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<h2 class="hd hd-2 unit-title">Introduction to Waves in Media</h2>
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We've discussed many aspects of electromagnetic waves—for instance, how they are produced, their behavior in vacuum, and how they reflect off perfect conductors (at normal incidence). </p><p>
Now we begin to consider how EM waves behave in media. In this lesson, we introduce the concept of "dielectric" materials, and discuss how Maxwell's equations govern the properties of electric and magnetic fields in such materials. </p><p>
In the lesson after this one, we will use what we have learned about Maxwell's equations in media to find the appropriate boundary conditions for waves incident on the boundary between two different media. Then, we will be able to examine how electromagnetic waves are transmitted and reflected at such boundaries. </p>
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<h2 class="hd hd-2 unit-title">L33v1: Maxwell's Equation in a Dielectric</h2>
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<h2 class="hd hd-2 unit-title">L33Q1: Maxwell's Equation in Media I - Electric Field</h2>
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Dielectrics are systems consisting of bound charges. The total field in a material is a sum of the field due to bound charges, called the polarization field, and everything that is not the polarization field—the latter is called the displacement field. </p><p>
The polarization field is related to the bound charge in the system (denoted using a subscript [mathjaxinline]b[/mathjaxinline]) by the following: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\nabla \cdot \vec{P} = -\rho _{b}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The term "bound" refers to charges that can move a little but are not free to move anywhere throughout a material. For example, the electrons in a molecule could rearrange (while remaining "bound" to the molecule) so that there is an excess negative charge at one place in the molecule matched by a net positive charge elsewhere. </p><p>
We are familiar with Gauss's law, which relates the electric field due to the total charge (including charges that are either bound or free, with the latter denoted using a subscript [mathjaxinline]f[/mathjaxinline] ) in the following way: </p><table id="a0000000003" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000004"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \nabla \cdot \vec{E}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {\rho }{\epsilon _{0}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \epsilon _{0}\nabla \cdot \vec{E}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \rho _{f} + \rho _{b}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
Then we can define the displacement field [mathjaxinline]\vec{D}[/mathjaxinline], which is ONLY related to the field of the free charge. </p><table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000007"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{D}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \epsilon _{0} \vec{E} + \vec{P}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000008"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \nabla \cdot \vec{D}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \left(\rho _{f} + \rho _{b}\right) -\rho _{b}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \rho _{f}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
Thus, Maxwell's equations in matter which pertain to the electric field are: </p><table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\nabla \cdot \vec{D} = \rho _{f}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\nabla \times \vec{E} = -\dfrac {\partial \vec{B}}{\partial t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Maxwell&#39;s Equation in Media I
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<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]\vec{E}[/mathjaxinline]? </p>
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<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]\vec{D}[/mathjaxinline]? </p>
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<p><b class="bfseries">(Part c)</b> What is [mathjaxinline]\vec{P}[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">L33Q2: Maxwell's Equation in Media II - Magnetic Field</h2>
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Our next goal is to formulate how the Maxwell's equations which pertain to the magnetic field are different in matter. </p><p>
We are familiar with Maxwell-Ampere's law, which relates the magnetic field due to the total current in the system in the following way: </p><table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000003"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \nabla \times \vec{B}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \mu _{0}\left(\vec{J} +\epsilon _{0}\dfrac {\partial \vec{E}}{\partial t} \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000004"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \dfrac {1}{\mu _{0}}\nabla \times \vec{B}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \vec{J}_{f} + \vec{J}_{b} + \epsilon _{0}\dfrac {\partial \vec{E}}{\partial t}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
where we have now divided the current into components due to bound and free charges (denoted using subscripts [mathjaxinline]b[/mathjaxinline] and [mathjaxinline]f[/mathjaxinline], respectively). We will define the "magnetization" to be the field due to the bound currents and polarization field: [mathjaxinline]\nabla \times \vec{M} = \vec{J}_{b} - \dfrac {\partial \vec{P}}{\partial t}[/mathjaxinline]. </p><table id="a0000000005" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000006"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \dfrac {1}{\mu _{0}}\nabla \times \vec{B}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \vec{J}_{f} + \nabla \times \vec{M} + \dfrac {\partial \vec{P}}{\partial t} + \epsilon _{0}\dfrac {\partial \vec{E}}{\partial t}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000007"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \vec{J}_{f} + \nabla \times \vec{M} + \dfrac {\partial \vec{D}}{\partial t}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
Then, we can define the auxiliary magnetic field [mathjaxinline]\vec{H}[/mathjaxinline], which is ONLY related to the field of the free currents. </p><table id="a0000000008" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{H}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {1}{\mu _{0}}\vec{B} - \vec{M}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000010"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \nabla \times \vec{H}[/mathjaxinline]
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[mathjaxinline]\displaystyle = \vec{J}_{f} + \dfrac {\partial \vec{D}}{\partial t}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
Thus, Maxwell's equations in matter which pertain to the magnetic field are: </p><table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\nabla \cdot \vec{B} = 0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\nabla \times \vec{H} = \vec{J}_{f} + \dfrac {\partial \vec{D}}{\partial t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Maxwell&#39;s Equation in Media II
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<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]\vec{B}[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">L33Q3: Maxwell's Equation in Media III - Relative Permittivity</h2>
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The induced polarization field is related to the total field [mathjaxinline]\vec{E}[/mathjaxinline] in the following way: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\vec{P} = \epsilon _{0} \chi _{e} \vec{E}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]\chi _{e}[/mathjaxinline] is the electric susceptibility, which governs the response of the system and is an inherent material property. Note, the field [mathjaxinline]\vec{E}[/mathjaxinline] is the total electric field and is also composed of the polarization field! </p><p>
The displacement field is also related to the total field through a constant of proportionality (for systems with linear response): </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\vec{D}=\epsilon \vec{E}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]\epsilon =\epsilon _{0}\varepsilon _{r}[/mathjaxinline] and [mathjaxinline]\varepsilon _{r}[/mathjaxinline] is a dimensionless constant called the relative permittivity. </p>
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Maxwell&#39;s Equation in Media III
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What is the relation between [mathjaxinline]\chi _{e}[/mathjaxinline] and [mathjaxinline]\varepsilon _{r}[/mathjaxinline]? Express your answer in terms of <code>varepsilon_r</code> for [mathjaxinline]\varepsilon _{r}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\chi _{e} =[/mathjaxinline] </p>
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<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L33Q4: Maxwell's Equation in Media IV - Relative Permeability</h2>
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Similarly, we can also relate the induced magnetization to the auxiliary magnetic field: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\vec{M} = \chi _{m} \vec{H}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]\chi _{m}[/mathjaxinline] is the magnetic susceptibility, which governs the response of the system and is an inherent material property. </p><p>
The total magnetic field is also related to the auxiliary magnetic field through a constant of proportionality (for systems with linear response): </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\vec{B}=\mu \vec{H}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]\mu =\mu _{0}\mu _{r}[/mathjaxinline] and [mathjaxinline]\mu _{r}[/mathjaxinline] is a dimensionless constant called the relative permeability. </p>
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Maxwell&#39;s Equation in Media IV
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What is the relation between [mathjaxinline]\chi _{m}[/mathjaxinline] and [mathjaxinline]\mu _{r}[/mathjaxinline]? Express your answer in terms of <code>mu_r</code> for [mathjaxinline]\mu _{r}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\chi _{m} =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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</tr>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
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<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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