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Chapter 8: Problem 1

Use the general form of Maxwell's equations (8.2.17) to (8.2.20), together with Gauss' and Stokes' theorems, to obtain the corresponding integral equations for a material medium, $$ \begin{aligned} &\oint_{s} \mathbf{D} \cdot d \mathbf{S}=q_{t r e t} \\ &\oint_{L} \mathbf{E} \cdot d \mathrm{l}=-\frac{d \Phi_{m}}{d t} \\ &\oint_{S} \mathbf{B} \cdot d \mathbf{S}=0 \\ &\oint_{L} \mathbf{H} \cdot d \mathbf{l}=I_{\text {froe }}+\frac{d \Phi_{\bullet}}{d t} \end{aligned} $$ where \(\Phi_{m}=\int \mathrm{B} \cdot d \mathrm{~S}\) and \(\Phi_{*}=\int \mathrm{D} \cdot d \mathrm{~S}\) are the magnetic and electric fluxes linking the closed line \(L\). These equations are generalizations of \((8.2 .11)\) to \((8.2 .14)\).

Short Answer

Expert verified
Question: Using Gauss' and Stokes' theorems, derive the integral equations for a material medium from Maxwell's equations (8.2.17 to 8.2.20). Answer: Following the steps outlined in the provided solution, we can derive the integral equations for a material medium as: 1. $$\oint_{S} \mathbf{D}\cdot d\mathbf{S} = q_{\text {tret}}$$ 2. $$ \oint_{L} \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_{m}}{dt} $$ 3. $$\oint_{S} \mathbf{B} \cdot d\mathbf{S} = 0$$ 4. $$ \oint_{L} \mathbf{H} \cdot d\mathbf{l} = I_{\text {froe}} + \frac{d\Phi_{\bullet}}{dt} $$

Step by step solution

01

Recall Maxwell's equations and theorems

First, we need to recall the general form of Maxwell's equations (8.2.17 to 8.2.20) which are differential equations, and also Gauss' theorem and Stokes' theorem, which are used to convert these differential equations into integral equations.
02

Apply Gauss' theorem to the first equation

We will now apply Gauss' theorem to Maxwell's first equation. Gauss' theorem states that the volume integral of the divergence of a vector field is equal to the surface integral of the vector field over the closed surface surrounding the volume. Mathematically, it is given by: $$\int_{V} \nabla \cdot \mathbf{D} dV = \oint_{S} \mathbf{D} \cdot d\mathbf{S}$$ Applying Gauss's equation to the first we get: $$\oint_{S} \mathbf{D}\cdot d\mathbf{S} = q_{\text {tret}}$$
03

Apply Stokes' theorem to the second equation

We will now apply Stokes' theorem to Maxwell's second equation. Stokes' theorem states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of the vector field over the surface enclosed by the loop. Mathematically, it is given by: $$\oint_{L} \mathbf{E} \cdot d\mathbf{l} = \int_S \nabla \times \mathbf{E} \cdot d\mathbf{S}$$ Applying Stokes's equation to the second we get: $$ \oint_{L} \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_{m}}{dt} $$
04

Apply Gauss' theorem to the third equation

Repeat Step 2, this time applying Gauss' theorem to Maxwell's third equation. We have: $$\oint_{S} \mathbf{B} \cdot d\mathbf{S} = 0$$
05

Apply Stokes' theorem to the fourth equation

Repeat Step 3, this time applying Stokes' theorem to Maxwell's fourth equation. We have: $$ \oint_{L} \mathbf{H} \cdot d\mathbf{l} = I_{\text {froe}} + \frac{d\Phi_{\bullet}}{dt} $$ After all the steps, we have obtained the integral equations for a material medium, which are generalizations of Maxwell's equations (8.2.11 to 8.2.14).

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Most popular questions from this chapter

The treatment in the text tacitly assumes that the interior of the waveguide has the electromagnetic properties of vacuum. Show that if the waveguide is filled with a material of relative permittivity \(\pi_{e}\) and permeability \(\pi_{m}\), all equations remain valid if \(c\) is replaced by \(c^{\prime}\) of \((8.2 .21)\) and \(\lambda_{9}\) in \((8.7 .20)\) is replaced by \(\lambda^{\prime}=2 \pi c^{\prime} / \omega\).

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

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 \(\mathbf{J}\).

Use Gauss' and Stokes' theorems (Appendix A) to convert Maxwell's differential equations for vacuum, \((82.1)\) to \((8.2 .4)\), to their integral form $$ \begin{aligned} &\oint_{S} \mathbf{E} \cdot d \mathbf{S}=\frac{q}{\epsilon_{0}} \\ &\oint_{L} \mathbf{E} \cdot d \mathbf{l}=-\frac{d \Phi_{m}}{d t} \\ &\oint_{S} \mathbf{B} \cdot d \mathbf{S}=0 \\ &\oint_{L} \mathbf{B} \cdot d \mathbf{l}=\mu_{0} I+\mu_{0} \frac{d \Phi_{*}}{d t} \end{aligned} $$ † See Sec. \(5.4\) and Prob. 8.2.4. where the closed surface \(S\) contains the net charge \(q\) and the closed line (loop) \(L\) is linked by the net current \(I\), the magnetic flux \(\Phi_{m}=\int \mathbf{B} \cdot d \mathbf{S}\), and the electric flux \(\Phi_{e}=\epsilon_{0} \int \mathbf{E} \cdot d \mathbf{S}\). Note: The corresponding equations for a general electromagnetic medium are developed in Prob. \(8.6 .1 .\)

Consider an E-field line of force, i.e., a continuous line everywhere parallel to the local direction of \(\mathbf{E}\), deflected at the boundary between two uniform media. Show that the exit line of force lies in the plane determined by the entrance line and the normal to the boundary surface and that the angles of incidence \(\theta_{1}\) and exit \(\theta_{2}\), measured with respect to the normal, are related by the Snell's law equation $$ \frac{1}{\kappa_{\theta 1}} \tan \theta_{1}=\frac{1}{K_{\theta 2}} \tan \theta_{2} \text {. } $$ What are the corresponding equations for \(\mathbf{B}, \mathbf{D}\), and \(\mathbf{H}\) ?
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