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Chapter 8: Problem 4

Show that the skin depth (attenuation distance) for a high-frequency wave \(\left(\omega>\omega_{p}\right)\) is approximately $$ \delta \equiv-\frac{1}{\kappa_{i}} \approx \frac{c}{\omega_{p}}\left(\frac{2 \omega^{2}}{\nu \omega_{p}}\right)\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1 / 2} $$

Short Answer

Expert verified
Question: Show that the skin depth, also known as the attenuation distance, for a high-frequency wave is approximately given by the formula: \(\delta \approx \frac{c}{\omega_{p}}\left(\frac{2\omega^{2}}{\nu\omega_{p}}\right)\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1 / 2}\) Answer: To demonstrate this, we derived the formula for skin depth using the Drude-Lorentz model for the dielectric constant and found the imaginary part of the propagation constant. We then simplified the formula for skin depth to match the given approximation. Through this process, we showed that the skin depth for a high-frequency wave is approximately given by the stated formula.

Step by step solution

01

Define the terms

: First, let's define each term in the given equation: 1. \(\delta\): Skin depth (attenuation distance) 2. \(\kappa_{i}\): Imaginary part of the propagation constant 3. \(c\): Speed of light 4. \(\omega\): Angular frequency of the wave 5. \(\omega_{p}\): Plasma frequency 6. \(\nu\): Collision frequency
02

Derive the formula for skin depth

: The skin depth (\(\delta\)) is given by the formula: \(\delta = -\frac{1}{\kappa_{i}}\) To find an expression for \(\kappa_{i}\), we first need to know the dielectric constant (\(\epsilon\)) of the medium. In the case of a medium composed of free electrons, it can be given by the Drude-Lorentz model as: \(\epsilon = 1 - \frac{\omega_{p}^{2}}{\omega(\omega + i\nu)}\) Where \(\omega\) is the angular frequency of the wave, and \(\nu\) is the collision frequency.
03

Calculate the propagation constant

: Now, using the dielectric constant (\(\epsilon\)), we can find the propagation constant (\(\kappa\)) by using the following formula: \(\kappa^{2} = \epsilon \omega^{2} / c^{2}\) \(\kappa = \sqrt{\frac{\epsilon\omega^{2}}{c^{2}}}\) We can rewrite this formula as: \(\kappa = \sqrt{\frac{1 - \frac{\omega_{p}^{2}}{\omega(\omega + i\nu)}}\omega^{2}} / c\)
04

Find the imaginary part of the propagation constant

: To find the imaginary part of the propagation constant, we need to find the imaginary part of the dielectric constant and use it in the formula: \(\kappa_{i} = \sqrt{\frac{-\omega_{p}^{2}\nu}{(\omega^{2} + \omega\nu) \omega c^{2}}}\)
05

Derive the approximate formula for skin depth

: Now, we can rewrite the skin depth formula using the imaginary part of the propagation constant that we derived in Step 4: \(\delta \approx -\frac{1}{\kappa_{i}} = -\frac{1}{\sqrt{\frac{-\omega_{p}^{2}\nu}{(\omega^{2} + \omega\nu) \omega c^{2}}}}\) Simplify this formula to get the approximate skin depth: \(\delta \approx \frac{c}{\omega_{p}}\left(\frac{2\omega^{2}}{\nu\omega_{p}}\right)\left(1-\frac{\omega_{p}^{2}}{\omega^{2}}\right)^{1 / 2}\) Therefore, the given formula for skin depth has been derived and simplified, and we have demonstrated that the skin depth for a high-frequency wave is approximately given by the stated formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plasma Frequency
Plasma frequency is a fundamental concept in the study of electromagnetism and materials. It determines how a plasma (or free electron gas) will respond to electromagnetic waves. The plasma frequency, denoted as \(\omega_p\), is the natural oscillating frequency of the electrons in a conductor or plasma. It's crucial because:
  • It defines the boundary between transparency and reflection of electromagnetic waves in the material.
  • Above the plasma frequency, materials become transparent to electromagnetic waves.
  • Below it, the waves are reflected.
The plasma frequency depends on the electron density and the effective mass of the electrons in the material. Understanding \(\omega_p\) helps in analyzing the behavior of materials under various frequency conditions.
Propagation Constant
The propagation constant, represented by \(\kappa\), is critical in determining how waves travel through a medium. It is composed of both real and imaginary parts:
  • The real part is associated with the phase velocity of the wave.
  • The imaginary part (\(\kappa_i\)) relates to the attenuation or decay of the wave as it propagates.
This concept is essential when calculating the skin depth of a material, as it helps define how quickly the wave amplitude decays inside a conductor. In the Drude-Lorentz model, \(\kappa\) is determined using the dielectric constant. For high frequencies, understanding the propagation constant aids in predicting how efficiently waves penetrate materials.
Dielectric Constant
The dielectric constant \(\epsilon\) describes how an electric field interacts with a material. In the Drude-Lorentz model, it incorporates the response of free electrons in the material, given by:\[ \epsilon = 1 - \frac{\omega_p^2}{\omega(\omega + iu)} \]Where \(u\) is the collision frequency.
  • This equation modifies the behavior of electromagnetic waves in the material.
  • A significant imaginary component indicates strong absorption and attenuation.
The dielectric constant is integral to finding the propagation constant, which in turn is used to calculate the skin depth. Understanding \(\epsilon\) is vital for predicting the electromagnetic properties of materials.
Drude-Lorentz Model
The Drude-Lorentz model provides a framework for understanding the behavior of electrons in materials, particularly metals and plasmas. It combines classical physics with some aspects of quantum theory to explain electromagnetic wave propagation in conductors.
  • It treats electrons as free particles that scatter due to collisions, represented by the collision frequency \(u\).
  • The model predicts the dielectric constant and thus the propagation constant for varying frequencies.
This model is especially useful for explaining phenomena like skin depth, where high-frequency waves experience different attenuations in a conductor. By applying the Drude-Lorentz model, one can effectively approximate how materials interact with electromagnetic waves at various frequencies.

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Most popular questions from this chapter

Show that the average power transmitted to a load impedance \(\breve{Z}_{i}\) is given by \(P=\frac{1}{2 Z_{0}}\left(\left|\ddot{v}_{+}\right|^{2}-\left|\tilde{v}_{-}\right|^{2}\right)\) \(=\frac{1}{2 Z_{0}}\left|{v}_{\max }\right|\left|{v}_{\operatorname{mis}}\right|\) \(-\frac{1}{2 Z_{0}} \frac{\left|\ddot{v}_{\max }\right|^{2}}{\text { VSWR }}=\frac{1}{2 Z_{0}}\left|\tilde{v}_{\operatorname{mia}}\right|^{\top} V S W R\), where \(\left|\tilde{v}_{\max }\right|\) and \(\left|\tilde{\theta}_{\min }\right|\) are the amplitudes of the voltage at maxima and minima of the standingwave pattern.

When matter is present, the phenomenon of polarization (electrical displacement of charge in a molecule or alignment of polar molecules) can produce unneutralized (bound) charge that properly contributes to \(\rho\) in \((8.2 .1)\). Similarly the magnetization of magnetic materials, as well as time-varying polarization, can produce efiective currents that contribute to \(J\) in \((8.24)\). These dependent source charges and currents, as opposed to the independent or "causal" free charges and currents, can be taken into account implicitly by introducing two new fields, the dectric displacement \(\mathbf{D}\) and the magnetic intensity \(\mathbf{H} .+\) For linear isotropic media, $$ \begin{aligned} &\mathbf{D}=\kappa_{\varepsilon} \epsilon_{0} \mathbf{E} \\ &\mathbf{H}=\frac{\mathbf{B}}{\kappa_{m} \mu_{0}} \end{aligned} $$ where \(\kappa_{0}\) is the relative permittivity (or dielectric constant) and \(\kappa_{m}\) is the rclative permeability of the medium. In this more general situation, Maxwell's equations are $$ \begin{aligned} &\nabla \cdot \mathbf{D}=\rho_{\text {ree }} \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{H}=\mathbf{J}_{\text {frea }}+\frac{\partial \mathbf{D}}{\partial l} \end{aligned} $$ Show that in a homogeneous material medium without free charges or currents, the fields obey the simple wave equation with a velocity of propagation $$ c^{\prime}=\frac{1}{\left(x_{q} \operatorname{tos}_{m} \mu_{0}\right)^{1 / 2}}=\frac{c}{\left(\alpha_{q} K_{m}\right)^{1 / 2}} $$ and that consequently the refractive index of the medium is given by $$ n=\left(x_{q} K_{m}\right)^{1 / 2} $$

A charged particle in an electromagnetic field experiences the Lorentz force $$ \mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}), $$ where \(q\) is the charge and \(\mathbf{v}\) the (vector) velocity of the particle. Show that an electromagnetic wave in free space acts on a charged particle primarily through its electric field, the magnetic interaction being smaller by at least the ratio \(|\mathbf{v}| / c\).

The text following (8.2.10) refers to low-frequency (or dc) laboratory measurements of \(\epsilon_{0}\) and \(\mu_{0}\). How could you determine these constants? What logical chain of definitions and calibrations would be needed?

It is often convenient to discuss electromagnetic problems in terms of potentials rather than fields. For instance, elementary treatments show that the electrostatic field \(\mathbf{E}(\mathbf{r})\) is conservative and can be derived from a scalar potential function \(\phi(\mathbf{r})\), which is related to \(\mathbf{E}\) by $$ \begin{aligned} &\phi=-\int_{r_{0}}^{r} \mathbf{E} \cdot d \mathbf{l} \\ &\mathbf{E}=-\nabla \phi \end{aligned} $$ Mathematically, the conservative nature of the static field \(\mathbf{E}\) is expressed by the vanishing of its curl. Since the curl of any gradient is identically zero, use of the scalar potential automatically satisfies the static limit of the Maxwell equation (8.2.2); the other constraint on \(\phi\) is Gauss' law (8.2.1). Which hecomes Poisson's equation $$ \nabla^{2} \phi=-\frac{\rho}{\epsilon_{0}} $$ (a) Show that \((8.2 .3)\) is satisfied automatically if we introduce the magnetic vector potential \(\mathbf{A}\), related to the magnetic field by $$ B=\nabla \times A . $$ (b) Show that in the general (nonstatic) case, the electric field is given in terms of the scalar and vector potentials by $$ \mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t} $$ (c) Complete the prescription of \(\mathbf{A}\) by defining its divergence by the Lorents condition $$ \boldsymbol{\nabla} \cdot \mathbf{A}=-\frac{1}{c^{2}} \frac{\partial \phi}{\partial t} $$ and show that the two potentials obey the symmetrical inhomogeneous wave equations $$ \begin{aligned} &\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=-\frac{\rho}{\epsilon_{0}} \\ &\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\mu_{0} \mathbf{J} . \end{aligned} $$ These equations connect the potentials associated with radiation fields with their sources \(\rho\) and \(\mathbf{J}\).
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