Question

Consider a left circularly polarized wave in free space that propagates in the forward \(z\) direction. The electric field is given by the appropriate form of Eq. (100). Determine ( \(a\) ) the magnetic field phasor, \(\mathbf{H}_{s} ;(b)\) an expression for the average power density in the wave in \(\mathrm{W} / \mathrm{m}^{2}\) by direct application of Eq. (77).

Short answer

Based on the given information about the electric field of a left circularly polarized wave propagating in the forward z direction, determine the magnetic field phasor and an expression for the average power density in the wave. Solution: 1. The magnetic field phasor is given by: \(\textbf{H} = \frac{1}{\omega\mu}(E_y\textbf{u}_x - jE_x\textbf{u}_z)e^{j(\omega t - kz)}\) 2. The average power density of the wave is: \(S_{av} = \frac{1}{2}(E_xE_y^* - E_yE_x^*)\textbf{u}_y\) W/m²

Step-by-step solution

(Step 1: Recall the Equation (100) for the electric field phasor)

For a left circularly polarized wave, the electric field phasor, \(\textbf{E}\), is given by Equation (100) as: \(\textbf{E}(\textbf{r}, t) = E_x\textbf{u}_x\ e^{j(\omega t - kz)} + jE_y\textbf{u}_y\ e^{j(\omega t - kz)}\) where \(E_x\) and \(E_y\) are the amplitudes of the electric field components, \(k\) is the wave number, \(\omega\) is the angular frequency, and \(\textbf{u}_x\) and \(\textbf{u}_y\) are the unit vectors along the x and y axes.

(Step 2: Find the magnetic field phasor using Faraday's Law)

Faraday's Law relates the electric field to the magnetic field as follows: \(\nabla \times \textbf{E} = -j\omega\mu\textbf{H}\) So, we can write the relation between the magnetic field phasor, \(\textbf{H}\), and the electric field phasor, \(\textbf{E}\), as: \(\textbf{H} = -\frac{1}{j\omega\mu}(\nabla \times \textbf{E})\)

(Step 3: Determine the curl of the electric field phasor)

Calculate the curl \(\nabla \times \textbf{E}\): \(\nabla \times \textbf{E} = \Big(\frac{\partial}{\partial y}(jE_y e^{j(\omega t - kz)}) - \frac{\partial}{\partial z}(E_x e^{j(\omega t - kz)})\Big)\textbf{u}_x - \Big(\frac{\partial}{\partial x}(jE_y e^{j(\omega t - kz)})\Big)\textbf{u}_y + \Big(\frac{\partial}{\partial x}(E_x e^{j(\omega t - kz)})\Big)\textbf{u}_z\) Using the chain rule, we have: \(\nabla \times \textbf{E} = (jE_yjk e^{j(\omega t - kz)})\textbf{u}_x - 0\textbf{u}_y - (E_xjk e^{j(\omega t - kz)})\textbf{u}_z\)

(Step 4: Calculate the magnetic field phasor)

Substitute the curl of the electric field phasor into the expression for the magnetic field phasor: \(\textbf{H} = -\frac{1}{j\omega\mu}(jE_y k\textbf{u}_x\ e^{j(\omega t - kz)} - E_x k\textbf{u}_z\ e^{j(\omega t - kz)})\) Hence, the magnetic field phasor is given by \(\textbf{H} = \frac{1}{\omega\mu}(E_y\textbf{u}_x - jE_x\textbf{u}_z)e^{j(\omega t - kz)}\)

(Step 5: Recall the equation (77) for average power density)

The average power density, \(S_{av}\), is given by Equation (77) as: \(S_{av} = \frac{1}{2}Re(\textbf{E} \times \textbf{H}^*)\)

(Step 6: Calculate the cross product of electric and conjugate magnetic fields)

Find the cross product of \(\textbf{E}\) and \(\textbf{H}^*\): \(\textbf{E} \times \textbf{H}^* = (E_y^* \textbf{u}_y + jE_x^* \textbf{u}_z)e^{-j(\omega t - kz)} \times \frac{1}{\omega\mu}(E_y^*\textbf{u}_x + jE_x^*\textbf{u}_z)e^{-j(\omega t - kz)}\) Using the cross product rule, we have: \(\textbf{E} \times \textbf{H}^* = \frac{1}{\omega\mu}(jE_xE_y^* - jE_yE_x^*)e^{-j2(\omega t - kz)}\textbf{u}_y\)

(Step 7: Find the average power density)

The average power density is given by, \(S_{av} = \frac{1}{2}Re(jE_xE_y^* - jE_yE_x^*)\textbf{u}_y\) Therefore, the average power density of the given left circularly polarized wave is: \(S_{av} = \frac{1}{2}(E_xE_y^* - E_yE_x^*)\textbf{u}_y\) W/m²

Key concepts

Circular Polarization

Circular polarization is a fascinating phenomenon in which the electric field of an electromagnetic wave rotates in a circle as the wave propagates. In contrast to linear polarization, where the electric field oscillates in a straight line, circular polarization involves a rotating field, creating a spiral motion as the wave travels. This can occur either in the right or left direction, leading to right or left circular polarization.
For left circular polarization, specifically, the electric field vectors follow a left-handed helical pattern in the direction of the wave's propagation. This means that if you look along the direction the wave is traveling, the electric field seems to rotate counterclockwise.
This is significant in many practical applications, including satellite communication, where circular polarization aids in reducing signal degradation that stems from atmospheric interference and alignment issues of antennas. Such applications leverage the unique rotational property of circularly polarized waves to maintain signal quality and reliability.

Faraday's Law

Faraday's Law is a cornerstone principle of electromagnetism, detailing how electric fields and magnetic fields interact. The law states that a changing magnetic field induces an electric field, which can be quantified mathematically as the curl of the electric field being equal to the negative time rate of change of the magnetic field.
In the context of electromagnetic waves, Faraday's Law is used to derive the relationship between the electric field (\(\textbf{E}\)) and the magnetic field (\(\textbf{H}\)). It helps to determine the wave's propagation characteristics and provides a basis for calculating the magnetic field phasor from a given electric field phasor.

• The magnetic field can be expressed as \(\textbf{H} = -\frac{1}{j\omega\mu}(abla \times \textbf{E})\), enabling us to find the magnetic field's magnitude and direction.
• It's crucial to understand and apply this law because it bridges electric and magnetic field dynamics, helping us appreciate how energy is transferred through electromagnetic waves.

Power Density

Power density is a measure of power flow per unit area, typically expressed in watts per square meter (W/m²), and is a critical concept in understanding the intensity of electromagnetic waves. It describes how much power is being carried by the wave through a given area and is often used to evaluate how much energy is available for work or conversion in communicative and electronic systems.
To find the power density in an electromagnetic wave, we use the formula:
\[ S_{av} = \frac{1}{2}Re(\textbf{E} \times \textbf{H}^*) \]
Where \(\textbf{E}\) is the electric field, and \(\textbf{H}^*\) is the conjugate of the magnetic field phasor.
This formula arises from the Poynting vector, which represents the directional energy flux (or power per unit area) of an electromagnetic field.

• A higher power density indicates more energy being transmitted, which can be critical for effective communication, especially in wireless and satellite networks.
• Understanding power density allows for evaluating and ensuring that systems are working correctly and efficiently.

Magnetic Field Phasor

A phasor is a representation of a sinusoidal function that simplifies the analysis of linear time-invariant systems, like wave equations in electromagnetics. The magnetic field phasor represents the magnetic field component of an electromagnetic wave in its complex exponential form, simplifying wave calculations and providing insights into the wave's behavior.
In the presence of circular polarization, the magnetic field phasor can be derived using the relation:
\[ \textbf{H} = \frac{1}{\omega\mu}(E_y\textbf{u}_x - jE_x\textbf{u}_z)e^{j(\omega t - kz)} \]
Here, \(\omega\) is the angular frequency, \(\mu\) is the permeability of free space, \(E_y\) and \(E_x\) are the electric field component amplitudes, and \(\textbf{u}_x\) and \(\textbf{u}_z\) are unit vectors.
The phasor form makes it easier to handle the trigonometric functions in the electromagnetic wave equation by working with exponential functions of complex numbers, providing a clearer insight into phase relationships and amplitudes.