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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">Compass Angle Hell</p>
<p class="paragraph-P2">by Tyler Clarke</p>
<p class="paragraph-P2"> </p>
<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>
<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>
<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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<!--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>
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<mi>,</mi>
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</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>
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</math> </span><span class="text-T3"> (one of the basic trig relationships). Because we know</span>
<!--Next 'span' is a draw:frame. -->
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<msub>
<mi>b</mi>
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</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>
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<span id="Object5"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
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<mi>b</mi>
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<mi>b</mi>
<mi mathvariant="italic">earth</mi>
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<mi>tan</mi>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
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<mi>θ</mi>
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</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">
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<mi>r</mi>
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</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">
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</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>
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<span id="Object8"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
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<mn>1.2592681e-4</mn>
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</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>
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</math> </span><span class="text-T4"> is just the magnetic constant </span>
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<span id="Object10"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
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</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>
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<mo stretchy="false">=</mo>
<mrow>
<mo stretchy="false">−</mo>
<mn>4.25082884311</mn>
</mrow>
</mrow>
</mrow>
</math> </span><span class="text-T4">. </span><span class="text-T6">Yay for horrible trigonometry!</span></div>
<p class="paragraph-P6"><span class="text-T4"/></p>
</body>
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