Question

A uniform plane wave has electric field \(\mathbf{E}_{s}=\left(E_{y 0} \mathbf{a}_{y}-E_{z 0} \mathbf{a}_{z}\right) e^{-\alpha x} e^{-j \beta x} \mathrm{~V} / \mathrm{m} .\) The intrinsic impedance of the medium is given as \(\eta=|\eta| e^{j \phi}\), where \(\phi\) is a constant phase. \((a)\) Describe the wave polarization and state the direction of propagation. \((b)\) Find \(\mathbf{H}_{s} \cdot(c)\) Find \(\mathcal{E}(x, t)\) and \(\mathcal{H}(x, t) .(d)\) Find \(<\mathbf{S}>\) in \(\mathrm{W} / \mathrm{m}^{2} \cdot(e)\) Find the time-average power in watts that is intercepted by an antenna of rectangular cross-section, having width \(w\) and height \(h\), suspended parallel to the \(y z\) plane, and at a distance \(d\) from the wave source.

Short answer

Answer: The time-average power intercepted by the antenna is given by \(P = \frac{wh}{2|\eta|}E_{y0}E_{z0}\cos \phi e^{-2\alpha d}\), where \(d\) is the distance from the wave source, \(E_{y0}\) and \(E_{z0}\) are the amplitudes of the electric field along the \(y\) and \(z\) axes, and \(|\eta|\) and \(\phi\) represent the magnitude and phase of the intrinsic impedance of the medium.

Step-by-step solution

1. Wave Polarization and Direction of Propagation

Since the electric field is given only along the \(y\) and \(z\) axes, this indicates that the wave is polarized in the \(yz\) plane. The direction of propagation is found from the exponential term: \(e^{-\alpha x}e^{-j \beta x}\). The propagation is along the \(x\) axis, since the exponential term involves the variable \(x\).

2. Calculate \(\mathbf{H}_{s}\)

To find the magnetic field component, we need to use the intrinsic impedance \(\eta = |\eta| e^{j \phi}\) and the equation relating electric and magnetic fields: \(\mathbf{H}_{s} = \frac{1}{\eta} \times \mathbf{E}_{s}\). We obtain: \(\mathbf{H}_{s} = \frac{1}{|\eta|e^{j \phi}}\begin{pmatrix} 0 \\ E_{y0} \\ -E_{z0} \end{pmatrix}e^{-\alpha x}e^{-j \beta x}\) \(\mathbf{H}_{s} = \frac{1}{|\eta|}\begin{pmatrix} 0 \\ E_{y0}e^{-j \phi} \\ -E_{z0}e^{-j \phi} \end{pmatrix}e^{-\alpha x}e^{-j \beta x}\)

3. Find \(\mathcal{E}(x, t)\) and \(\mathcal{H}(x, t)\)

The electric field and magnetic field components in time-domain are given by: \(\mathcal{E}(x, t) = \mathbf{E}_{s} e^{j\omega t} \) and \(\mathcal{H}(x, t) = \mathbf{H}_{s} e^{j\omega t}\). Therefore, \(\mathcal{E}(x, t) = \left(E_{y 0} \mathbf{a}_{y} - E_{z 0} \mathbf{a}_{z}\right) e^{-\alpha x} e^{j(\omega t - \beta x)}\) \(\mathcal{H}(x, t) = \frac{1}{|\eta|}\begin{pmatrix} 0 \\ E_{y0}e^{-j \phi} \\ -E_{z0}e^{-j \phi} \end{pmatrix} e^{-\alpha x}e^{j (\omega t - \beta x) }\)

4. Calculate the average Poynting vector, \(\)

The average Poynting vector is given by: \(<\mathbf{S}> = \frac{1}{2}Re\{\mathbf{E}_s \times \mathbf{H}_{s}^*\}\). We first compute the cross product: \(\mathbf{E}_{s} \times \mathbf{H}_{s}^* = \begin{pmatrix} 0 \\ E_{y0} \\ -E_{z0} \end{pmatrix} e^{-\alpha x}e^{-j \beta x} \times \left(\frac{1}{|\eta|}\begin{pmatrix} 0 \\ E_{y0}e^{j \phi} \\ -E_{z0}e^{j \phi} \end{pmatrix}e^{-\alpha x}e^{j \beta x}\right)^*\) \(\mathbf{E}_{s} \times \mathbf{H}_{s}^* = \frac{1}{|\eta|}\begin{pmatrix} E_{y0}E_{z0}e^{j \phi} \\ 0 \\ 0 \end{pmatrix}e^{-2\alpha x}\) Now, we take the real part and divide by 2: \(<\mathbf{S}> = \frac{1}{2}Re\{\mathbf{E}_{s} \times \mathbf{H}_{s}^*\} = \frac{1}{2|\eta|}\begin{pmatrix} E_{y0}E_{z0}\cos \phi \\ 0 \\ 0 \end{pmatrix}e^{-2\alpha x}\)

5. Calculate the time-average power intercepted by an antenna

The time-average power intercepted by an antenna with width \(w\) and height \(h\) is given by: \(P = <\)area of antenna\(> \cdot <\mathbf{S}>\). For an antenna with rectangular cross-section suspended parallel to the \(yz\) plane: \(P = wh \cdot <\mathbf{S}> = wh\left(\frac{1}{2|\eta|}E_{y0}E_{z0}\cos \phi e^{-2\alpha x} \right)\) Hence, the time-average power intercepted by the antenna is \(P = \frac{wh}{2|\eta|}E_{y0}E_{z0}\cos \phi e^{-2\alpha d}\), where \(d\) is the distance from the wave source.

Key concepts

Wave Polarization

Polarization of an electromagnetic wave describes the direction in which the electric field vector oscillates as the wave propagates. For the given uniform plane wave problem, the electric field components are along the y and z directions and vary with the variable x. This implies that the wave is polarized in the plane formed by the y and z axes. The absence of an x-component suggests that the polarization is linear. As the wave's electric field has components along both y and z, we can surmise that the wave could possibly have an elliptical or circular polarization if these components were out of phase. However, the problem indicates that the wave is moving along the x-axis. To comprehend wave polarization fully, one must consider the relative phases and magnitudes of the electric field components. This critical understanding of wave polarization is beneficial in various applications such as in designing antennas and understanding light behavior through certain materials.

In essence, wave polarization provides insight into the geometry of the electric field oscillation, which is paramount for studying how waves interact with matter.

Poynting Vector

The Poynting vector is a fundamental concept in electromagnetics that provides the direction and rate at which energy is transferred by an electromagnetic field. It is denoted as S and is calculated as the cross product of the electric field E and the complex conjugate of the magnetic field H*. Mathematically, it's represented as \(<\mathbf{S}> = \frac{1}{2}Re\{\mathbf{E} \times \mathbf{H}^*\}\). In the step-by-step solution, the average Poynting vector for the given uniform plane wave is determined. This Poynting vector is especially crucial because it represents the time-averaged power flow per unit area. For the provided problem, the Poynting vector direction is along the x-axis and diminishes with distance due to the e^{-2\alpha x} term, indicative of the power being absorbed or scattered as the wave propagates through the medium. Understanding the Poynting vector is key to many practical aspects such as determining the power delivered to antennas or even calculating sunlight's energy reaching Earth.

Intrinsic Impedance

Intrinsic impedance, often denoted as \(\eta\), is a property of a medium that describes the relationship between the electric field E and the magnetic field H within an electromagnetic wave propagating through that medium. In the given context, the intrinsic impedance is complex, including a magnitude and a phase component \(\eta = |\eta| e^{j \phi}\). The phase component plays a significant role in determining the phase difference between E and H, which can affect the wave's polarization state and the distribution of energy between the electric and magnetic fields. . High intrinsic impedance might suggest a medium is more 'resistive' to the electric field, whereas low intrinsic impedance implies the medium is less 'resistive' or more 'permissive' for the magnetic field. This concept is crucial in designing transmission lines and understanding wave behavior in different materials - for instance, it plays a role in how radio waves penetrate into different types of soil or rock.

Uniform Plane Wave

A uniform plane wave, often referred to in electromagnetic theory, is a wave whose electric and magnetic field amplitudes are constant across planes perpendicular to the direction of propagation. Plane waves are an idealization that applies to waves traveling over large distances compared to the wavelength, or in far-field regions. For the given problem, the wave's mathematical representation, \(\mathbf{E}_{s} = \left(E_{y 0} \mathbf{a}_{y} - E_{z 0} \mathbf{a}_{z}\right) e^{-\alpha x} e^{-j \beta x} \), indicates a uniform plane wave, as the field amplitudes do not vary with y or z but only with x. It's termed 'uniform' because its magnitude does not change across the yz-plane, reflecting a consistent wavefront. Uniform plane waves are a fundamental concept because their simple form makes them easy to analyze mathematically, and they are a good approximation for many practical wave propagation scenarios, such as in free space or within waveguides.