Given a \(100-\mathrm{MHz}\) uniform plane wave in a medium known to be a good dielectric, the phasor electric field is \(\mathcal{E}_{s}=4 e^{-0.5 z} e^{-j 20 z} \mathbf{a}_{x} \mathrm{~V} / \mathrm{m}\). Determine (a) \(\epsilon^{\prime} ;(b) \epsilon^{\prime \prime} ;(c) \eta ;(d) \mathbf{H}_{s} ;(e)\langle\mathbf{S}\rangle ;(f)\) the power in watts that is incident on a rectangular surface measuring \(20 \mathrm{~m} \times 30 \mathrm{~m}\) at \(z=10 \mathrm{~m}\).

Permittivity is a fundamental property of materials that quantifies how an electric field affects, and is affected by, a dielectric medium. It is represented by the symbol \(\epsilon\) and is an essential factor in the propagation of electromagnetic waves, determining how much the material opposes the electrical field. In the exercise, permittivity is critical to identify as it allows us to understand the behavior of the electromagnetic wave as it travels through the material.

The real part of the permittivity (\(\epsilon^{\prime}\)) represents the storage capability of the electric energy, while the imaginary part (\(\epsilon^{\prime \prime}\)) is associated with the loss of energy, or dissipation, within the medium, often due to the absorption by the material and conversion to other forms of energy such as heat. The complex nature of permittivity in a good dielectric, as given in the problem, indicates that both storage and loss occur as the wave propagates. These values can be calculated using Maxwell's equations in combination with the wave's attenuation and phase constants.

Intrinsic impedance, denoted as \(\eta\), is an important concept for understanding how electromagnetic waves interact with materials. It represents the ratio of the electric field \(\mathcal{E}_s\) to the magnetic field \(\mathbf{H}_s\) in a medium and gives insight into how easily the wave can propagate through that medium.

Intrinsic impedance is a complex quantity that depends on both the permittivity and the permeability of the medium, with the real part indicating resistance to the wave and the imaginary part relating to the material's reactance. In the solution to our exercise, we calculated \(\eta\) for a good dielectric medium. A low intrinsic impedance suggests that the medium supports stronger electric fields for a given magnetic field, allowing an electromagnetic wave to travel with less resistance. However, as the medium's properties influence \(\eta\), changes in permittivity and permeability can significantly affect the wave's behavior.

The Poynting vector, expressed as \(\langle\mathbf{S}\rangle\), represents the flow of electromagnetic power per unit area in the direction of wave propagation. It is a vector quantity calculated as the cross product of the electric field vector \(\mathcal{E}_s\) and the complex conjugate of the magnetic field vector \(\mathbf{H}_s^{*}\).

In the context of our exercise, determining the average Poynting vector is essential for evaluating the amount of power that an electromagnetic wave carries as it travels through a medium. The Poynting vector is utilized to find the intensity of the wave at a certain distance, and by integrating this intensity over a surface, we can calculate the total power incident on that surface. Understanding the Poynting vector is key for practical applications, such as determining power losses in transmission lines and calculating radiation pressure in antenna design.