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<h2 class="hd hd-2 unit-title">Introduction to Radiation</h2>
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So far, we've been discussing electromagnetic waves without ever addressing where they come from! How is light created? </p><p>
In this lesson, we discuss how electric and magnetic fields may be produced from charges. We will discuss several cases:<br/>(1) stationary charges<br/>(2) charges moving at constant velocity<br/>(3) accelerating charges </p><p>
We will see how [mathjaxinline]\vec{E}[/mathjaxinline] and [mathjaxinline]\vec{B}[/mathjaxinline] fields are produced in each of the above cases. We also discuss the particular example of dipole radiation and its applications. </p>
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<h2 class="hd hd-2 unit-title">L32v1: Creation of EM Waves</h2>
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<h3 class="hd hd-2">L32v1: Creation of EM waves</h3>
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<h2 class="hd hd-2 unit-title">L32Q1: Charge Moving in Various Ways</h2>
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Charge Moving in Various Ways
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<p>
To see light far away from a source, the power that is emitted by the source should be nonzero very far away (note, "far away" is a relative term). Another way to say this is that the integrated Poynting flux through a surface that encloses a source should be CONSTANT. </p>
<p>
In each of the following cases, determine the behavior of the power (integrated Poynting flux through a surface) emitted as a function of distance to the surface at which the power is measured (where [mathjaxinline]C[/mathjaxinline] is a constant). </p>
<p><b class="bfseries">(Part a)</b> A stationary charge. </p>
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<text> a) [mathjaxinline]P \propto r[/mathjaxinline]</text>
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<text> b) [mathjaxinline]P \propto C[/mathjaxinline]</text>
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<text> c) [mathjaxinline]P=0[/mathjaxinline]</text>
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<text> d) [mathjaxinline]P \propto \dfrac {1}{r}[/mathjaxinline]</text>
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<text> e) [mathjaxinline]P \propto \dfrac {1}{r^{2}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> A charge moving at constant velocity. </p>
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<text> a) [mathjaxinline]P \propto r[/mathjaxinline]</text>
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<text> b) [mathjaxinline]P \propto C[/mathjaxinline]</text>
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<text> c) [mathjaxinline]P=0[/mathjaxinline]</text>
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<text> d) [mathjaxinline]P \propto \dfrac {1}{r}[/mathjaxinline]</text>
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<text> e) [mathjaxinline]P \propto \dfrac {1}{r^{2}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part c)</b> An accelerating charge. </p>
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<text> a) [mathjaxinline]P \propto r[/mathjaxinline]</text>
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<text> b) [mathjaxinline]P \propto C[/mathjaxinline]</text>
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<text> c) [mathjaxinline]P=0[/mathjaxinline]</text>
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<text> d) [mathjaxinline]P \propto \dfrac {1}{r}[/mathjaxinline]</text>
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<text> e) [mathjaxinline]P \propto \dfrac {1}{r^{2}}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L32v2: Accelerating Charges</h2>
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<h3 class="hd hd-2">L32v2: Accelerating charges</h3>
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<h2 class="hd hd-2 unit-title">L32Q2: Accelerating Charge I</h2>
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Accelerating Charge I
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Consider the following questions related to the time-dependent behavior of the accelerating charge that we are investigating in this lesson. </p>
<p><b class="bfseries">(Part a)</b> What happens between the times [mathjaxinline]t=0[/mathjaxinline] and [mathjaxinline]t=\Delta t[/mathjaxinline]? </p>
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<text> a) the charge is stationary</text>
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<text> b) the charge moves with speed [mathjaxinline]u[/mathjaxinline]</text>
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<text> c) the charge accelerates upwards</text>
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<text> d) the charge oscillates up and down</text>
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<p><b class="bfseries">(Part b)</b> What happens after the time [mathjaxinline]t=\Delta t[/mathjaxinline]? </p>
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<text> a) the charge is stationary</text>
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<text> b) the charge moves with speed [mathjaxinline]u[/mathjaxinline]</text>
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<text> c) the charge accelerates upwards</text>
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<text> d) the charge oscillates up and down</text>
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<p><b class="bfseries">(Part c)</b> An observer a distance [mathjaxinline]r[/mathjaxinline] away from a light source will receive information about an event that occurs at time [mathjaxinline]t=t_{1}[/mathjaxinline] some time [mathjaxinline]\Delta T[/mathjaxinline] after this event occurs (note, the "retarded time" is defined to be [mathjaxinline](t - \Delta T)[/mathjaxinline]. What is [mathjaxinline]\Delta T[/mathjaxinline] in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], and other relevant numerical constants? </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta T =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<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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<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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<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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<code>abs, ln, sqrt</code>
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<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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<code>sin, cos, tan, sec, csc, cot</code>
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<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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<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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<h2 class="hd hd-2 unit-title">L32Q3: Accelerating Charge II</h2>
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Accelerating Charge II
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The "kink" in the electric field of an accelerating charge has which of the following properties (check all that apply): </p>
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<text>a) the kink travels at the speed of light</text>
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<text>b) the kink only has a component perpendicular to the direction of propagation</text>
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<text>c) the kink only has a component parallel to the direction of propagation</text>
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<text>d) the kink has components that are both perpendicular and parallel to the direction of propagation</text>
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<h2 class="hd hd-2 unit-title">L32Q4: Electric Field Components from an Accelerated Charge</h2>
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Electric Field Components from an Accelerated Charge
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<p>
Consider the diagrams in the previous video to answer the following questions. </p>
<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]u_{\perp }[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_03_2_1">
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<text> a) the velocity of the particle</text>
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<text> b) the velocity of the wave front</text>
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<text> c) the component of the velocity of the particle that is perpendicular to the line of sight to the observer</text>
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<text> d) the component of the velocity of the particle that is perpendicular to the wavefront</text>
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<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]a_{\perp }[/mathjaxinline]? </p>
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<text> a) the acceleration of the particle</text>
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<text> b) the acceleration of the wave front</text>
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<text> c) the component of the acceleration of the particle that is perpendicular to the line of sight to the observer</text>
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<text> d) the component of the acceleration of the particle that is perpendicular to the wavefront</text>
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<p><b class="bfseries">(Part c)</b> Our ultimate objective in this analysis is to calculate the Poynting vector [mathjaxinline]\vec{S}=\frac{1}{\mu _{0}}\vec{E}\times \vec{B}[/mathjaxinline], but we must determine the form of [mathjaxinline]\vec{E}[/mathjaxinline]. We have begun to do this diagrammatically. </p>
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Which component of [mathjaxinline]\vec{E}_{\mathrm{kink}}[/mathjaxinline] will contribute to the Poynting flux? </p>
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<text> b) [mathjaxinline]\vec{E}_{\mathrm{\perp }}[/mathjaxinline]</text>
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<text> c) Both [mathjaxinline]\perp[/mathjaxinline] and [mathjaxinline]||[/mathjaxinline] components</text>
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<h2 class="hd hd-2 unit-title">L32Q5: Radiating Electric Field Derivation</h2>
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Radiating Electric Field Derivation
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In the preceding derivation, we used Gauss's law to determine [mathjaxinline]E_{||}[/mathjaxinline]. Specifically, we drew a pillbox around a point in space and concluded that [mathjaxinline]E_{||}=E_{\mathrm{out}}[/mathjaxinline], where [mathjaxinline]E_{\mathrm{out}}[/mathjaxinline] is the magnitude of the electric field a distance [mathjaxinline]r[/mathjaxinline] away from a point charge. </p>
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Which of the following factors contribute to this equivalence? Select ALL that apply. </p>
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<text>a) there is no charge enclosed within the surface</text>
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<text>b) the net flux due to [mathjaxinline]E_{\perp }[/mathjaxinline] is zero</text>
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<text>c) the pillbox is very far away from the source, so [mathjaxinline]E_{||} \approx E_{\mathrm{out}}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L32Q6: Electric Field from Charge Acceleration</h2>
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Electric Field from Charge Acceleration - part a
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<p><b class="bfseries">(Part a)</b> Consider a positive charge [mathjaxinline]q[/mathjaxinline] that is constantly accelerating in the [mathjaxinline]\hat{j}[/mathjaxinline] direction, [mathjaxinline]\vec{a}(t)=a_{0}\hat{j}[/mathjaxinline]. Find an expression for the magnitude of the radiative component of the electric field, [mathjaxinline]|\vec{E}_{\mathrm{rad}}|[/mathjaxinline], as a function of [mathjaxinline]\theta[/mathjaxinline], the angle relative to the j direction. Consider times [mathjaxinline]t \ll c/a_{0}[/mathjaxinline], such that the particle moves nonrelativistically. </p>
<p>
Use the formula [mathjaxinline]\vec{E}_{\mathrm{rad}}(\vec{r},t) = \dfrac {-k q \vec{a}_{\perp }(t-r/c)}{c^2r}[/mathjaxinline] for the electric field at a location a distance [mathjaxinline]r[/mathjaxinline] away from the charge, where [mathjaxinline]k=\dfrac {1}{4\pi \epsilon _0}[/mathjaxinline] and [mathjaxinline]\vec{a}_{\perp }(t-r/c)[/mathjaxinline] is the perpendicular component of the acceleration at a time [mathjaxinline](t-r/c)[/mathjaxinline]. Express your answer in terms of <code>k</code>, <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>a_0</code> for [mathjaxinline]a_{0}[/mathjaxinline], and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]&#8212;make sure that your expression for the magnitude is positive for all values of [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{E}_{\mathrm{rad}}| =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
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enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter (english) name of letter</td>
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enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<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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<td class="formulainput">matrix</td>
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Electric Field from Charge Acceleration - part b
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<p><b class="bfseries">(Part b)</b> Find an expression for the magnitude of the radiative component of the magnetic field, [mathjaxinline]|\vec{B}_{\mathrm{rad}}|[/mathjaxinline]. Express your answer in terms of <code>k</code>, <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>a_0</code> for [mathjaxinline]a_{0}[/mathjaxinline], and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{B}_{\mathrm{rad}}| =[/mathjaxinline] </p>
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<td class="formulainput"><code>^</code> (raise to a power)</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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<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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<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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<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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<code>sin, cos, tan, sec, csc, cot</code>
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<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>
</tr>
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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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Electric Field from charge acceleration - part c
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<p><b class="bfseries">(Part c)</b> Find an expression for the magnitude of the Poynting vector [mathjaxinline]|\vec{S}_{\mathrm{rad}}|[/mathjaxinline]. Express your answer in terms of <code>k</code>, <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>a_0</code> for [mathjaxinline]a_{0}[/mathjaxinline], <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>mu_0</code> for [mathjaxinline]\mu _{0}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{S}_{\mathrm{rad}}| =[/mathjaxinline] </p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
<td class="formulainput">
<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]
</td>
</tr>
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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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<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>
</tr>
<tr class="formulainput">
<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>
</tr>
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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>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
</tr>
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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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We used a geometric argument to see that light is emitted from an accelerated charge in a direction that is antiparallel to the acceleration and falls off like [mathjaxinline]1/r[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/bf03d234b4a432d6b2a9b9df607609ac/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_21_review.svg" width="400" style="width:"/></center><p>
Specifically we have: </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 \vec{E}_{\mathrm{rad}}(\vec{r},t) = \dfrac {-q \vec{a}_{\perp }(t')}{4 \pi \epsilon _0 c^2 r}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{B}_{\mathrm{rad}}(\vec{r},t) = \left(\frac{1}{c}\right)\left(\hat{r}\times \vec{E}_{\mathrm{rad}}(\vec{r},t)\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
where [mathjaxinline]t'[/mathjaxinline] is the retarded time, [mathjaxinline]t' = t - r/c[/mathjaxinline]. </p><p>
This means that the Poynting vector, or energy flux per unit area, </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{S}_{\mathrm{rad}}(\vec{r},t) = \frac{1}{\mu _0} \vec{E}_{\mathrm{rad}}(\vec{r},t) \times \vec{B}_{\mathrm{rad}}(\vec{r},t) \propto \frac{1}{r^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
falls off like [mathjaxinline]1/r^2[/mathjaxinline], and so is detectable far away from the radiating source. </p>
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<h3 class="hd hd-2">L32v5: Dipole Antenna [DEMO]</h3>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L32v5">Downloads and transcripts</h3>
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<h4 class="hd hd-5">Transcripts</h4>
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