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Chapter 11: Problem 12

Describe how the attenuation coefficient of a liquid medium, assumed to be a good conductor, could be determined through measurement of wavelength in the liquid at a known frequency. What restrictions apply? Could this method be used to find the conductivity as well?

Short Answer

Expert verified
Answer: The attenuation coefficient is a parameter that measures how quickly an electromagnetic wave loses its intensity as it travels through a conducting medium. To determine the attenuation coefficient for a liquid medium, we follow these steps: 1. Set up an experimental arrangement with transmitting and receiving antennas. 2. Measure the wavelength in the liquid at a known frequency by recording the maximum and minimum signal intensities. 3. Calculate the angular frequency using the known frequency. 4. Use the obtained values of angular frequency and wavelength to calculate the attenuation coefficient using the equation α = √(ω²μ₀σ/2). However, since conductivity is unknown, rewrite the equation in terms of conductivity: σ = (2α²)/(ω²μ₀). 5. Consider the method's restrictions, such as uniform density and composition, precise positioning, and negligible energy dissipation. 6. Finally, use the calculated values to find the conductivity of the liquid medium.

Step by step solution

01

Attenuation coefficient is a parameter that measures how quickly an electromagnetic wave loses its intensity as it travels through a conducting medium. In a liquid medium, the attenuation coefficient is directly proportional to the conductivity and inversely proportional to the wavelength of the wave. The equation for attenuation coefficient (α) in a good conductor can be given as: α = √(ω²μ₀σ/2) where: ω = angular frequency μ₀ = permeability of free space σ = conductivity #Step 2: Measure the wavelength in the liquid at known frequency#

To perform the measurement, we first need to set up an experimental arrangement. Create an oscillating electric field inside the liquid using an antenna connected to a signal generator. The frequency of oscillation should be known. Use a second antenna (connected to a signal detector) to receive the electromagnetic wave transmitted through the liquid. Move the receiving antenna away from the transmitting antenna along the path of the wave and record the maximum and minimum signal intensities (peaks and troughs). The difference in distance between two successive maximums (or minimums) in the detected signal will correspond to the wavelength (λ) of the wave in the liquid medium. #Step 3: Calculate the angular frequency#
02

As the frequency (f) of oscillation is known, we can find the angular frequency (ω) using the following equation: ω = 2πf #Step 4: Calculate the attenuation coefficient#

With the obtained values of angular frequency (ω) and wavelength (λ), we can determine the attenuation coefficient (α) using the above-defined equation: α = √(ω²μ₀σ/2) However, we only know μ₀, ω, and λ. As σ is unknown, we can re-write the equation in terms of conductivity: σ = (2α²)/(ω²μ₀) #Step 5: Restrictions#
03

There are certain restrictions to this method: 1. The liquid medium should have a fairly uniform density and composition in order to avoid variations in the measured attenuation coefficient. 2. The precise positioning of the antennas and accurate determination of signal intensities is crucial to accurately measure the wavelength. 3. This method works best when the dissipation of energy through heat is negligible. #Step 6: Finding the conductivity#

This method does allow us to find the conductivity of the liquid medium. As mentioned earlier, once we have the attenuation coefficient (α), wavelength (λ), and angular frequency (ω), we can use the equation for conductivity: σ = (2α²)/(ω²μ₀) to find the conductivity of the liquid medium.

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Most popular questions from this chapter

A uniform plane wave in free space has electric field vector given by \(\mathbf{E}_{s}=\) \(10 e^{-j \beta x} \mathbf{a}_{z}+15 e^{-j \beta x} \mathbf{a}_{y} \mathrm{~V} / \mathrm{m} .(a)\) Describe the wave polarization. (b) Find \(\mathbf{H}_{s} .(c)\) Determine the average power density in the wave in \(\mathrm{W} / \mathrm{m}^{2}\).

The phasor magnetic field intensity for a \(400 \mathrm{MHz}\) uniform plane wave propagating in a certain lossless material is \(\left(2 \mathbf{a}_{y}-j 5 \mathbf{a}_{z}\right) e^{-j 25 x} \mathrm{~A} / \mathrm{m}\). Knowing that the maximum amplitude of \(\mathbf{E}\) is \(1500 \mathrm{~V} / \mathrm{m}\), find \(\beta, \eta, \lambda, v_{p}\), \(\epsilon_{r}, \mu_{r}\), and \(\mathcal{H}(x, y, z, t) .\)

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.

The cylindrical shell, \(1 \mathrm{~cm}<\rho<1.2 \mathrm{~cm}\), is composed of a conducting material for which \(\sigma=10^{6} \mathrm{~S} / \mathrm{m}\). The external and internal regions are nonconducting. Let \(H_{\phi}=2000 \mathrm{~A} / \mathrm{m}\) at \(\rho=1.2 \mathrm{~cm}\). Find \((a) \mathbf{H}\) everywhere; \((b) \mathbf{E}\) everywhere; \((c)\langle\mathbf{S}\rangle\) everywhere.

(a) Most microwave ovens operate at \(2.45 \mathrm{GHz}\). Assume that \(\sigma=1.2 \times\) \(10^{6} \mathrm{~S} / \mathrm{m}\) and \(\mu_{r}=500\) for the stainless steel interior, and find the depth of penetration. \((b)\) Let \(E_{s}=50 \angle 0^{\circ} \mathrm{V} / \mathrm{m}\) at the surface of the conductor, and plot a curve of the amplitude of \(E_{s}\) versus the angle of \(E_{s}\) as the field propagates into the stainless steel.
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