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<h2 class="hd hd-2 unit-title">Introduction to Energy in Inductance</h2>
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<p><b>Energy in Inductors</b></p><p> When we studied capacitors we observed that they could store energy in the electric fields created by the stored charge. Similarly, we are going to see that inductors can store energy in the magnetic field created by the current flowing through them.</p>
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<p>Textbook Links</p>
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<li><a href="https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.3x+1T2019/pdfbook/0/#viewer-frame" target="[object Object]">Chapter 11.3: Energy in Inductors </a></li>
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<h2 class="hd hd-2 unit-title">L27v1: Energy in and Inductor</h2>
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<h3 class="hd hd-2">L27v1: Energy in and Inductor</h3>
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<h2 class="hd hd-2 unit-title">L27Q1: Inductors</h2>
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Inductors
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Any arrangement of conductors with the ability to allow current to flow has an inductance. Any circuit is suffused with its own magnetic field, so there is inductance distributed over the entire circuit. However, often a certain amount of inductance is desired in a certain part of the circuit. Then a special element called an "inductor" is used. Inductors usually are made by winding a conductor into a coil, such as a solenoid. The magnetic field produced by the inductor is then largely localized inside the coil, which minimizes the usually unwanted mutual inductance with other parts of the circuit. For essentially all typical cases, the inductance of the physical inductors in the circuit is much, much larger than the inductance of the remaining wires and other circuit elements and also much, much larger than any mutual inductances. </p>
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Which of the following statements is (are) true? </p>
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<text>(a) Inductors store charge.</text>
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<text>(b) Inductors oppose changes in current.</text>
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<text>(c) An inductor stores energy in its magnetic field.</text>
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<text>(d) A coil wound of [mathjaxinline]N[/mathjaxinline] turns has an inductance linearly proportional to [mathjaxinline]N[/mathjaxinline].</text>
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<text>(e) The work done to establish a current [mathjaxinline]I[/mathjaxinline] in an inductor with inductance [mathjaxinline]L[/mathjaxinline] is [mathjaxinline]\frac{1}{2}L\, I^2[/mathjaxinline].</text>
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<h2 class="hd hd-2 unit-title">L27Q2: Scaling of Energy Density</h2>
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Scaling of Energy Density
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<p> Consider the two solenoids shown in the figure. Approximate their magnetic fields to the field of an infinite solenoid to answer the questions below. In all cases, assume that both solenoids carry the same current</p>
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<p><b>(Part a)</b> What is the ratio of the magnetic field in C1 to that in C2?</p>
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<h2 class="hd hd-2 unit-title">L27v4: Worked Example - Inductance of Solenoid using Energy</h2>
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Worked Example - Inductance of solenoid using energy
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The solenoid in the figure has radius [mathjaxinline]R[/mathjaxinline], height [mathjaxinline]h[/mathjaxinline], number of turns [mathjaxinline]N[/mathjaxinline], and number of turns per unit length [mathjaxinline]n=N/h[/mathjaxinline]. A current [mathjaxinline]I[/mathjaxinline] is flowing through the coils. Consider its height to be much larger than its radius, [mathjaxinline]h&gt;&gt;R[/mathjaxinline], so that we can assume the magnetic field to be uniform inside the solenoid and zero outside it. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]B[/mathjaxinline], the magnitude of the magnetic field produced by [mathjaxinline]I[/mathjaxinline] inside the solenoid. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]n[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]B=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]u_ B[/mathjaxinline], the magnetic energy density inside the solenoid. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]n[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p><b class="bfseries">(Part c)</b> Integrate the magnetic energy density in volume to obtain [mathjaxinline]U_ B[/mathjaxinline], the magnetic energy stored in the solenoid. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]n[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]U_ B =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part d)</b> The energy stored in an inductor of inductance [mathjaxinline]L[/mathjaxinline] is given by [mathjaxinline]U_ B= \frac{1}{2}LI^2[/mathjaxinline]. Use the result of part (c) to calculate the inductance of the solenoid. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]n[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<h2 class="hd hd-2 unit-title">L27v5: Worked Example - Inductance of a Toroid using Flux</h2>
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Worked Example - Inductance of a toroid using flux
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The top view of a toroid of inner radius [mathjaxinline]a[/mathjaxinline], outer radius [mathjaxinline]b[/mathjaxinline], and [mathjaxinline]N[/mathjaxinline] turns is shown in the left figure above. The toroid has a rectangular cross section of height [mathjaxinline]h[/mathjaxinline] as shown in the right figure. A current [mathjaxinline]I[/mathjaxinline] is flowing through the coils. Assume the coils are uniformly distributed and the magnetic field outside is zero. </p>
<p><b class="bfseries">(Part a)</b> Consider the coordinate system shown in the figure and let [mathjaxinline]r[/mathjaxinline] be the magnitude of the position vector measured from the center of the toroid. Calculate [mathjaxinline]\vec{B}[/mathjaxinline], the magnetic field produced by [mathjaxinline]I[/mathjaxinline] inside the toroid. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], hattheta for [mathjaxinline]\hat{\theta }[/mathjaxinline], and hatr for [mathjaxinline]\hat{r}[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\vec{B}=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]\Phi _ B[/mathjaxinline], the magnetic flux through one turn of the rectangular cross section of the toroid. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], [mathjaxinline]I[/mathjaxinline],and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\Phi _ B =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> Use the result of part (b) to calculate [mathjaxinline]L[/mathjaxinline], the self inductance of the toroid. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], [mathjaxinline]I[/mathjaxinline],and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]L =[/mathjaxinline]</p>
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<h3 class="hd hd-2">L27v5: Worked Example - Inductance of a Toroid using Flux</h3>
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<h2 class="hd hd-2 unit-title">L27v6: Worked Example - Inductance of a Toroid using Energy</h2>
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Worked Example - Inductance of a toroid using energy
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The top view of a toroid of inner radius [mathjaxinline]a[/mathjaxinline], outer radius [mathjaxinline]b[/mathjaxinline], and [mathjaxinline]N[/mathjaxinline] turns is shown in the left figure above. The toroid has a rectangular cross section of height [mathjaxinline]h[/mathjaxinline] as shown in the right figure. A current [mathjaxinline]I[/mathjaxinline] is flowing through the coils. Assume the coils are uniformly distributed and the magnetic field outside is zero. Consider the coordinate system shown in the figure and let [mathjaxinline]r[/mathjaxinline] be the magnitude of the position vector measured from the center of the toroid. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]u_ B[/mathjaxinline], the magnetic energy density inside the toroid. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], [mathjaxinline]I[/mathjaxinline],and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]u_ B =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> Integrate the magnetic energy density in volume to obtain [mathjaxinline]U_ B[/mathjaxinline], the magnetic energy stored in the toroid. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], [mathjaxinline]I[/mathjaxinline],and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]U_ B =[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> The energy stored in the toroid can be expressed as [mathjaxinline]U_ B = \frac{1}{2}L I^2[/mathjaxinline]. Use the result above to calculate [mathjaxinline]L[/mathjaxinline], the self inductance of the toroid. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], [mathjaxinline]I[/mathjaxinline],and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]L =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L27Q3: Superconducting Magnets</h2>
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Superconducting Magnets
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<p>To make a strong uniform magnetic field, a superconducting solenoid is frequently employed. Once the current is "loaded" into the superconducting solenoid, the two ends can be tied together and then the current will continue to circulate forever. Calculate the inductance and the total energy stored in the following two superconducting magnets. Ignore the fields outside the magnets.</p>
<p><b>(Part a)</b> The strongest constant field that can be created with currently available laboratory equipment is [mathjaxinline]45 \, T[/mathjaxinline]. This field is created with a current of [mathjaxinline]150 \, A[/mathjaxinline], inside a core with a [mathjaxinline]5 \, cm[/mathjaxinline] diameter and an active length (where the field is basically uniform) of about [mathjaxinline]10 \, cm[/mathjaxinline].</p>
<p>What is the inductance of the magnet, and the total energy stored? </p>
<p style="display:inline">[mathjaxinline]L =[/mathjaxinline] </p>
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<p style="display:inline"> (in [mathjaxinline]H[/mathjaxinline])</p>
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<p><b>(Part b)</b> A typical MRI magnet is [mathjaxinline]2 \, m[/mathjaxinline] long, [mathjaxinline]0.75 \, m[/mathjaxinline] in diameter and has a field strength of [mathjaxinline]4 \, T[/mathjaxinline] with a current of [mathjaxinline]100 \, A[/mathjaxinline]. In designing MRI magnets, people work very hard to make the field uniform throughout the core of the magnet (that is to say, it's not just a straight solenoid).</p>
<p>What is the inductance of such a magnet, and the total energy stored?</p>
<p style="display:inline">[mathjaxinline]L =[/mathjaxinline] </p>
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