Question

A A \(2 \mathrm{GHz}\) uniform plane wave has an amplitude \(E_{y 0}=1.4 \mathrm{kV} / \mathrm{m}\) at \((0,0,0,\), \(t=0\) ) and is propagating in the \(\mathbf{a}_{z}\) direction in a medium where \(\epsilon^{\prime \prime}=1.6 \times\) \(10^{-11} \mathrm{~F} / \mathrm{m}, \epsilon^{\prime}=3.0 \times 10^{-11} \mathrm{~F} / \mathrm{m}\), and \(\mu=2.5 \mu \mathrm{H} / \mathrm{m}\). Find \((\) a \() E_{y}\) at \(P(0,0,1.8 \mathrm{~cm})\) at \(0.2 \mathrm{~ns} ;(b) H_{x}\) at \(P\) at \(0.2 \mathrm{~ns}\).

Short answer

(b) What is the magnetic field component Hx at point P(0,0,1.8 cm) at time 0.2 ns? To find the values of Ey and Hx at the given point and time, follow these steps: 1. Calculate the complex permittivity, complex propagation constant, attenuation constant, phase constant, and intrinsic impedance using the given ε', ε'' and µ values. 2. Use these values to calculate Ey at point P and time t, using the equation: \(E_y = E_{y0} e^{(-\alpha z)}\cos{(\omega t - \beta z)}\). 3. Calculate Hx at point P and time t, using the equation: \(H_x(z, t) = \frac{E_y(z, t)}{\eta}\). After performing these calculations, you will find the values of Ey and Hx at the specified point and time.

Step-by-step solution

Phase 1: Find Attenuation Constant, Phase Constant, Impedance, and Phase Velocity

We will first find the attenuation constant, phase constant, impedance, and phase velocity of the wave using the given values of ε', ε'' and µ. Let's start by finding the complex permittivity ε*, complex propagation constant γ, and intrinsic impedance η. \(\epsilon^{*} = \epsilon' - j\epsilon'' = 3.0 \times 10^{-11} - j1.6 \times 10^{-11}\) F/m We also know the angular frequency, ω, and should calculate it: \(\omega = 2\pi f = 2\pi(2 \times 10^9)\) rad/s Now, let's compute the complex propagation constant, γ, using: \(\gamma = j\omega\sqrt{(\mu\epsilon^*)} = j(2\pi \times 2 \times 10^9)\sqrt{(2.5 \times 10^{-6})(3.0 \times 10^{-11} - j1.6 \times 10^{-11})}\) After calculating the complex propagation constant, we can find the attenuation constant, α, and phase constant, β, using: \(\alpha = \operatorname{Re}(\gamma)\) and \(\beta = \operatorname{Im}(\gamma)\) Next, let's find the intrinsic impedance, η, using: \(\eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon'}}\) Now that we have α, β, and η, we can proceed to phase 2.

Phase 2: Find Ey and Hx at the given point and time

Using the values calculated in Phase 1, we can find Ey and Hx at the given point (0,0,1.8 cm) and time (0.2 ns). (a) Ey at point P and time t can be calculated using: \(E_y = E_{y0} e^{(-\alpha z)}\cos{(\omega t - \beta z)}\) Substitute the known values: \(E_{y0} = 1.4 \times 10^3\) V/m, z = 1.8 × 10⁻² m, and t = 0.2 × 10⁻⁹ s, and use α, β, and ω derived in Phase 1. \(E_{y} = (1.4 \times 10^3) e^{(-\alpha \times 1.8 \times 10^{-2})}\cos{((2\pi \times 2 \times 10^9)(0.2 \times 10^{-9}) - \beta \times 1.8 \times 10^{-2})}\) Solve for Ey. (b) Now, let's find the magnetic field component Hx at point P at time 0.2 ns. Using the given values and the values derived in Phase 1, Hx can be calculated using: \(H_x(z, t) = \frac{E_y(z, t)}{\eta}\) Substitute the derived value of Ey and η into the equation: \(H_{x}(1.8 \times 10^{-2}, 0.2 \times 10^{-9}) = \frac{E_{y}(1.8 \times 10^{-2}, 0.2 \times 10^{-9})}{\eta}\) Solve for Hx. By following these calculations step-by-step, you can find the values of Ey and Hx at the specified point and time.

Key concepts

Attenuation Constant

When electromagnetic waves travel through a medium, they gradually lose their intensity. This reduction in strength is quantified by the **attenuation constant** (\(\alpha\)). It determines how much the amplitude of the wave decreases as it propagates through the medium.
In essence, the higher the attenuation constant, the faster the wave's energy diminishes.

• The attenuation constant is a part of the complex propagation constant \(\gamma\), calculated as \(\alpha = \operatorname{Re}(\gamma)\).
• It is affected by the properties of the medium, such as conductivity and permittivity.
• In practical terms, understanding \(\alpha\) helps engineers design systems with minimal signal loss.

Recognizing its role is crucial to optimizing communication technologies and systems.

Phase Constant

The **phase constant** (\(\beta\)) is another critical part of wave propagation. It measures how quickly the phase of the wave changes with distance. This constant is vital in determining the oscillatory nature of the wave as it travels.

• Defined as \(\beta = \operatorname{Im}(\gamma)\), it is a component of the complex propagation constant \(\gamma\).
• \(\beta\) influences the wavelength and the speed at which the wavefronts move.
• High values of \(\beta\) mean that the wave oscillates rapidly, which is a factor in phase velocity calculations.

Understanding \(\beta\) is essential for accurately predicting wave behavior in various media.

Intrinsic Impedance

The **intrinsic impedance** (\(\eta\)) of a medium is a measure of how much a wave resists the electrical and magnetic fields as it propagates. It's critical for assessing how the wave interacts with the medium.

• Intrinsic impedance is calculated using \(\eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon'}}\), where \(\mu\) is permeability, \(\epsilon'\) is permittivity, and \(\sigma\) is conductivity.
• \(\eta\) determines the ratio between electric and magnetic field intensities.
• It plays a key role in calculating the transmission and reflection coefficients at boundaries.

Having a grasp of \(\eta\) helps in matching systems to characterized media properties, ensuring effective signal transmission.

Plane Wave Propagation

**Plane wave propagation** describes how uniform electromagnetic waves move through space. These waves maintain a consistent amplitude and phase across planes perpendicular to the direction of travel.

• Plane waves are often used as an ideal model for studying wave behavior in homogeneous media.
• Their direction of propagation is indicated with a vector, often represented as \(\mathbf{a}_z\).
• The wave equation for plane waves involves quantities like attenuation and phase constants to predict behavior over time and space.

Understanding plane waves aids in simplifying the analysis of complex wave phenomena, making them essential in fields such as antennas, optics, and wireless communication.