some food stuffs, in the microwave band, vary with temperature.

Some food stuffs in Table 1 follow the predicted trend of increasing dielectric constant as temperature increases; however, it is apparent that the dielectric constant of other entries in Table 1 decline with increasing temperature rather than increasing with temperature. This is linked to their water content, because the dielectric constant of water at a fixed frequency decreases with increasing temperature (Figure 4). Figure 5 shows how the dielectric properties of water vary with temperature, over a wider range of frequencies.

#### 2.2.1 Density dependence of dielectric properties

Because a dielectric material's influence over electromagnetic waves depends on the amount of the material present in the space occupied by the material, it follows that the density of the material must influence the bulk dielectric properties of

Figure 4. Dielectric properties of pure water as a function of temperature at 2.45 GHz.


#### Table 1.

Dielectric properties of some common foods at 2.8 GHz as a function of temperature (source: [19]).

#### Figure 5.

Dielectric properties of pure water as a function of frequency and temperature (data sources: [20]; model for dielectric properties based on: [21]).

materials. This is especially true of particulate materials, such as soil, grains or flours [18].

As an example, the dielectric properties of oven dry wood, with the electric field oriented perpendicular to the wood grain, are described by [22]:

$$\boldsymbol{\kappa}^{'} = \kappa\_{\infty} + \frac{(\kappa\_{\boldsymbol{\sigma}} - \kappa\_{\infty}) \left[ \mathbf{1} + a \boldsymbol{\sigma}^{(1-a)} \operatorname{Cov} \left( \boldsymbol{\pi} \frac{1-a}{2} \right) \right]}{\left[ \mathbf{1} + a \boldsymbol{\sigma}^{2(1-a)} + 2a \boldsymbol{\sigma}^{(1-a)} \operatorname{Cov} \left( \boldsymbol{\pi} \frac{1-a}{2} \right) \right]} \tag{6}$$

and

$$\kappa'' = \frac{(\kappa\_o - \kappa\_\infty) \left[1 + a\sigma^{(1-a)} \operatorname{Cost} \left(\pi \frac{1-a}{2}\right)\right]}{2\left[1 + a\sigma^{2(1-a)} + 2a\sigma^{(1-a)} \operatorname{Cost} \left(\pi \frac{1-a}{2}\right)\right]} \tag{7}$$

Table 2 shows the values of κ<sup>o</sup> and κ as a function of wood density.

### 2.3 Wave propagation

If a material is homogeneous in terms of its electromagnetic properties, it is apparent that an incident electromagnetic wave would be partly reflected at the material boundary and partly transmitted. The transmitted energy would be dissipated due to any losses within the medium. If a plane wave is propagating through space in the x-direction, it can be described by:

$$E(\varkappa, t) = E\_o e^{j(k\varkappa - \alpha t)} \tag{8}$$

where, in general, the wavenumber (k) can be described by:

$$k = k\_o(\beta + j\alpha) \tag{9}$$

where ko is the wavenumber of free space:

$$k\_o = \frac{2\pi f}{c} \tag{10}$$

Van Remmen, et al. [23] show that the components of the wavenumber are given by:

$$\beta = \sqrt{\kappa' \frac{\sqrt{\mathbf{1} + \left(\frac{\kappa'}{\kappa'}\right)^2} + \mathbf{1}}{2}} \tag{11}$$

and:

$$a = \sqrt{\kappa' \frac{\sqrt{\mathbf{1} + \left(\frac{\kappa'}{\kappa'}\right)^2} - \mathbf{1}}{2}} \tag{12}$$

The term α is associated with wave attenuation with distance travelled through a medium. For free space (or air) α = 0 at most frequencies.

For a wave that is perpendicularly incident onto the surface of a material, the reflection coefficient, which is the ratio of the reflected wave amplitude to the incident wave amplitude, is given by:

$$\Gamma = \frac{(\beta\_1 + \beta\_2)(\beta\_1 - \beta\_2) + (a\_1 + a\_2)(a\_1 - a\_2)}{\left(\beta\_1 + \beta\_2\right)^2 + (a\_1 + a\_2)^2} + j\frac{(\beta\_1 + \beta\_2)(a\_1 - a\_2) - (\beta\_1 - \beta\_2)(a\_1 + a\_2)}{\left(\beta\_1 + \beta\_2\right)^2 + (a\_1 + a\_2)^2} \tag{13}$$


Table 2.

Values of κ<sup>s</sup> and κ for oven dried wood of various densities when the electric field is perpendicular to the wood grain (based on data from: [22]).

Electromagnetic Fields and Waves

where the subscripts refer to medium 1 and medium 2 across the medium interface. The transmission coefficient, which is the ratio of the transmitted wave amplitude to the incident wave amplitude, is given by:

$$\pi = 2 \left\{ \frac{\beta\_1^{-2} + \beta\_1 \beta\_2 + a\_1 a\_2 + a\_1^2}{\left(\beta\_1 + \beta\_2\right)^2 + \left(a\_1 + a\_2\right)^2} + j \frac{\beta\_2 a\_1 - \beta\_1 a\_2}{\left(\beta\_1 + \beta\_2\right)^2 + \left(a\_1 + a\_2\right)^2} \right\} \tag{14}$$

Therefore, the wave, which propagates across a boundary from one medium (or vacuum) to another, is described by:

$$E(\mathbf{x},t) = E\_o \mathbf{r} \bullet \mathbf{e}^{j(k\_o \beta \mathbf{x} - \alpha t)} \bullet \mathbf{e}^{-k\_o \alpha \mathbf{x}} \tag{15}$$

## 3. Transmission through a material

If the electromagnetic wave passes through a material, the wave emerging on the other side will be described by:

$$E(\mathbf{x},t) = E\_o \tau\_{1,2} \bullet \tau\_{2,1} \bullet e^{j(k\_o\beta\_1\mathbf{x} + k\_o\beta\_2L - \alpha t)} \bullet e^{-k\_o a\_2 L} \tag{16}$$

where τ1,2 and τ2,1 are the transmission coefficients of the two material interfaces and L is the thickness of the material through which the wave passes. Therefore, the wave is delayed, phase shifted (because of the complex value of the transmission coefficient) and attenuated by the material, in comparison to a similar wave propagating though free space. This is illustrated in Figure 6.
