Question

A \(10 \mathrm{GHz}\) uniform plane wave propagates in a lossless medium for which \(\epsilon_{r}=8\) and \(\mu_{r}=2 .\) Find \((a) v_{p} ;(b) \beta ;(c) \lambda ;(d) \mathbf{E}_{s} ;(e) \mathbf{H}_{s} ;(f)\langle\mathbf{S}\rangle .\)

Short answer

Based on the given information, we calculated the following parameters: a) Phase velocity (v_p) = 7.5 x 10^7 m/s b) Wavelength (λ) = 7.5 x 10^-3 m c) Phase constant (β) = 838.48 rad/m d) Ratio of electric field magnitude to magnetic field magnitude = 377 Ω However, due to the lack of information about the source or boundary conditions, we cannot determine the exact magnitudes of the electric and magnetic fields, nor the average Poynting vector.

Step-by-step solution

Calculate the phase velocity (v_p)

Using the formula mentioned above, we can calculate the phase velocity: \(v_p = \frac{c}{\sqrt{\epsilon_{r} * \mu_{r}}} = \frac{3 \times 10^{8}}{\sqrt{8 * 2}}\) \(v_p = \frac{3 \times 10^{8}}{\sqrt{16}} = \frac{3 \times 10^{8}}{4} = 7.5 \times 10^7\,\text{m/s}\)

Calculate the wavelength (λ)

Now, let's calculate the wavelength using the formula: \(\lambda = \frac{v_p}{f} = \frac{7.5 \times 10^7}{10 \times 10^9}\) \(\lambda = 7.5 \times 10^{-3}\,\text{m}\)

Calculate the phase constant (β)

Using the formula: \(\beta = \frac{2\pi}{\lambda} = \frac{2\pi}{7.5 \times 10^{-3}}\) \(\beta = 838.48\,\text{rad/m}\)

Calculate the magnitude ratio of the electric field to the magnetic field (|E_s|/|H_s|)

Wave impedance \(Z\) for the lossless medium can be calculated using the formula: \(Z = \sqrt{\frac{\mu_{r} * \mu_0}{\epsilon_{r} * \epsilon_0}}\) \(Z = \sqrt{\frac{2 * 4\pi \times 10^{-7}}{8 * 8.85 \times 10^{-12}}} = 377\,\Omega\) Therefore, the ratio of the electric field magnitude to the magnetic field magnitude is: \(|E_s|/|H_s| = Z = 377\,\Omega\)

Calculate the electric field (E_s) and magnetic field (H_s) vectors

As we only know the ratio of the magnitudes, we can't find the exact values of the electric and magnetic fields, but we can write their relationship as: \(\textbf{E_s} = 377\,\textbf{H_s}\) If we know the situation of the source or boundary conditions, we can determine the exact magnitudes of the electric and magnetic fields.

Calculate the average Poynting vector (⟨S⟩)

The average Poynting vector can be calculated using the formula: \(\langle\textbf{S}\rangle = 0.5 * |E_s| * |H_s|\) However, since we don't know the exact magnitudes of E_s and H_s due to the missing information, we cannot calculate the exact value of the average Poynting vector ⟨S⟩.

Key concepts

Phase Velocity

Phase velocity is a fundamental property of a wave that describes the rate at which the phase of the wave propagates in space. One could imagine this as tracking a specific peak or trough of a wave as it moves along. For a uniform plane wave like the one mentioned in our exercise, the phase velocity, denoted as \(v_p\), can be determined by the formula \(v_p = \frac{c}{\sqrt{\epsilon_r \mu_r}}\), where \(c\) is the speed of light in vacuum, \(\epsilon_r\) is the relative permittivity, and \(\mu_r\) is the relative permeability of the medium.

In our example, plugging in the given values results in a phase velocity of \(7.5 \times 10^7\) meters per second. This tells us how quickly the wave's phase, or in other words, its characteristic points, are moving through the medium.

Wavelength

Wavelength is the spatial period of a wave, meaning the distance over which the wave's shape repeats. It is usually denoted by the Greek letter \(\lambda\). The wavelength of a uniform plane wave can be found using the relationship: \(\lambda = \frac{v_p}{f}\), where \(v_p\) is the phase velocity, as previously discussed, and \(f\) is the frequency of the wave.

For the 10 GHz wave in our exercise, the wavelength calculated comes out to be \(7.5 \times 10^{-3}\) meters. Understanding wavelength is important because it affects the design and functionality of many wireless systems, such as antennas and circuits.

Poynting Vector

The Poynting vector describes the power per unit area carried by an electromagnetic wave. Mathematically, it is represented as a vector product of the electric field \(\textbf{E}\) and the magnetic field \(\textbf{H}\): \(\textbf{S} = \textbf{E} \times \textbf{H}\). In practical terms, this concept tells us the direction and magnitude of the energy flow in an electromagnetic field.

For a uniform plane wave, the average Poynting vector is often considered, denoted as \(\langle\textbf{S}\rangle\), where the averaging accounts for the oscillatory nature of the fields. While we may not have the exact field magnitudes from our exercise, the relationship between the electric and magnetic fields is still essential for understanding power transmission through the medium.

Wave Impedance

Wave impedance, often symbolized as \(Z\), is essentially the ratio of the electric field strength to the magnetic field strength and shows how much resistance an electromagnetic wave experiences as it propagates through a medium. For a lossless medium like in our exercise, the formula \(Z = \sqrt{\frac{\mu_r \mu_0}{\epsilon_r \epsilon_0}}\) is used, where \(\mu_0\) and \(\epsilon_0\) are the permeability and permittivity of free space, respectively.

In the context of the exercise, the wave impedance is calculated to be 377 ohms, which is a standard value for impedance in free space. Wave impedance is crucial in the design and analysis of transmission lines and antennas for impedance matching to minimize reflections.

Phase Constant

The phase constant, represented by the Greek letter \(\beta\), provides a measure of how much the phase of the wave changes per unit length at a given point in space. It is closely related to the wavelength and is given by the formula \(\beta = \frac{2\pi}{\lambda}\). The phase constant is integral in describing the propagation characteristics of a wave and is used in formulas that describe the wave's behavior over distance.

In the exercise, by finding the phase constant we acquire knowledge of the wave’s phase shift per meter, which is useful, for instance, in engineering when determining how a wave will behave over a certain distance or when designing phase-based devices like phase shifters.