# 1.7: Electromagnetic Fields and Materials

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The discussion in this section follows the description by Max Born [11] (who was instrumental in the development of quantum mechanics). Extensions are made to relates quantum descriptions to effects important to microwave engineering. In particular the descriptions here relate phenomena at the discrete particle level to continuum properties such as permittivity, \(\epsilon\).

## 1.7.1 Electromagnetic Fields in a Dielectric

A dielectric has no electrons that are free to move through a material under the influence of an electric field. When an EM field is incident on a dielectric the electric field moves charge centers. For a crystal these charge centers are at the scale of the lattice, and for composite materials like plastics the centers are at the molecular scale. With some materials, which we know as materials having high permittivity, the charge centers are normally separated but by less than the intra-atom spacing. With all dielectric materials, the major impact of an applied electric field is movement of the charge centers, alternately moving them apart and moving them together as the applied electric field alternates. Now being a dielectric, the charges cannot move freely through the material and the energy that is transferred to the charges is stored as electric potential energy in stretched bonds or, sometimes, as a distorted lattice (such as with the piezoelectric effect). This energy storage is much like storing mechanical energy in a spring. Now the charges move and thus excite an electric field of their own. However the charges move sluggishly and so the phase of the EM signal they produce is out of phase with respect to an externally applied alternating EM field. The charge centers are moving the fastest at the peak of the applied sinusoidal field and the net effect is a \(90^{\circ}\) phase lag.

Another phenomenon is that the combined effect of the reradiated fields from the moving charge centers produces an EM wave with the same frequency as the applied field but with a smaller phase velocity (smaller because the phase velocity is averaged over many paths). The oscillating charge centers radiate in all directions (which is called scattering) and so some of the EM energy will be scattered in the direction from which the applied EM wave came.

An ideal dielectric has no loss. That is, there is no dielectric relaxation loss associated with heating as the charge centers move, and there is no conductivity due to moving free charges in the dielectric.

## 1.7.2 Refractive Index

Dielectrics were first characterized by their refractive index long before the concept of EM waves and permittivity were developed. The refractive index, \(n\), of a medium is defined as the ratio of the speed of light (i.e., of an EM wave) in a vacuum, \(c\), of an EM wave to the phase velocity, \(v_{p}\), of the wave in the medium:

\[\label{eq:1}n=\frac{c}{v_{p}}=\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}\frac{\sqrt{\epsilon\mu}}{1} \]

For a loss-free medium

\[\label{eq:2}n=\sqrt{\varepsilon_{r}\mu_{r}} \]

For a lossy medium the complex index of refraction is

\[\label{eq:3}\overline{n}=n+\jmath\kappa \]

where \(n\) is called the refractive index and is directly related to the phase velocity, \(v_{p}\), and \(\kappa\) is called the extinction coefficient, which describes loss when the EM wave propagates through the material.

Conversion between refractive index and permittivity is as follows [12, 13]; the complex permittivity

\[\label{eq:4}\varepsilon =\epsilon_{1}+\jmath\epsilon_{2}=(n+\jmath\kappa)^{2} \]

where

\[\label{eq:5}\varepsilon_{1}=n^{2}-\kappa^{2}\quad\text{and}\quad\varepsilon_{2}=2n\kappa \]

The components of the complex index of refraction are then

\[\label{eq:6}n=\sqrt{\frac{\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}}+\varepsilon_{1}}{2}}\quad\text{and}\quad\kappa =\sqrt{\frac{\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}}-\epsilon_{1}}{2}} \]

The permittivity is just the square of the (complex) refractive index in a nonmagnetic medium \((\mu_{r} = 1)\). The refractive index is used with optics and the permittivity is used when working with Maxwell’s equations and with electronics.

The refractive index, and thus permittivity, of a dielectric can vary significantly between microwave and optical frequencies, a result of how quickly different types of charge centers can move. For water at \(20^{\circ}\text{C}\) and at the standard optical wavelength of \(589\text{ nm}\) (the yellow doublet sodium D-line), \(n = 1.333\) (\(\epsilon_{r} = 1.78\)) [14]. At \(1\text{ GHz}\) \(n = 8.94\) (\(\epsilon_{r} = 80\)) [15].

## 1.7.3 Electromagnetic Fields in a Metal

The discussion here refers to the behavior of EM fields in metals at frequencies from DC to \(10\text{ THz}\). Above these frequencies the EM wave changes direction much faster than electrons can move. Metals have a large number of free charges that can move through a metal under the influence of an electric field. On transmission lines the energy is contained in the EM fields between metal guides, and electric and magnetic fields are present at the surface of the metal. The main effect of the EM fields at the surface of the metal is to accelerate the free electrons at the surface, with the electrons accelerating in the direction opposite to that of the electric field. While some of the EM energy propagates into the metal, the overwhelming effect is transfer of energy from the EM photons to kinetic energy of the free electrons. There is also some transfer of energy to the bound electrons, however, this effect is smaller as the bound electrons are shielded by the sea of free electrons. Electrons have mass and accelerate relatively slowly. Even when the electric field is reversed, the electrons will continue moving considerably in the same direction as the electric field before reversing. The moving and accelerating electrons also produce electric fields of their own that in turn influence the movement of other electrons. Electrons are sluggish and their position is almost \(180^{\circ}\) out of phase with the applied \(E\) field. The net effect is that the electric field produced by the electrons almost completely cancels

Metal |
Photon energy \((\text{eV})\) |
Wavelength \(\lambda\:(\mu\text{m})\) |
Frequency \(f\:(\text{THz})\) |
\(n\) |
\(\kappa\) |
\(\varepsilon_{1}\) |
\(\varepsilon_{2}\) |
---|---|---|---|---|---|---|---|

Gold | \(0.1\) | \(12.398\) | \(24.197\) | \(8.17\) | \(82.83\) | \(-6794\) | \(1353\) |

Copper | \(0.1\) | \(12.398\) | \(24.197\) | \(29.69\) | \(71.57\) | \(-4240\) | \(4250\) |

Aluminum | \(0.1\) | \(12.398\) | \(24.197\) | \(98.595\) | \(203.7\) | \(-31663\) | \(40168\) |

Table \(\PageIndex{1}\): Electromagnetic properties of metals at optical frequencies. (\(n\) and \(\kappa\) are measured, \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are derived using Equation \(\eqref{eq:5}\).

the incoming electric field. It is therefore almost meaningless to talk about the (group or phase) velocity of the EM fields in the metal. The overriding effect is transfer of energy to moving electrons that eventually make the lattice move and transfer their energy to the lattice, producing heat.

The accelerating electrons radiate electric field in all directions and in turn this field accelerates other electrons. There is also a damping force as the free electrons collide with atoms. So there is a rapidly diminishing component of the EM field that travels through the metal at the speed of light. Rather than energy being stored in the electric field through the movement of charge centers as in dielectrics, electric energy in the forward-propagating field is almost entirely lost in the scattering and collision processes. So at microwave frequencies the effective permittivity of a metal is almost entirely imaginary (corresponding to loss), and this imaginary component of the relative permittivity is \(8–10\) orders of magnitude greater than the real component, which is believed to be \(1\) [16]. The real component of the relative permittivity is \(1\) because the charge centers are masked by the sea of free electrons. It is not possible to directly measure the real part of the permittivity of metals at microwave frequencies. An ideal conductor has infinite conductivity, so there are no EM fields inside the metal and net charges are confined to the surface of the conductor. At optical frequencies metals are known to have a negative permittivity (see Table \(\PageIndex{1}\)).