91影视

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 $$ n=\left(x_{q} K_{m}\right)^{1 / 2} $$

Step by step solution

01

General form of the wave equation

The general form of the wave equation is given by: $$ \nabla^2 \mathbf{F} - \frac{1}{v^2}\frac{\partial^2 \mathbf{F}}{\partial t^2} = 0 $$ where \(\mathbf{F}\) represents a generic field, and \(v\) is the propagation velocity.

02

Substitute the expressions of \(\mathbf{D}\) and \(\mathbf{H}\) in Maxwell's equations

Let's substitute the expression for \(\mathbf{D}\) and \(\mathbf{H}\) in the Maxwell's equations: $$ \begin{aligned} &\nabla \cdot (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})=\rho_{\text {fee }} \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right)=\mathbf{J}_{\text {free }}+\frac{\partial (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t} \end{aligned} $$

03

Consider no free charges and currents

As there are no free charges or currents in the material medium, we set \(蟻_{\text{free}} = 0\) and \(J_{\text {free}} = 0\): $$ \begin{aligned} &\nabla \cdot (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})=0 \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right)=\frac{\partial (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t} \end{aligned} $$

04

Use the curl of the curl identity

We will use the following identity for the curl of the curl: $$ \nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2 \mathbf{F} $$ Applying this identity to the second and fourth equation: $$ \begin{aligned} &-\nabla \times (\nabla \times \mathbf{E}) = \frac{\partial^2 \mathbf{B}}{\partial t^2} \\ &\nabla(\nabla \cdot \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right)) - \nabla^2 \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right) = \frac{\partial^2 (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t^2} \end{aligned} $$

05

Show that the resulting equations satisfy the wave equation

Now, we have the following equations: $$ \begin{aligned} &-\nabla \times (\nabla \times \mathbf{E}) = \frac{\partial^2 \mathbf{B}}{\partial t^2} \\ &\nabla^2 \left(\frac{\mathbf{B}}{\kappa_{m} \mu_{0}}\right) = \frac{\partial^2 (\kappa_{\varepsilon} \epsilon_{0} \mathbf{E})}{\partial t^2} \end{aligned} $$ These equations have the form of the wave equation, as shown in Step 1.

06

Find the propagation velocity and refractive index

From the wave equation, we can find the propagation velocity \(c'\) and the refractive index \(n\) of the medium: $$ c^{\prime}=\frac{1}{\sqrt{\kappa_{\varepsilon} \kappa_{m} \epsilon_{0} \mu_{0}}}=\frac{c}{\sqrt{\kappa_{\varepsilon} \kappa_{m}}} $$ and $$ n=\sqrt{\kappa_{\varepsilon} \kappa_{m}} $$

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polarization

In the context of electromagnetic waves, polarization refers to the orientation of the electric field vector in the wave as it travels. When electromagnetic waves engage with materials, they can induce polarization: charges within molecules may become displaced or polar molecules may align. This phenomenon plays a crucial role in understanding how electromagnetic fields interact with matter.

- **Polarization Types**: Linear, circular, and elliptical polarization are the main types, determined by how the electric field behaves over time. Linear polarization occurs when the electric field oscillates along a single line.
- **Effect of Materials**: Different materials can affect the degree and type of polarization experienced by a wave. Materials with polar molecules can shift the wave's characteristics significantly, affecting the overall electromagnetic field within the material.

By understanding polarization, one can better comprehend how waves are modified by different environments, which is crucial when considering wave propagation through various media.

Magnetic Materials

Magnetic materials have intrinsic properties that allow them to influence magnetic fields. They are characterized by their ability to be magnetized themselves and to change the behavior of magnetic fields passing through them.

- **Types of Magnetism**: Magnetic materials can exhibit various types of magnetism, including ferromagnetism, paramagnetism, and diamagnetism. Ferromagnetic materials like iron can be strongly magnetized, holding their magnetic properties even after external magnetic fields are removed.
- **Magnetization**: When magnetic materials are introduced to a magnetic field, they can become magnetized, aligning magnetic domains within the material towards the external field.

Magnetic materials allow for the existence of effective currents, as their inherent properties modify how magnetic fields behave, making them essential in electromagnetic applications and technologies.

Refractive Index

The refractive index of a medium (\(n\)) is a measure of how much the speed of light is reduced inside a material compared to vacuum. It is fundamental for understanding light propagation and wave behaviors in different media.

- **Definition**: Refractive index is defined as \(n = \frac{c}{v}\), where \(c\) is the speed of light in a vacuum and \(v\) is the speed of light in the medium. Higher refractive indices imply slower light speeds within the medium.
- **Calculation**: In material such as those described in the exercise, the refractive index can be determined using \(n = \sqrt{\kappa_{\varepsilon} \kappa_{m}}\), where \(\kappa_{\varepsilon}\) and \(\kappa_{m}\) are the material's relative permittivity and permeability, respectively.

Understanding the refractive index is crucial for designing optical systems and understanding how materials interact with light.

Wave Equation

The wave equation is a pivotal concept in physics, describing how waves propagate through various media. It provides a mathematical framework for understanding the behavior of waves, ranging from sound waves to electromagnetic waves.

- **General Form**: The wave equation is often expressed as \(abla^2 \mathbf{F} - \frac{1}{v^2}\frac{\partial^2 \mathbf{F}}{\partial t^2} = 0\), where \(\mathbf{F}\) is the wave field and \(v\) is the velocity of propagation.
- **Wave Equation in Electromagnetism**: In the context of the original exercise, it's shown that electromagnetic fields in homogeneous, charge-free materials obey a simple wave equation form. The propagation velocity \(c'\) can be crucially linked to the material properties like permittivity and permeability.

The wave equation captures the essence of how waves travel and is fundamental to a myriad of scientific and engineering applications.