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<p class="paragraph-P1">Compass Angle Hell</p>
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||||
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<p class="paragraph-P2">by Tyler Clarke</p>
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<p class="paragraph-P2"> </p>
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||||
<p class="paragraph-P2">Well, <span class="text-T1">je suis retourné</span><span class="text-T2">! The exam went well, and although I’m not sure what I got exactly, I think it was probably good. One thing I noticed was an uncomfortable amount of trigonometry: I had to derive polar coordinate forms for a translated circle, among other things. Fortunately, trig was always a strong suit.</span></p>
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||||
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||||
<p class="paragraph-P3">My other classes seem to be picking up on the trig craze too, which is in fact why I’m here: the very first question on my homework set for this week in Physics (yeah, I know I’m doing it pretty late, shoot me) requires finding the magnetic dipole of a bar magnet… knowing nothing but the local strength of the earth’s magnetic field, and the angle of a compass affected by both.</p>
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||||
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||||
<p class="paragraph-P3">The problem is not too hard from a trigonometry perspective, but there are a few stumbling blocks. Rather than trust my geometric intuition, I drew this handy lil’ diagram:</p>
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truncated
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T2"/></p>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P4"><span class="text-T3">We know that the magnetic field of the Earth is 3e-5 Tesla at the observation point (the origin), the bar magnet is at </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">[</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mn>0.189</mn>
|
||||
</mrow>
|
||||
<mi>,</mi>
|
||||
<mn>0</mn>
|
||||
<mi>,</mi>
|
||||
<mn>0</mn>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">]</mo>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T3">, and the angle between the total field and the magnet is 13.4deg. To find the magnetic dipole, we’ll work backwards from magnetic field and distance. The equation relating the values is </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>tan</mi>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mfrac>
|
||||
<msub>
|
||||
<mi>b</mi>
|
||||
<mi mathvariant="italic">earth</mi>
|
||||
</msub>
|
||||
<msub>
|
||||
<mi>b</mi>
|
||||
<mi mathvariant="italic">mag</mi>
|
||||
</msub>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T3"> (one of the basic trig relationships). Because we know</span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object4"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<msub>
|
||||
<mi>b</mi>
|
||||
<mi mathvariant="italic">earth</mi>
|
||||
</msub>
|
||||
</math> </span><span class="text-T3">, and we know </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object3"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mi>θ</mi>
|
||||
</math> </span><span class="text-T3">, this can be algebra’d to get </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>b</mi>
|
||||
<mi mathvariant="italic">mag</mi>
|
||||
</msub>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mfrac>
|
||||
<msub>
|
||||
<mi>b</mi>
|
||||
<mi mathvariant="italic">earth</mi>
|
||||
</msub>
|
||||
<mrow>
|
||||
<mi>tan</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</mfrac>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mfrac>
|
||||
<mn>3e-5</mn>
|
||||
<mrow>
|
||||
<mi>tan</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mn>13.4</mn>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T3">. This evaluates “nicely” to 1.2592681e-4. Great.</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P5"><span class="text-T4">Unfortunately, this isn’t the final answer: we need the magnitude of the </span><span class="text-T5">magnetic dipole moment</span><span class="text-T4">, not the </span><span class="text-T5">magnetic field</span><span class="text-T4">. There’s a handy fact about magnets that becomes useful here: for a distance </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object6"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>r</mi>
|
||||
<mo stretchy="false">≫</mo>
|
||||
<mi>s</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4"> (where </span><span class="text-T5">s</span><span class="text-T4"> is the length of the magnet), the magnetic field on the parallel axis is just </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object7"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>k</mi>
|
||||
<mi>m</mi>
|
||||
</msub>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mn>2</mn>
|
||||
<mi>u</mi>
|
||||
</mrow>
|
||||
<msup>
|
||||
<mi>r</mi>
|
||||
<mn>3</mn>
|
||||
</msup>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4">, where </span><span class="text-T5">u</span><span class="text-T4"> is the magnitude of the magnetic dipole moment and </span><span class="text-T5">r</span><span class="text-T4"> is the distance (which, handily, we know). This works </span><span class="text-T4">out to an equation </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object8"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>1.2592681e-4</mn>
|
||||
<mo stretchy="false">=</mo>
|
||||
<msub>
|
||||
<mi>k</mi>
|
||||
<mi>m</mi>
|
||||
</msub>
|
||||
</mrow>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mn>2</mn>
|
||||
<mi>u</mi>
|
||||
</mrow>
|
||||
<msup>
|
||||
<mi>r</mi>
|
||||
<mn>3</mn>
|
||||
</msup>
|
||||
</mfrac>
|
||||
<mo stretchy="false">→</mo>
|
||||
<mrow>
|
||||
<mi>u</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mfrac>
|
||||
<mn>1.2592681e-4</mn>
|
||||
<mrow>
|
||||
<mn>2</mn>
|
||||
<msub>
|
||||
<mi>k</mi>
|
||||
<mi>m</mi>
|
||||
</msub>
|
||||
</mrow>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
<msup>
|
||||
<mi>r</mi>
|
||||
<mn>3</mn>
|
||||
</msup>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4">. </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object9"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<msub>
|
||||
<mi>k</mi>
|
||||
<mi>m</mi>
|
||||
</msub>
|
||||
</math> </span><span class="text-T4"> is just the magnetic constant </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object10"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<msub>
|
||||
<mi>u</mi>
|
||||
<mn>0</mn>
|
||||
</msub>
|
||||
<mrow>
|
||||
<mn>4</mn>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
</mfrac>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>1e-7</mn>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4">, and </span><span class="text-T5">r</span><span class="text-T4"> is -</span><span class="text-T6">0.189</span><span class="text-T4"> (see above), so the magnitude of the dipole moment must be </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object11"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mi>u</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
<msup>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mn>0.189</mn>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
<mn>3</mn>
|
||||
</msup>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<mn>1.2592681e-4</mn>
|
||||
<mn>2e-7</mn>
|
||||
</mfrac>
|
||||
<mo stretchy="false">=</mo>
|
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|
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|
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Notice a mistake? Contact me at <a href="mailto:plupy44@gmail.com">plupy44@gmail.com</a><br>
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Most of the posts have been written in LibreOffice. The new site design (fancy Sitix templates!) interferes somethin' awful with them, so I'm moving them over by hand... ugh. All of the unupdated posts are listed below:
|
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|
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<a href="/phys2exam2reviwe.html">Physics 2 Exam 2 Reviwe</a>
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||||
<a href="/a-average-math2551-spring25-exam1">Multivariable Calculus Exam 1 (Thomas 12.1 -> 14.2)</a>
|
||||
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|
||||
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|
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|
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|
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|
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<p class="paragraph-P1">Mathin’ a Cos Rose</p>
|
||||
|
||||
<p class="paragraph-P2">by Tyler Clarke</p>
|
||||
<p class="paragraph-P2"> </p>
|
||||
<p class="paragraph-P2">Got me a multivariable calculus exam coming up soon. Most of the material is pretty simple – lagrange multipliers, for instance (I’ll be writing something about those soon) – but I’ve got one particular problem regarding integrating with radial coordinates that’s hard and interesting.</p>
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<div class="paragraph-P2">I’m asked to find the area of a single loop of a <span class="text-T1">rose</span><span class="text-T2"> described in radial coordinates by </span>
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|
||||
<span id="Object1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mi>r</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>5</mn>
|
||||
</mrow>
|
||||
<mi>cos</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T2">. </span><span class="text-T3">This is difficult to visualize, but luck hath smiled upon us, and Desmos supports radial coordinates:</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P2">
|
||||
<!--Next 'div' is a draw:frame. -->
|
||||
<div style="height:3.5909in;width:3.8638in; padding:0; float:left; position:relative; left:4.0749cm; top:0.161cm; " class="graphic-fr2" id="Image1"><img style="height:9.120900000000001cm;width:9.8141cm;" alt="" src="data:image/png;base64,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 truncated
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P2"><span class="text-T3"/></p>
|
||||
|
||||
<p class="paragraph-P3"><span class="text-T2">Not too bad. The maximum extent would appear to be 5, which makes sense, because cosine maxes out at 1 and we multiply by 5. I was a bit curious to see if the multiplier of theta controls the number of loops – as it turns out, the relation is a bit complicated, and I won’t get into that now.</span></p>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P4"><span class="text-T4">It’s looking to me like the best solution here is to integrate for </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mo stretchy="false">≤</mo>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">≤</mo>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4">, to grab that petal in the first and fourth quadrants, but I’m worried that that isn’t a </span><span class="text-T5">general</span><span class="text-T4"> enough solution: other roses that I’ve got to graph might not be so conveniently situated. The more general way I can think of is integrating for all theta and dividing by the number of petals – which isn’t necessarily easily predictable for all theta, but is a damn sight better than just </span><span class="text-T5">hoping</span><span class="text-T4"> my hand-drawn sketches appropriately capture the useful quadrants. </span><span class="text-T6">Regardless, let’s do the first idea first, and see how it works out.</span><span class="text-T4"> In this case, we </span><span class="text-T5">know</span><span class="text-T4"> there are 3 petals, so the integral should be </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object3"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<munderover>
|
||||
<mo stretchy="false">∫</mo>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</munderover>
|
||||
<mrow>
|
||||
<munderover>
|
||||
<mo stretchy="false">∫</mo>
|
||||
<mn>0</mn>
|
||||
<mrow>
|
||||
<mn>5</mn>
|
||||
<mi>cos</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</munderover>
|
||||
<mi>r</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mi mathvariant="italic">dr</mi>
|
||||
<mi>d</mi>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4">. Why </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object4"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>r</mi>
|
||||
<mi mathvariant="italic">dr</mi>
|
||||
<mi>d</mi>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4"> instead of just </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mn>1</mn>
|
||||
<mi mathvariant="italic">dr</mi>
|
||||
<mi>d</mi>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T4">? Because of the shape of the mesh we’re integrating; I won’t elaborate, but only because </span><a href="https://tutorial.math.lamar.edu/classes/calciii/dipolarcoords.aspx" class="text-Internet_20_link"><span class="text-T4">Paul's Online Math Notes</span></a> <span class="text-T7">already did.</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P5">Integrating this isn’t too hard. The inner integral is simply
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object6"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<munderover>
|
||||
<mo stretchy="false">∫</mo>
|
||||
<mn>0</mn>
|
||||
<mrow>
|
||||
<mn>5</mn>
|
||||
<mi>cos</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</munderover>
|
||||
<mi>r</mi>
|
||||
</mrow>
|
||||
<mi mathvariant="italic">dr</mi>
|
||||
<mo stretchy="false">→</mo>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<msubsup>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<msup>
|
||||
<mi>r</mi>
|
||||
<mn>2</mn>
|
||||
</msup>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||||
</mrow>
|
||||
<mn>0</mn>
|
||||
<mrow>
|
||||
<mn>5</mn>
|
||||
<mi>cos</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</msubsup>
|
||||
<mo stretchy="false">→</mo>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<msup>
|
||||
<mi>cos</mi>
|
||||
<mn>2</mn>
|
||||
</msup>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span>, giving us
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object7"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mrow>
|
||||
<munderover>
|
||||
<mo stretchy="false">∫</mo>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</munderover>
|
||||
<msup>
|
||||
<mi>cos</mi>
|
||||
<mn>2</mn>
|
||||
</msup>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
<mi>d</mi>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</math> </span>. <span class="text-T8">This is kind of a nasty trig integral, one which I very much don’t want to derive myself, but that’s what Wolfram is for! Wolfram kindly tells me that my disgusting integral is </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object8"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mn>25</mn>
|
||||
<mn>24</mn>
|
||||
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|
||||
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|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
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|
||||
<mrow>
|
||||
<mn>6</mn>
|
||||
<mrow>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">+</mo>
|
||||
<mi>sin</mi>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
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|
||||
<mn>6</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
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|
||||
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|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
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|
||||
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|
||||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||||
</mrow>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</msubsup>
|
||||
</math> </span><span class="text-T8">. Because </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object9"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>sin</mi>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>0</mn>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T8">, this works out nicely to… </span>
|
||||
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|
||||
<span id="Object10"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>24</mn>
|
||||
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|
||||
<mn>6</mn>
|
||||
<mrow>
|
||||
<mi>π</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>4</mn>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T8">. </span><span class="text-T9">This is wrong.</span></div>
|
||||
|
||||
<p class="paragraph-P6"><span class="text-T10">I’ve kept the wrong work up because it’s </span><span class="text-T11">subtly</span><span class="text-T12"> wrong, and (I think) does a pretty good job of illustrating one of the most annoying problems with polar coordinates. Do you see it?</span></p>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P7"><span class="text-T13">The problem lies in how cosine behaves with multipliers. </span>
|
||||
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|
||||
<span id="Object11"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>cos</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T13"> makes a full trip from 1 to 0 in </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object12"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mn>0</mn>
|
||||
<mo stretchy="false">≤</mo>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">≤</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T13">; </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object13"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>cos</mi>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T15"> makes </span><span class="text-T16">three</span><span class="text-T15"> such trips in the same span. And our rose has three peaks.</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P6"><span class="text-T17">By some horrible mathemagic, integrating what should be a half-turn integrates </span><span class="text-T18">the whole pattern</span><span class="text-T17">. As it turns out, this is true for every cosine rose: regardless of the multiplier, a full span of pi (not 2pi!) covers the entire shape. Because we only want one petal, we have to cut our</span><span class="text-T18"> bounds</span><span class="text-T19"> </span><span class="text-T17">in third: </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object14"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mrow>
|
||||
<munderover>
|
||||
<mo stretchy="false">∫</mo>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
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|
||||
<mn>6</mn>
|
||||
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|
||||
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|
||||
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|
||||
<mn>6</mn>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mn>2</mn>
|
||||
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|
||||
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|
||||
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|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
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|
||||
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|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
<mi>d</mi>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">→</mo>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<msubsup>
|
||||
<mrow>
|
||||
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|
||||
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|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>24</mn>
|
||||
</mfrac>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>6</mn>
|
||||
<mrow>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">+</mo>
|
||||
<mi>sin</mi>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>6</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
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|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
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|
||||
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|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
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|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||||
</mrow>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>6</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>6</mn>
|
||||
</mfrac>
|
||||
</msubsup>
|
||||
<mo stretchy="false">→</mo>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>12</mn>
|
||||
</mfrac>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T20">. </span><span class="text-T21">And that’s right!</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P8"><span class="text-T17">What if we integrated the entire shape and divided by three, as mentioned above? This gives us actually the same integral, with a different multiplier and different bounds: </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object15"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
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|
||||
<mfrac>
|
||||
<mn>1</mn>
|
||||
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|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mrow>
|
||||
<munderover>
|
||||
<mo stretchy="false">∫</mo>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</munderover>
|
||||
<msup>
|
||||
<mi>cos</mi>
|
||||
<mn>2</mn>
|
||||
</msup>
|
||||
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|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>3</mn>
|
||||
<mi>θ</mi>
|
||||
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|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
<mi>d</mi>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">→</mo>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<msubsup>
|
||||
<mrow>
|
||||
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|
||||
<mrow>
|
||||
<mfrac>
|
||||
<mn>1</mn>
|
||||
<mn>3</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>24</mn>
|
||||
</mfrac>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>6</mn>
|
||||
<mrow>
|
||||
<mi>θ</mi>
|
||||
<mo stretchy="false">+</mo>
|
||||
<mi>sin</mi>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>6</mn>
|
||||
<mi>θ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
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|
||||
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|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
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|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="true">|</mo>
|
||||
</mrow>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<mo stretchy="false">−</mo>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
<mfrac>
|
||||
<mi>π</mi>
|
||||
<mn>2</mn>
|
||||
</mfrac>
|
||||
</msubsup>
|
||||
<mo stretchy="false">→</mo>
|
||||
<mfrac>
|
||||
<mn>25</mn>
|
||||
<mn>12</mn>
|
||||
</mfrac>
|
||||
<mi>π</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T17">. Exactly the same! Math is weird like that.</span></div>
|
||||
|
||||
<p class="paragraph-P8"><span class="text-T17">To avoid boundary hell in the future, I’m just going to integrate whole shapes and divide by petals. It’s much easier to wrap my head around why that actually works.</span></p>
|
||||
|
||||
<p class="paragraph-P8"><span class="text-T17"/></p>
|
||||
|
||||
<p class="paragraph-P8"><span class="text-T17">Stay tuned for Lagrange multipliers!</span></p>
|
||||
</body>
|
||||
|
||||
</html>
|
||||
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|
||||
[!]
|
||||
[=title "News"]
|
||||
[=content-]
|
||||
<h1>Deadly Boring Math News</h1>
|
||||
<p>
|
||||
Essentially a tiny blog about dbmus.
|
||||
</p>
|
||||
<p>
|
||||
<b>2024-3-11</b><br>
|
||||
Site redesign! Now I've gotta move all the libreoffice exports to actual sitix pages...
|
||||
</p>
|
||||
[/]
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||||
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<body dir="ltr" style="max-width:8.5in;margin-top:0.7874in; margin-bottom:0.7874in; margin-left:0.7874in; margin-right:0.7874in; ">
|
||||
|
||||
<p class="paragraph-P1">Some circuit math</p>
|
||||
|
||||
<p class="paragraph-P2">by Tyler Clarke</p>
|
||||
<p class="paragraph-P3"> </p>
|
||||
<p class="paragraph-P3">I was given a pretty interesting problem in Physics 2, and figured I oughta share my solutions for all the enterprising students trying to learn about circuits.</p>
|
||||
|
||||
<p class="paragraph-P3">We’re provided a series circuit going like,</p>
|
||||
<ul><li><p class="P4" style="margin-left:0cm;"><span class="Bullet_20_Symbols" style="display:block;float:left;min-width:0.635cm;">•</span>A battery, pushing 1.4 volts<span class="odfLiEnd"/> </p></li><li><p class="P4" style="margin-left:0cm;"><span class="Bullet_20_Symbols" style="display:block;float:left;min-width:0.635cm;">•</span>A 0.28m long thick wire with a cross sectional area of 1.1 * 10<span class="text-T1">-6 </span><span class="text-T2">m²</span><span class="odfLiEnd"/> </p></li><li><p class="P4" style="margin-left:0cm;"><span class="Bullet_20_Symbols" style="display:block;float:left;min-width:0.635cm;">•</span>A 0.063m long thin wire with cross sectional area 5.7 * 10<span class="text-T1">-8</span><span class="text-T2"> m</span><span class="text-T1">2</span><span class="odfLiEnd"/> </p></li><li><p class="P4" style="margin-left:0cm;"><span class="Bullet_20_Symbols" style="display:block;float:left;min-width:0.635cm;">•</span>Another thick wire identical to the first, going back to the battery<span class="odfLiEnd"/> </p></li></ul>
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P3">The question first asks for a voltage conservation <span class="text-T3">loop rule</span><span class="text-T4"> representing this circuit. Essentially, the loop rule states that for any given loop circuit, the sum of potential differences across every element </span><span class="text-T3">must</span><span class="text-T4"> be 0. Any other value will produce perpetual motion and a violent explosion. The loop rule describing </span><span class="text-T3">this</span><span class="text-T4"> circuit is quite simple: </span>
|
||||
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|
||||
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||||
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|
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||||
<mn>0</mn>
|
||||
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|
||||
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|
||||
</math> </span><span class="text-T4">. The potential difference across every wire segment has to add to -1.4V to counteract the battery.</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P3"><span class="text-T4">We are then asked for a charge conservation (node rule) equation. The node rule essentially states that, across any circuit branches, the </span><span class="text-T3">sum </span><span class="text-T4">of the current on both branches is equal to the current flowing into the branch. Since this is a series circuit, the rules for parallel circuits don’t apply, so we can just use the standard wisdom that current must be the same at every point in a series circuit: </span>
|
||||
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|
||||
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||||
<mi>i</mi>
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||||
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||||
<mn>1</mn>
|
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||||
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||||
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<mi>i</mi>
|
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<mi mathvariant="italic">thin</mi>
|
||||
</msub>
|
||||
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|
||||
</math> </span></div>
|
||||
<p class="paragraph-P5"> </p>
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P6"><span class="text-T5">Next</span>, we’re asked to find the electron current in the middle of the thin wire. This is a hard one. The obvious-ish way I can see is to calculate the total resistance (electron mobility and supply is given, and we know wire dimensions, so this shouldn’t be too hard) and use Ohm’s law. Gemma2 running locally seems to agree, and helpfully informed me that the resistance is just
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object6"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>R</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mrow>
|
||||
<mi>p</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mi>L</mi>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mi>A</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span>. L and A are already well-known (length and cross-sectional area) and p is the resistivity, which can be found with
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object7"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>p</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
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|
||||
<mn>1</mn>
|
||||
<mo stretchy="false">/</mo>
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|
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||||
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||||
<mi>n</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mi>e</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mi>μ</mi>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span> where n is the electron density, e is the charge of an electron, and μ is the electron mobility. This combines nicely to the behemoth:
|
||||
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|
||||
<span id="Object8"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
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||||
<mi>R</mi>
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||||
<mo stretchy="false">=</mo>
|
||||
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||||
<mi>L</mi>
|
||||
<mo stretchy="false">/</mo>
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||||
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|
||||
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||||
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<mo stretchy="false">∗</mo>
|
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<mi>n</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mi>e</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mi>μ</mi>
|
||||
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|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
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|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T6">.</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P7"><span class="text-T6">T</span><span class="text-T4">he electron mobility here is 0.00052 m²/(V * s), and there are 3.7 * 10</span><span class="text-T7">28</span><span class="text-T8"> mobile electrons per cubic meter, in all of the wires. An expression for the resistance of a thick wire is thus </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object9"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>R</mi>
|
||||
<mi mathvariant="italic">thick</mi>
|
||||
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|
||||
<mo stretchy="false">=</mo>
|
||||
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|
||||
<mn>0.28</mn>
|
||||
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|
||||
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|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
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|
||||
<mrow>
|
||||
<mn>1.1e-6</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>0.00052</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>1.602e-19</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>3.7e28</mn>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mn>0.08258435224</mn>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T10">. </span><span class="text-T8">That’s a… surprisingly normal number! I was half expecting it to be something ridiculous. 0.0826 ohms seems about right for the wires in this configuration. Now for the thin wire: </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
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|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>R</mi>
|
||||
<mi mathvariant="italic">thin</mi>
|
||||
</msub>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mrow>
|
||||
<mn>0.063</mn>
|
||||
<mo stretchy="false">/</mo>
|
||||
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|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
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|
||||
<mrow>
|
||||
<mn>5.7e-8</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>0.00052</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>1.602e-19</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>3.7e28</mn>
|
||||
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|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
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|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>0.35858995054</mn>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T8">.</span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P7"><span class="text-T8">Also a pretty normal, reasonable number. Because resistance sums in a circuit, we can now apply Ohm’s law to get current: </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object11"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mi>V</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
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|
||||
<mi>I</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
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|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mi mathvariant="italic">thick</mi>
|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
<mi>I</mi>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mi mathvariant="italic">thin</mi>
|
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>2.67298685488</mn>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span></div>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P7"><span class="text-T8">Nice. A pretty normal number! Unfortunately, this is not electron current. This is amperage in coulombs per second, but we need electrons per second. Fortunately, because there are 1.602e-19 coulombs per electron, we can just divide: </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object12"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mi>i</mi>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mrow>
|
||||
<mn>2.67298685488</mn>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mn>1.602e-19</mn>
|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>1.6685311e+19</mn>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T8">. That is a much less normal number.</span></div>
|
||||
|
||||
<p class="paragraph-P7"><span class="text-T8"/></p>
|
||||
|
||||
<!--Next 'div' was a 'text:p'.-->
|
||||
<div class="paragraph-P8"><span class="text-T11">Finally, we’re asked to find the electric field at the middle of the thin wire and at the middle of a thick wire (it is the same in both thick wires). The electric field strength in a wire is just resistivity times current density, and current density is current divided by area, so we have a simple formula: </span>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mi>E</mi>
|
||||
<mi mathvariant="italic">field</mi>
|
||||
</msub>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mrow>
|
||||
<mi>p</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mi>I</mi>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mi>A</mi>
|
||||
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|
||||
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|
||||
</math> </span><span class="text-T11">. Expanding resistivity, we get </span>
|
||||
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|
||||
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|
||||
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||||
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|
||||
<mi>E</mi>
|
||||
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|
||||
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|
||||
<mo stretchy="false">=</mo>
|
||||
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|
||||
<mi>I</mi>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<mi>A</mi>
|
||||
<mo stretchy="false">∗</mo>
|
||||
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|
||||
<mo stretchy="false">∗</mo>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
</mrow>
|
||||
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|
||||
</mrow>
|
||||
</mrow>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T11">. For the thin wire, this is just </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
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|
||||
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|
||||
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|
||||
<mi>E</mi>
|
||||
<mi mathvariant="italic">thin</mi>
|
||||
</msub>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mrow>
|
||||
<mn>2.67298685488</mn>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>5.7e-8</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>0.00052</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>1.602e-19</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>3.7e28</mn>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>15.2143845096</mn>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mi>N</mi>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mi>C</mi>
|
||||
</mrow>
|
||||
<mi>.</mi>
|
||||
</mrow>
|
||||
</math> </span><span class="text-T11"> </span><span class="text-T12">For the thick wire, </span>
|
||||
<!--Next 'span' is a draw:frame. -->
|
||||
<span id="Object13"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
|
||||
<mrow>
|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>E</mi>
|
||||
<mi mathvariant="italic">thick</mi>
|
||||
</msub>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mrow>
|
||||
<mn>2.67298685488</mn>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mrow>
|
||||
<mo fence="true" form="prefix" stretchy="false">(</mo>
|
||||
<mrow>
|
||||
<mrow>
|
||||
<mn>1.1e-6</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>0.00052</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>1.602e-19</mn>
|
||||
<mo stretchy="false">∗</mo>
|
||||
<mn>3.7e28</mn>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo fence="true" form="postfix" stretchy="false">)</mo>
|
||||
</mrow>
|
||||
</mrow>
|
||||
<mo stretchy="false">=</mo>
|
||||
<mn>0.78838174277</mn>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mi>N</mi>
|
||||
<mo stretchy="false">/</mo>
|
||||
<mi>C</mi>
|
||||
</mrow>
|
||||
<mi>.</mi>
|
||||
</mrow>
|
||||
</math> </span> <span class="text-T13">Easy!</span></div>
|
||||
|
||||
<p class="paragraph-P9"><span class="text-T14">Attribution: Gemma2 running locally was quite helpful reminding me which equations to use.</span></p>
|
||||
</body>
|
||||
|
||||
</html>
|
||||
@@ -0,0 +1,6 @@
|
||||
[?]
|
||||
[=content-]
|
||||
<h1>[^title]</h1>
|
||||
[^post]
|
||||
[/]
|
||||
[#default.html]
|
||||
@@ -0,0 +1,9 @@
|
||||
[!]
|
||||
[=post-]
|
||||
hi!
|
||||
[/]
|
||||
[=title "Multivariable Exam 1 Review"]
|
||||
[=author "Tyler Clarke"]
|
||||
[=date "unknown"]
|
||||
[=subject "Calculus"]
|
||||
[#post.html]
|
||||
Reference in new issue
Block a user