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<h2 class="label"><span class="chapnum"><strong>1</strong></span></h2>
<h2 class="h2a"><a id="page_xii"></a><a id="page_1"></a><a id="ch1"></a>The electromagnetic field</h2>
<p class="noindent">In this introductory chapter some basic relations and concepts of the classic electromagnetic field are briefly reviewed for the sake of easy reference and to make clear the significance of the symbols.</p>
<h3 class="h3"><a id="ch1.1"></a><strong>1.1 Maxwell’s Equations in Simple Media</strong></h3>
<p class="noindent">In the mks, or Giorgi, system of units, which we shall use throughout this book, Maxwell’s field equations<sup><a id="ifn1"></a><a href="9780486656786_13_foot.xhtml#fn1">1</a></sup> are</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.1"></a><img src="images/img_0001_0001.jpg" width="582" height="44" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.2"></a><img src="images/img_0001_0002.jpg" width="581" height="45" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.3"></a><img src="images/img_0001_0003.jpg" width="582" height="30" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.4"></a><img src="images/img_0001_0004.jpg" width="580" height="25" alt="image" /></p>
<table cellspacing="0" cellpadding="0" width="100%">
<tr>
<td valign="top"><p class="table">where <strong>E</strong>(<strong>r</strong>, <em>t</em>) =</p></td>
<td valign="top"><p class="table1">electric field intensity vector, volts per meter</p></td>
</tr>
<tr>
<td valign="top"><p class="table"><strong>H</strong>(<strong>r</strong>, <em>t</em>) =</p></td>
<td valign="top"><p class="table1">magnetic field intensity vector, amperes per meter</p></td>
</tr>
<tr>
<td valign="top"><p class="table"><a id="page_2"></a><strong>D</strong>(<strong>r</strong>,<em>t</em>) =</p></td>
<td valign="top"><p class="table1">electric displacement vector, coulombs per meter<sup>2</sup></p></td>
</tr>
<tr>
<td valign="top"><p class="table"><strong>B</strong>(<strong>r</strong>,<em>t</em>) =</p></td>
<td valign="top"><p class="table1">magnetic induction vector, webers per meter<sup>2</sup></p></td>
</tr>
<tr>
<td valign="top"><p class="table"><strong>J</strong>(<strong>r</strong>,<em>t</em>) =</p></td>
<td valign="top"><p class="table1">current-density vector, amperes per meter<sup>2</sup></p></td>
</tr>
<tr>
<td valign="top"><p class="table"><em>ρ</em>(<strong>r</strong>,<em>t</em>) =</p></td>
<td valign="top"><p class="table1">volume density of charge, coulombs per meter<sup>3</sup></p></td>
</tr>
<tr>
<td valign="top"><p class="table"><strong>r</strong> =</p></td>
<td valign="top"><p class="table1">position vector, meters</p></td>
</tr>
<tr>
<td valign="top"><p class="table"><em>t</em> =</p></td>
<td valign="top"><p class="table1">time, seconds</p></td>
</tr>
</table>
<p class="noindent">The equation of continuity</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.5"></a><img src="images/img_0002_0001.jpg" width="687" height="48" alt="image" /></p>
<p class="noindent">which expresses the conservation of charge is a corollary of <a href="9780486656786_06_ch1.xhtml#eq_1.4">Eq. (4)</a> and the divergence of <a href="9780486656786_06_ch1.xhtml#eq_1.2">Eq. (2)</a>.</p>
<p class="indent">The quantities <strong>E</strong>(<strong>r</strong>,<em>t</em>) and <strong>B</strong>(<strong>r</strong>,<em>t</em>) are defined in a given frame of reference by the density of force <strong>f</strong>(<strong>r</strong>,<em>t</em>) in newtons per meter<sup>3</sup> acting on the charge and current density in accord with the Lorentz force equation</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.6"></a><img src="images/img_0002_0002.jpg" width="687" height="30" alt="image" /></p>
<p class="noindent">In turn <strong>D</strong>(<strong>r</strong>,<em>t</em>) and <strong>H</strong>(<strong>r</strong>,<em>t</em>) are related respectively to <strong>E</strong>(<strong>r</strong>,<em>t</em>) and <strong>B</strong>(<strong>r</strong>,<em>t</em>) by constitutive parameters which characterize the electromagnetic nature of the material medium involved. For a homogeneous isotropic linear medium, viz., a “simple” medium, the constitutive relations are</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.7"></a><img src="images/img_0002_0003.jpg" width="686" height="27" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.8"></a><img src="images/img_0002_0004.jpg" width="686" height="51" alt="image" /></p>
<p class="noindent">where the constitutive parameters <img class="mid" src="images/img_epsi.jpg" width="9" height="12" alt="image" /> in farads per meter and <em>μ</em> in henrys per meter are respectively the dielectric constant and the permeability of the medium.</p>
<p class="indent">In simple media, Maxwell’s equations reduce to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.9"></a><img src="images/img_0002_0005.jpg" width="685" height="46" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.10"></a><img src="images/img_0002_0006.jpg" width="685" height="45" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="page_3"></a><a id="eq_1.11"></a><img src="images/img_0003_0001.jpg" width="686" height="34" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.12"></a><img src="images/img_0003_0002.jpg" width="685" height="48" alt="image" /></p>
<p class="noindent">The curl of <a href="9780486656786_06_ch1.xhtml#eq_1.9">Eq. (9)</a> taken simultaneously with <a href="9780486656786_06_ch1.xhtml#eq_1.10">Eq. (10)</a> leads to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.13"></a><img src="images/img_0003_0003.jpg" width="682" height="47" alt="image" /></p>
<p class="noindent">Alternatively, the curl of <a href="9780486656786_06_ch1.xhtml#eq_1.10">Eq. (10)</a> with the aid of <a href="9780486656786_06_ch1.xhtml#eq_1.9">Eq. (9)</a> yields</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.14"></a><img src="images/img_0003_0004.jpg" width="686" height="45" alt="image" /></p>
<p class="noindent">The vector wave <a href="9780486656786_06_ch1.xhtml#eq_1.13">equations (13)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.14">(14)</a> serve to determine <strong>E</strong>(<strong>r</strong>,<em>t</em>) and <strong>H</strong>(<strong>r</strong>,<em>t</em>) respectively when the source quantity <strong>J</strong>(<strong>r</strong>,<em>t</em>) is specified and when the field quantities are required to satisfy certain prescribed boundary and radiation conditions. Thus it is seen that in the case of simple media, Maxweir’s equations determine the electromagnetic field when the current density <strong>J</strong>(<strong>r</strong>,<em>t</em>) is a given quantity. Moreover, this is true for any linear medium, i.e., any medium for which the relations connecting <strong>B</strong>(<strong>r</strong>,<em>t</em>) to <strong>H</strong>(<strong>r</strong>,<em>t</em>) and <strong>D</strong>(<strong>r</strong>,<em>t</em>) to <strong>E</strong>(<strong>r</strong>,<em>t</em>) are linear, be it anisotropic, inhomogeneous, or both.</p>
<p class="indent">To form a complete field theory an additional relation connecting <strong>J</strong>(<strong>r</strong>,<em>t</em>) to the field quantities is necessary. If <strong>J</strong>(<strong>r</strong>,<em>t</em>) is purely an ohmic conduction current in a medium of conductivity <em>σ</em> in mhos per meter, then Ohm’s law</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.15"></a><img src="images/img_0003_0005.jpg" width="687" height="29" alt="image" /></p>
<p class="noindent">applies and provides the necessary relation. On the other hand, if <strong>J</strong>(<strong>r</strong>,<em>t</em>) is purely a convection current density, given by</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.16"></a><img src="images/img_0003_0006.jpg" width="687" height="28" alt="image" /></p>
<p class="noindent">where <strong>v</strong>(<strong>r</strong>,<em>t</em>) is the velocity of the charge density in meters per second, the necessary relation is one that connects the velocity with the field. To find such a connection in the case where the convection current is made up of charge carriers in motion (discrete case), we must calculate <a id="page_4"></a>the total force <strong>F</strong>(<strong>r</strong>,<em>t</em>) acting on a charge carrier by first integrating the force density <strong>f</strong>(<strong>r</strong>,<em>t</em>) throughout the volume occupied by the carrier, i.e.,</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.17"></a><img src="images/img_0004_0001.jpg" width="686" height="29" alt="image" /></p>
<p class="noindent">where <em>q</em> is the total charge, and then equating this force to the force of inertia in accord with Newton’s law of motion</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.18"></a><img src="images/img_0004_0002.jpg" width="686" height="45" alt="image" /></p>
<p class="noindent">where <em>m</em> is the mass of the charge carrier in kilograms. In the case where the convection current is a charged fluid in motion (continuous case), the force density <strong>f</strong>(<strong>r</strong>,<em>t</em>) is entered directly into the equation of motion of the fluid.</p>
<p class="indent">Because Maxwell’s equations in simple media form a linear system, no generality is lost by considering the “monochromatic” or “steady” state, in which all quantities are simply periodic in time. Indeed, by Fourier’s theorem, any linear field of arbitrary time dependence can be synthesized from a knowledge of the monochromatic field. To reduce the system to the monochromatic state we choose exp (–<em>iωt</em>) for the time dependence and adopt the convention</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.19"></a><img src="images/img_0004_0003.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">where <em>C</em>(<strong>r</strong>,<em>t</em>) is any real function of space and time, <em>C<sub>ω</sub></em>(<strong>r</strong>) is the concomitant complex function of position (sometimes called a “phasor”), which depends parametrically on the frequency <em>f</em>(= <em>ω</em>/2<em>π</em>) in cycles per second, and Re is shorthand for “real part of.” Application of this convention to the quantities entering the field <a href="9780486656786_06_ch1.xhtml#eq_1.1">equations (1)</a> through <a href="9780486656786_06_ch1.xhtml#eq_1.4">(4)</a> yields the monochromatic form of Maxwell’s equations:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.20"></a><img src="images/img_0004_0004.jpg" width="685" height="25" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.21"></a><img src="images/img_0004_0005.jpg" width="685" height="24" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.22"></a><img src="images/img_0004_0006.jpg" width="685" height="24" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.23"></a><img src="images/img_0004_0007.jpg" width="686" height="24" alt="image" /></p>
<p class="noindent"><a id="page_5"></a>In a similar manner the monochromatic form of the equation of continuity</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.24"></a><img src="images/img_0005_0001.jpg" width="686" height="23" alt="image" /></p>
<p class="noindent">is derived from <a href="9780486656786_06_ch1.xhtml#eq_1.5">Eq. (5)</a>.</p>
<p class="indent">The divergence of <a href="9780486656786_06_ch1.xhtml#eq_1.20">Eq. (20)</a> yields <a href="9780486656786_06_ch1.xhtml#eq_1.22">Eq. (22)</a>, and the divergence of <a href="9780486656786_06_ch1.xhtml#eq_1.21">Eq. (21)</a> in conjunction with <a href="9780486656786_06_ch1.xhtml#eq_1.24">Eq. (24)</a> leads to <a href="9780486656786_06_ch1.xhtml#eq_1.23">Eq. (23)</a>. We infer from this that of the four monochromatic Maxwell equations only the two curl relations are independent. Since there are only two independent vectorial equations, viz., <a href="9780486656786_06_ch1.xhtml#eq_1.20">Eqs. (20)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.21">(21)</a>, for the determination of the five vectorial quantities <strong>E</strong><sub><em>ω</em></sub>(<strong>r</strong>), <strong>H</strong><sub><em>ω</em></sub>(<strong>r</strong>), <strong>D</strong><sub><em>ω</em></sub>(<strong>r</strong>), <strong>B</strong><sub><em>ω</em></sub>(<strong>r</strong>), and <strong>J</strong><sub><em>ω</em></sub>(<strong>r</strong>), the monochromatic Maxwell equations form an under determined system of first-order differential equations. If the system is to be made determinate, linear constitutive relations involving the constitutive parameters must be invoked. One way of doing this is first to assume that in a given medium the linear relations <strong>B</strong><sub><em>ω</em></sub>(<strong>r</strong>) = <em>α</em><strong>H</strong><sub><em>ω</em></sub>(<strong>r</strong>), <strong>D</strong><sub><em>ω</em></sub>(<strong>r</strong>) = <em>β</em><strong>E</strong><sub><em>ω</em></sub>(<strong>r</strong>), and <strong>J</strong><sub><em>ω</em></sub>(<strong>r</strong>) = <em>γ</em><strong>E</strong><sub><em>ω</em></sub>(<strong>r</strong>) are valid, then to note that with this assumption the system is determinate and possesses solutions involving the unknown constants <em>α</em>, <em>β</em>, and <em>γ</em>, and finally to choose the values of these constants so that the mathematical solutions agree with the observations of experiment. These appropriately chosen values are said to be the monochromatic permeability <em>μ<sub>ω</sub></em>, dielectric constant <img class="mid" src="images/img_epsi.jpg" width="9" height="12" alt="image" /><sub><em>ω</em></sub>, and conductivity <em>σ<sub>ω</sub></em> of the medium. Another way of defining the constitutive parameters is to resort to the microscopic point of view, according to which the entire system consists of free and bound charges interacting with the two vector fields <strong>E</strong><sub><em>ω</em></sub>(<strong>r</strong>) and <strong>B</strong><sub><em>ω</em></sub>(<strong>r</strong>) only. For simple media the constitutive relations are</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.25"></a><img src="images/img_0005_0002.jpg" width="687" height="22" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.26"></a><img src="images/img_0005_0003.jpg" width="686" height="24" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.27"></a><img src="images/img_0005_0004.jpg" width="688" height="25" alt="image" /></p>
<p class="noindent">In media showing microscopic inertial or relaxation effects, one or more of these parameters may be complex frequency-dependent quantities.</p>
<p class="indent">For the sake of notational simplicity, in most of what follows we shall drop the subscript <em>ω</em> and omit the argument <strong>r</strong> in the monochromatic <a id="page_6"></a>case, and we shall suppress the argument <strong>r</strong> in the time-dependent case. For example, <strong>E</strong>(<em>t</em>) will mean <strong>E</strong>(<strong>r</strong>,<em>t</em>) and <strong>E</strong> will mean <strong>E</strong><sub><em>ω</em></sub>(<strong>r</strong>). Accordingly, the monochromatic form of Maxwell’s equations in simple media is</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.28"></a><img src="images/img_0006_0001.jpg" width="688" height="24" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.29"></a><img src="images/img_0006_0002.jpg" width="688" height="23" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.30"></a><img src="images/img_0006_0003.jpg" width="688" height="23" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.31"></a><img src="images/img_0006_0004.jpg" width="688" height="44" alt="image" /></p>
<h3 class="h3"><a id="ch1.2"></a><strong>1.2 Duality</strong></h3>
<p class="noindent">In a region free of current (<strong>J</strong> = 0), Maxwell’s equations possess a certain duality in <strong>E</strong> and <strong>H</strong>. By this we mean that if two new vectors <strong>E′</strong> and <strong>H′</strong> are defined by</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.32"></a><img src="images/img_0006_0005.jpg" width="688" height="51" alt="image" /></p>
<p class="noindent">then as a consequence of Maxwell’s equations (source-free)</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.33"></a><img src="images/img_0006_0006.jpg" width="687" height="60" alt="image" /></p>
<p class="noindent">it follows that <strong>E′</strong> and <strong>H′</strong> likewise satisfy Maxwell’s equations (source-free)</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.34"></a><img src="images/img_0006_0007.jpg" width="686" height="62" alt="image" /></p>
<p class="noindent">and thereby constitute an electromagnetic field <strong>E′</strong>, <strong>H′</strong> which is the “dual” of the original field.</p>
<p class="indent">This duality can be extended to regions containing current by employing the mathematical artifice of magnetic charge and magnetic <a id="page_7"></a>current.<sup><a id="ifn2"></a><a href="9780486656786_13_foot.xhtml#fn2">2</a></sup> In such regions Maxwell’s equations are</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.35"></a><img src="images/img_0007_0001.jpg" width="686" height="84" alt="image" /></p>
<p class="noindent">and under the transformation <a href="9780486656786_06_ch1.xhtml#eq_1.32">(32)</a> they become</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.36"></a><img src="images/img_0007_0002.jpg" width="686" height="125" alt="image" /></p>
<p class="noindent">Formally these relations are Maxwell’s equations for an electromagnetic field <strong>E′</strong>, <strong>H′</strong> produced by the “magnetic current” <img class="mid" src="images/img_0007_0003.jpg" width="99" height="26" alt="image" /> and the “magnetic charge” <img class="mid" src="images/img_0007_0004.jpg" width="98" height="28" alt="image" />. These considerations suggest that complete duality is achieved by generalizing Maxwell’s equations as follows:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.37"></a><img src="images/img_0007_0005.jpg" width="684" height="86" alt="image" /></p>
<p class="noindent">where <strong>J</strong><sub><em>m</em></sub> and <em>ρ<sub>m</sub></em> are the magnetic current and charge densities. Indeed, under the duality transformation</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.38"></a><img src="images/img_0007_0006.jpg" width="684" height="118" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.39"></a><img src="images/img_0007_0007.jpg" width="684" height="89" alt="image" /></p>
<p class="noindent"><a id="page_8"></a>Thus to every electromagnetic field <strong>E</strong>, <strong>H</strong> produced by electric current <strong>J</strong> there is a dual field <strong>H′, E′</strong> produced by a fictive magnetic current <strong>J′</strong><sub><em>m</em></sub>.</p>
<h3 class="h3"><a id="ch1.3"></a><strong>1.3 Boundary Conditions</strong></h3>
<p class="noindent">The electromagnetic field at a point on one side of a smooth interface between two simple media, 1 and 2, is related to the field at the neighboring point on the opposite side of the interface by boundary conditions which are direct consequences of Maxwell’s equations.</p>
<p class="indent">We denote by <strong>n</strong> a unit vector which is normal to the interface and directed from medium 1 into medium 2, and we distinguish quantities in medium 1 from those in medium 2 by labeling them with the subscripts 1 and 2 respectively. From an application of Gauss’ divergence theorem to Maxwell’s divergence equations, ∇ · <strong>B</strong> = <em>ρ<sub>m</sub></em> and ∇ · <strong>D</strong> = <em>ρ</em>, it follows that the normal components of <strong>B</strong> and <strong>D</strong> are respectively discontinuous by an amount equal to the magnetic surface-charge density <em>η<sub>m</sub></em> and the electric surface-charge density <em>η</em> in coulombs per meter<sup>2</sup>:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.40"></a><img src="images/img_0008_0001.jpg" width="686" height="22" alt="image" /></p>
<p class="noindent">From an application of Stokes’ theorem to Maxwell’s curl equations, ∇ × <strong>E</strong> = –<strong>J</strong><sub><em>m</em></sub> + <em>iωμ</em><strong>H</strong> and ∇ – <strong>H</strong> = <strong>J</strong> –<em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" /><strong>E</strong>, it follows that the tangential components of <strong>E</strong> and <strong>H</strong> are respectively discontinuous by an amount equal to the magnetic surface-current density <strong>K</strong><sub><em>m</em></sub> and the electric surface-current density <strong>K</strong> in amperes per meter:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.41"></a><img src="images/img_0008_0002.jpg" width="686" height="23" alt="image" /></p>
<p class="noindent">In these relations <strong>K</strong><sub><em>m</em></sub> and <strong>K</strong> are magnetic and electric “current sheets” carrying charge densities <em>η<sub>m</sub></em> and <em>η</em> respectively. Such current sheets are mathematical abstractions which can be simulated by limiting forms of electromagnetic objects. For example, if medium 1 is a perfect conductor and medium 2 a perfect dielectric, i.e., if <em>σ</em><sub>1</sub> = ∞ and σ<sub>2</sub> = 0, then all the field vectors in medium 1 as well as <em>η<sub>m</sub></em> and <strong>K</strong><sub><em>m</em></sub> vanish identically and the boundary conditions reduce to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.42"></a><img src="images/img_0008_0003.jpg" width="686" height="23" alt="image" /></p>
<p class="noindent"><a id="page_9"></a>A surface having these boundary conditions is said to be an “electric wall.” By duality a surface displaying the boundary conditions</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.43"></a><img src="images/img_0009_0001.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">is said to be a “magnetic wall.”</p>
<p class="indent">At sharp edges the field vectors may become infinite. However, the order of this singularity is restricted by the Bouwkamp-Meixner<sup><a id="ifn3"></a><a href="9780486656786_13_foot.xhtml#fn3">3</a></sup> edge condition. According to this condition, the energy density must be integrable over any finite domain even if this domain happens to include field singularities, i.e., the energy in any finite region of space must be finite. For example, when applied to a perfectly conducting sharp edge, this condition states that the singular components of the electric and magnetic vectors are of the order δ<sup>–½</sup>, where δ is the distance from the edge, whereas the parallel components are always finite.</p>
<h3 class="h3"><a id="ch1.4"></a><strong>1.4 The Field Potentials and Antipotentials</strong></h3>
<p class="noindent">According to Helmholtz’s partition theorem<sup><a id="ifn4"></a><a href="9780486656786_13_foot.xhtml#fn4">4</a></sup> any well-behaved vector field can be split into an irrotational part and a solenoidal part, or, equivalently, a vector field is determined by a knowledge of its curl and divergence. To partition an electromagnetic field generated by a current <strong>J</strong> and a charge <em>ρ</em>, we recall Maxwell’s equations</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.44"></a><img src="images/img_0009_0002.jpg" width="686" height="22" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.45"></a><img src="images/img_0009_0003.jpg" width="685" height="23" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="page_10"></a><a id="eq_1.46"></a><img src="images/img_0010_0001.jpg" width="684" height="23" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.47"></a><img src="images/img_0010_0002.jpg" width="685" height="21" alt="image" /></p>
<p class="noindent">and the constitutive relations for a simple medium</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.48"></a><img src="images/img_0010_0003.jpg" width="685" height="24" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.49"></a><img src="images/img_0010_0004.jpg" width="686" height="24" alt="image" /></p>
<p class="indent">From the solenoidal nature of <strong>B</strong>, which is displayed by <a href="9780486656786_06_ch1.xhtml#eq_1.47">Eq. (47)</a>, it follows that <strong>B</strong> is derivable from a magnetic vector potential <strong>A</strong>:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.50"></a><img src="images/img_0010_0005.jpg" width="686" height="24" alt="image" /></p>
<p class="noindent">This relation involves only the curl of <strong>A</strong> and leaves free the divergence of <strong>A</strong>. That is, ∇ · <strong>A</strong> is not restricted and may be chosen arbitrarily to suit the needs of calculation. Inserting <a href="9780486656786_06_ch1.xhtml#eq_1.50">Eq. (50)</a> into <a href="9780486656786_06_ch1.xhtml#eq_1.45">Eq. (45)</a> we see that <strong>E</strong> – <em>iω</em><strong>A</strong> is irrotational and hence derivable from a scalar electric potential <em>ϕ</em>:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.51"></a><img src="images/img_0010_0006.jpg" width="686" height="23" alt="image" /></p>
<p class="noindent">This expression does not necessarily constitute a complete partition of the electric field because <strong>A</strong> itself may possess both irrotational and solenoidal parts. Only when <strong>A</strong> is purely solenoidal is the electric field completely partitioned into an irrotational part ∇<em>ϕ</em> and a solenoidal part <strong>A</strong>. The magnetic field need not be partitioned intentionally because it is always purely solenoidal.</p>
<p class="indent">By virtue of their form, expressions <a href="9780486656786_06_ch1.xhtml#eq_1.50">(50)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.51">(51)</a> satisfy the two Maxwell <a href="9780486656786_06_ch1.xhtml#eq_1.45">equations (45)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.47">(47)</a>. But in addition they must also satisfy the other two Maxwell equations, which, with the aid of the constitutive relations <a href="9780486656786_06_ch1.xhtml#eq_1.48">(48)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.49">(49)</a>, become</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.52"></a><img src="images/img_0010_0007.jpg" width="686" height="47" alt="image" /></p>
<p class="noindent">When relations <a href="9780486656786_06_ch1.xhtml#eq_1.50">(50)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.51">(51)</a> are substituted into these equations, the</p>
<p class="noindent"><a id="page_11"></a>following simultaneous differential equations are obtained,<sup><a id="ifn5"></a><a href="9780486656786_13_foot.xhtml#fn5">5</a></sup> relating <em>ϕ</em> and <strong>A</strong> to the source quantities <strong>J</strong> and <em>ρ</em>:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.53"></a><img src="images/img_0011_0001.jpg" width="686" height="25" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.54"></a><img src="images/img_0011_0002.jpg" width="684" height="24" alt="image" /></p>
<p class="noindent">where <em>k</em><sup>2</sup> = <em>ω</em><sup>2</sup><em>μ</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" />. Here ∇ · <strong>A</strong> is not yet specified and may be chosen to suit our convenience. Clearly a prudent choice is one that uncouples the equations, i.e., reduces the system to an equation involving <em>ϕ</em> alone and an equation involving <strong>A</strong> alone. Accordingly, we choose ∇ · <strong>A</strong> = <em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" /><em>μϕ</em> or ∇ · <strong>A</strong> = 0.</p>
<p class="indent">If we choose the Lorentz gauge</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.55"></a><img src="images/img_0011_0003.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">then <a href="9780486656786_06_ch1.xhtml#eq_1.53">Eqs. (53)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.54">(54)</a> reduce to the Helmholtz equations</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.56"></a><img src="images/img_0011_0004.jpg" width="685" height="23" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.57"></a><img src="images/img_0011_0005.jpg" width="684" height="23" alt="image" /></p>
<p class="noindent">The Lorentz gauge is the conventional one, but in this gauge the electric field is not completely partitioned. If complete partition is desired, we must choose the Coulomb gauge<sup><a id="ifn6"></a><a href="9780486656786_13_foot.xhtml#fn6">6</a></sup></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.58"></a><img src="images/img_0011_0006.jpg" width="686" height="22" alt="image" /></p>
<p class="noindent"><a id="page_12"></a>which reduces <a href="9780486656786_06_ch1.xhtml#eq_1.53">Eqs. (53)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.54">(54)</a> to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.59"></a><img src="images/img_0012_0001.jpg" width="686" height="44" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.60"></a><img src="images/img_0012_0002.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">We note that <a href="9780486656786_06_ch1.xhtml#eq_1.59">Eq. (59)</a> is Poisson’s equation and can be reduced no further. However, <a href="9780486656786_06_ch1.xhtml#eq_1.60">Eq. (60)</a> may be simplified by partitioning <strong>J</strong> into an irrotational part <strong>J</strong><sub><em>i</em></sub> and a solenoidal part <strong>J</strong><sub><em>s</em></sub>, and by noting that the irrotational part just cancels the term involving the gradient. To show this, <strong>J</strong> is split up as follows: <strong>J</strong> = <strong>J</strong><sub><em>i</em></sub> + <strong>J</strong><sub><em>s</em></sub>, where by definition ∇ × <strong>J</strong><sub><em>i</em></sub> = 0 and ∇ · <strong>J</strong><sub><em>s</em></sub> = 0. Since <strong>J</strong><sub><em>i</em></sub> is irrotational, it is derivable from a scalar function <em>ψ</em>, viz., <strong>J</strong><sub><em>i</em></sub> = ∇<em>ψ</em>. The divergence of this relation, ∇ · <strong>J</strong><sub><em>i</em></sub> = ∇<sup>2</sup><em>ψ</em>, when combined with the continuity equation ∇ · <strong>J</strong> = ∇ · (<strong>J</strong><sub><em>i</em></sub> + <strong>J</strong><sub><em>s</em></sub>) = ∇ · <strong>J</strong><sub><em>i</em></sub> = <em>iωρ</em>, leads to ∇<sup>2</sup><em>ψ</em> = <em>iωρ</em>. A comparison of this result with <a href="9780486656786_06_ch1.xhtml#eq_1.59">Eq. (59)</a> shows that <em>ψ</em> = –<em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" />ϕ and hence <strong>J</strong><sub><em>i</em></sub> = ∇<em>ψ</em> = –<em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" />∇ϕ. From this expression it therefore follows that –μ<strong>J</strong><sub><em>i</em></sub> – <em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" />μ∇ϕ vanishes and consequently <a href="9780486656786_06_ch1.xhtml#eq_1.60">Eq. (60)</a> reduces to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.61"></a><img src="images/img_0012_0003.jpg" width="684" height="22" alt="image" /></p>
<p class="noindent">Thus we see that in this gauge, <strong>A</strong> is determined by the solenoidal part <strong>J</strong><sub><em>s</em></sub> of the current distribution and <em>ϕ</em> by its irrotational part <strong>J</strong><sub><em>i</em></sub>. Since <em>ϕ</em> satisfies Poisson’s equation, its spatial distribution resembles that of an electrostatic potential and therefore contributes predominantly to the near-zone electric field. It is like an electrostatic field only in its space dependence; its time dependence is harmonic.</p>
<p class="indent">In regions free of current (<strong>J</strong> = 0) and charge (<em>ρ</em> = 0) we may supplement the gauge ∇ · <strong>A</strong> = 0 by taking <em>ϕ</em> ≡ 0. Then <a href="9780486656786_06_ch1.xhtml#eq_1.53">Eq. (53)</a> is trivially satisfied and <a href="9780486656786_06_ch1.xhtml#eq_1.54">Eq. (54)</a> reduces to the homogeneous Helmholtz equation</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.62"></a><img src="images/img_0012_0004.jpg" width="685" height="22" alt="image" /></p>
<p class="noindent">In this case the electromagnetic field is derived from the vector potential <strong>A</strong> alone.</p>
<p class="indent">Let us now partition the electromagnetic field generated by a magnetic current <strong>J</strong><sub><em>m</em></sub> and a magnetic charge <em>ρ</em><sub><em>m</em></sub>. We recall that Maxwell’s <a id="page_13"></a>equations for such a field are</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.63"></a><img src="images/img_0013_0001.jpg" width="686" height="22" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.64"></a><img src="images/img_0013_0002.jpg" width="685" height="22" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.65"></a><img src="images/img_0013_0003.jpg" width="684" height="22" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.66"></a><img src="images/img_0013_0004.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">and, as before, the constitutive relations <a href="9780486656786_06_ch1.xhtml#eq_1.48">(48)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.49">(49)</a> are valid. From <a href="9780486656786_06_ch1.xhtml#eq_1.65">Eq. (65)</a> it follows that <strong>D</strong> is solenoidal and hence derivable from an electric vector potential <strong>A</strong><sub><em>e</em></sub>:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.67"></a><img src="images/img_0013_0005.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">In turn it follows from <a href="9780486656786_06_ch1.xhtml#eq_1.63">Eq. (63)</a> that <strong>H</strong> –<em>iω</em><strong>A</strong><sub><em>e</em></sub> is irrotational and hence equal to –∇<em>ϕ<sub>m</sub></em>, where <em>ϕ<sub>m</sub></em> is a magnetic scalar potential:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.68"></a><img src="images/img_0013_0006.jpg" width="683" height="23" alt="image" /></p>
<p class="noindent">Substituting expressions <a href="9780486656786_06_ch1.xhtml#eq_1.67">(67)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.68">(68)</a> into <a href="9780486656786_06_ch1.xhtml#eq_1.64">Eqs. (64)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.66">(66)</a>, we get, with the aid of the constitutive relations, the following differential equations for <strong>A</strong><sub><em>e</em></sub> and <em>ϕ<sub>m</sub></em>:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.69"></a><img src="images/img_0013_0007.jpg" width="685" height="88" alt="image" /></p>
<p class="noindent">If we choose the conventional gauge</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.70"></a><img src="images/img_0013_0008.jpg" width="683" height="24" alt="image" /></p>
<p class="noindent">then <em>ϕ<sub>m</sub></em> and <strong>A</strong><sub><em>e</em></sub> satisfy</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.71"></a><img src="images/img_0013_0009.jpg" width="684" height="47" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.72"></a><img src="images/img_0013_0010.jpg" width="684" height="23" alt="image" /></p>
<p class="noindent">In this gauge <em>ϕ<sub>m</sub></em> and <strong>A</strong><sub><em>e</em></sub> are called “antipotentials.” Clearly we may <a id="page_14"></a>also choose the gauge ∇ · <strong>A</strong><sub><em>e</em></sub> = 0 which leads to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.73"></a><img src="images/img_0014_0001.jpg" width="686" height="89" alt="image" /></p>
<p class="noindent">where <strong>J</strong><sub><em>ms</em></sub> is the solenoidal part of the magnetic current; this gauge leads also to <em>ϕ<sub>m</sub></em> = 0 and</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.74"></a><img src="images/img_0014_0002.jpg" width="686" height="24" alt="image" /></p>
<p class="noindent">for regions where <strong>J</strong><sub><em>m</em></sub> = 0 and <em>ρ</em><sub><em>m</em></sub> = 0.</p>
<p class="indent">If the electromagnetic field is due to magnetic as well as electric currents and charges, then the field for the conventional gauge is given in terms of the potentials <strong>A</strong>, <em>ϕ</em> and the antipotentials <strong>A</strong><sub><em>e</em></sub>, <em>ϕ<sub>m</sub></em> by</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.75"></a><img src="images/img_0014_0003.jpg" width="684" height="44" alt="image" /></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.76"></a><img src="images/img_0014_0004.jpg" width="686" height="23" alt="image" /></p>
<h3 class="h3"><a id="ch1.5"></a><strong>1.5 Energy Relations</strong></h3>
<p class="noindent">The instantaneous electric and magnetic energy densities for a lossless medium are defined respectively by</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.77"></a><img src="images/img_0014_0005.jpg" width="686" height="45" alt="image" /></p>
<p class="noindent">where <strong>E</strong>(<em>t</em>) stands for <strong>E</strong>(<strong>r</strong>,<em>t</em>), <strong>D</strong>(<em>t</em>) for <strong>D</strong>(<strong>r</strong>,<em>t</em>), etc. In the present instance these expressions reduce to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.78"></a><img src="images/img_0014_0006.jpg" width="685" height="24" alt="image" /></p>
<p class="noindent">Both <em>w<sub>e</sub></em> and <em>w<sub>m</sub></em> are measured in joules per meter<sup>3</sup>. To transform these quadratic quantities into the monochromatic domain we recall <a id="page_15"></a>that</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.79"></a><img src="images/img_0015_0001.jpg" width="687" height="24" alt="image" /></p>
<p class="noindent">where <strong>E</strong> is shorthand for <strong>E</strong><sub><em>ω</em></sub>(<strong>r</strong>) and <strong>H</strong> for <strong>H</strong><sub><em>ω</em></sub>(<strong>r</strong>). Since <strong>E</strong> can always be written as <strong>E</strong> = <strong>E</strong><sub>1</sub> + <em>i</em><strong>E</strong><sub>2</sub> where <strong>E</strong><sub>1</sub> and <strong>E</strong><sub>2</sub> are respectively the real and imaginary parts of <strong>E</strong>, the first of <a href="9780486656786_06_ch1.xhtml#eq_1.79">Eqs. (79)</a> is equivalent to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.80"></a><img src="images/img_0015_0002.jpg" width="685" height="23" alt="image" /></p>
<p class="noindent">Inserting this representation into the first of <a href="9780486656786_06_ch1.xhtml#eq_1.78">Eqs. (78)</a> we obtain</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.81"></a><img src="images/img_0015_0003.jpg" width="684" height="63" alt="image" /></p>
<p class="noindent">which, when averaged over a period, yields the time-average electric energy density</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.82"></a><img src="images/img_0015_0004.jpg" width="684" height="24" alt="image" /></p>
<p class="noindent">where the bar denotes the time average. Since</p>
<p class="linespace"></p>
<p class="equation"><strong>E<sub>1</sub> · E<sub>1</sub> + <strong>E</strong><sub>2</sub> · E<sub>2</sub> = <strong>E</strong> · E*</strong></p>
<p class="noindent">where <strong>E*</strong> is the conjugate complex of <strong>E</strong>, we can express <img class="mid" src="images/img_0015_0005a.jpg" width="24" height="17" alt="image" /> in the equivalent form</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.83"></a><img src="images/img_0015_0005.jpg" width="686" height="24" alt="image" /></p>
<p class="noindent">By a similar procedure it follows from the second of <a href="9780486656786_06_ch1.xhtml#eq_1.78">Eqs. (78)</a> and the second of <a href="9780486656786_06_ch1.xhtml#eq_1.79">Eqs. (79)</a> that the time-average magnetic energy is given by</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.84"></a><img src="images/img_0015_0006.jpg" width="685" height="23" alt="image" /></p>
<p class="indent">The instantaneous Poynting vector <strong>S</strong>(<em>t</em>) is defined by</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.85"></a><img src="images/img_0015_0007.jpg" width="684" height="28" alt="image" /></p>
<p class="noindent">where <strong>S</strong>(<em>t</em>) stands for <strong>S</strong>(<strong>r</strong>,<em>t</em>) and is measured in watts per meter<sup>2</sup>. With <a id="page_16"></a>the aid of expressions <a href="9780486656786_06_ch1.xhtml#eq_1.79">(79)</a>, the time average of <a href="9780486656786_06_ch1.xhtml#eq_1.85">Eq. (85)</a> leads to the following expression for the complex Poynting vector:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.86"></a><img src="images/img_0016_0001.jpg" width="686" height="34" alt="image" /></p>
<p class="indent">If from the scalar product of <strong>H</strong>* and ∇ × <strong>E</strong> = <em>iωμ</em><strong>H</strong> the scalar product of <strong>E</strong> and ∇ × <strong>H</strong>* = <strong>J</strong>* + <em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" /><strong>E</strong>* (<img src="images/img_epsi.jpg" width="9" height="12" alt="image" /> is assumed to be real) is subtracted, and if use is made of the vector identity</p>
<p class="linespace"></p>
<p class="equation">∇ · (<strong>E</strong> X <strong>H</strong>*) = <strong>H</strong>* · ∇ × <strong>E</strong> – <strong>E</strong> · ∇ × <strong>H*</strong></p>
<p class="noindent">the following equation is obtained:</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.87"></a><img src="images/img_0016_0002.jpg" width="685" height="35" alt="image" /></p>
<p class="noindent">which, with the aid of definitions <a href="9780486656786_06_ch1.xhtml#eq_1.83">(83)</a>, <a href="9780486656786_06_ch1.xhtml#eq_1.84">(84)</a>, and <a href="9780486656786_06_ch1.xhtml#eq_1.86">(86)</a>, yields the monochromatic form of Poynting’s vector theorem<sup><a id="ifn7"></a><a href="9780486656786_13_foot.xhtml#fn7">7</a></sup></p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.88"></a><img src="images/img_0016_0003.jpg" width="685" height="30" alt="image" /></p>
<p class="noindent">The real part of this relation, i.e.,</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.89"></a><img src="images/img_0016_0004.jpg" width="684" height="30" alt="image" /></p>
<p class="noindent">expresses the conservation of time-average power, the term on the right representing a source (when positive) or a sink (when negative) and correspondingly the one on the left an outflow (when positive) or an inflow (when negative).</p>
<p class="indent">In Poynting’s vector theorem <a href="9780486656786_06_ch1.xhtml#eq_1.88">(88)</a> a term involving the difference <img class="mid" src="images/img_0016_0005.jpg" width="83" height="23" alt="image" /> appears. To obtain an energy relation (for the monochromatic state) which contains the sum <img class="mid" src="images/img_0016_0006.jpg" width="83" height="22" alt="image" /> instead of the difference <img class="mid" src="images/img_0016_0007.jpg" width="84" height="25" alt="image" /> we proceed as follows. From vector analysis we recall that the quantity</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.90"></a><img src="images/img_0016_0008.jpg" width="684" height="62" alt="image" /></p>
<p class="noindent"><a id="page_17"></a>is identically equal to</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.91"></a><img src="images/img_0017_0001.jpg" width="684" height="56" alt="image" /></p>
<p class="noindent">From Maxwell’s equations ∇ × <strong>E</strong> = <em>iωμ</em><strong>H</strong> and ∇ × <strong>H</strong> = <strong>J</strong> – <em>iω</em><img src="images/img_epsi.jpg" width="9" height="12" alt="image" /><strong>E</strong> it follows that</p>
<p class="linespace"></p>
<p class="equation"><img src="images/img_0017_0002.jpg" width="685" height="165" alt="image" /></p>
<p class="noindent">Substituting these relations into expression <a href="9780486656786_06_ch1.xhtml#eq_1.91">(91)</a> we obtain the desired energy relation</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.92"></a><img src="images/img_0017_0003.jpg" width="683" height="131" alt="image" /></p><p class="noindent">which we call the “energy theorem.” Here we interpret as the time-average electric and magnetic energy densities the quantities</p>
<p class="linespace"></p>
<p class="equation1"><a id="eq_1.93"></a><img src="images/img_0017_0004.jpg" width="684" height="48" alt="image" /></p>
<p class="noindent">which reduce respectively to expressions <a href="9780486656786_06_ch1.xhtml#eq_1.83">(83)</a> and <a href="9780486656786_06_ch1.xhtml#eq_1.84">(84)</a> when the medium is nondispersive, i.e., when ∂<img src="images/img_epsi.jpg" width="9" height="12" alt="image" />/∂<em>ω</em> = 0 and ∂<em>μ</em>/∂<em>ω</em> = 0.</p>
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