Physics 1 Problem Solutions 230

Physics 1 Problem Solutions 230 - I where the long side is...

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226 Chapt. 29 Magnetic Fields Causing Forces 029 qfull 00600 1 3 0 easy math: Earth current, Earth magnetic moment 4. The magnetic moment of the Earth is 8 . 0 × 10 22 J / T. As a simple model of the Earth’s current assume that it ±ows in a circular path of radius 3500 km in the Earth’s core. Calculate this current. 029 qfull 00630 3 5 0 super tough think: arbitrary loop magnetic torque 5. Prove (to general satisfaction if not with complete rigor) that the magnetic torque about an axis through the center of mass on an arbitrary planar loop of current in a uniform magnetic ²eld is v τ = IA ˆ n × v B , where I is the current in the loop, A is the area of the loop, and ˆ n the normal to the loop. The sense of the normal is ²xed by a right-hand rule: curl your right-hand ²ngers with the current direction and your thumb points in the direction of the normal. HINTS: Imagine replacing the loop by long, thin rectangular loops of current
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Unformatted text preview: I , where the long side is in the plane de²ned by the B-²eld and the normal and the short side is perpendicular to this plane. The currents of the rectangles all have the same sense as the current of the original loop. (“Sense” in mathematical jargon is direction when there are only two possibilities.) The rectangles tile the original loop with their short sides and end bits of long sides tracing the original loop as closely as possible. These rectangles are bound rigidly together by some unspeci²ed force. We only have to be concerned with torque component along the axis perpendicular to the magnetic ²eld and the normal to the loop: let us call this the pivot axis for brevity. All the other torque components cancel to zero though this isn’t so easy to see without a diagram. A diagram of the replacement loop would help....
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