Find expressions for the phase velocity and wavelength of an electromagnetic wave of angular frequency \omega propagating in a metal of conductivity \sigma. How do the values compare with the corresponding values in free space at the same frequency?

Phase velocity is the speed at which the phase of any one frequency component of the wave travels. In simple terms, it tells us how quickly the peaks and troughs of the wave (or any other point on the wave with constant phase) move through space. For electromagnetic waves in a vacuum, the phase velocity is the speed of light, denoted by 'c'. However, when these waves travel through a medium like metal, the phase velocity is generally reduced due to interactions with the material's atoms and electrons.

The phase velocity in a medium is given by the equation \(v_p = \frac{c}{\sqrt{\epsilon_r}}\), where \(\epsilon_r\) is the relative permittivity of the material. In metals, \(\epsilon_r\) becomes a complex number due to conductivity, significantly affecting the phase velocity. The formula then incorporates this complex permittivity, leading to a modified phase velocity which tends to be slower than in a vacuum.

This slowing down is critical in many applications such as in waveguides, where the reduced phase velocity can change the way electromagnetic waves are transmitted through the material.

Wavelength is another fundamental property of waves and is intimately related to the concept of phase velocity. It is the distance between two consecutive points that are in phase on a wave, such as two adjacent peaks or troughs. The wavelength is inversely proportional to the frequency of the wave; this means that as the frequency increases, the wavelength decreases.

For a wave traveling through a medium, the wavelength can be expressed as \(\lambda = \frac{v_p}{\omega}\), where \(\lambda\) is the wavelength, \(v_p\) is the phase velocity, and \(\omega\) is the angular frequency. When an electromagnetic wave enters a medium like a metal, its wavelength is reduced due to the decrease in phase velocity. In the context of metals, conductivity alters the complex relative permittivity, which in turn changes the wavelength in a way that differs from its propagation in free space. Understanding this change in wavelength is vital for designing and operating devices that use wave propagation, such as antennas and resonant circuits.

Relative permittivity, often represented as \(\epsilon_r\), is a measure of how much a material can 'permit' an electric field compared to the vacuum. In ideal insulators, it is a real number representing the material's dielectric properties. However, for materials with significant conductivity, like metals, \(\epsilon_r\) becomes a complex number to account for energy loss, represented by \(\epsilon_r = \epsilon_r' - i(\frac{\sigma}{\omega\epsilon_0})\).

The real part, \(\epsilon_r'\), relates to how the material stores electric energy, while the imaginary part accounts for the energy dissipation due to the metal’s conductivity. As the frequency of the electromagnetic wave or the conductivity of the metal increases, this imaginary part has a more significant impact, highlighting the material's resistance to the wave's propagation. The concept of complex permittivity is essential in understanding how electromagnetic waves interact with materials and in the design of materials for specific electrical or optical properties.

Conductivity, denoted as \(\sigma\), is a material property that describes its ability to allow the flow of electric charge. In the context of electromagnetic wave propagation, conductivity plays a critical role in how the wave behaves within a conducting medium like a metal.

Materials with high conductivity, such as copper or silver, allow electric charges to move freely, which leads to energy from the electromagnetic waves being rapidly absorbed and transformed into other forms, like heat. This absorption is expressed as the imaginary part of the complex relative permittivity, which directly influences the wave's phase velocity and wavelength within the metal. The interplay between conductivity and frequency affects not just these characteristics, but also the skin depth - the distance into the metal where the wave's amplitude falls to 1/e of its value at the surface. The insight into conductivity's effect on wave propagation is crucial for applications such as the design of microwave circuits, where precise control of wave behavior is necessary.