Problem: Find the magnetic field due to 79 coplanar loops, each of approximate radius .3000 meters, at their common center point, when 8.5 Amps flows clockwise in the loop.
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Solution: The center point is at the same distance from every point of the loop. Assume that the loop lies in the x-y plane. A clockwise current will result in every segment contributing a downward field component, so all contributions reinforce one another. A line from the center to a point of the circle will be perpendicular to the direction of the flow, so the displacement vector is perpendicular to the segment, and sin(`theta) = 1. The total length of the loops is the circumference 2`pi r = 2`pi ( .03000 m) = 496.1 m, multiplied by the number of loops 79, for a total of 39190 m. The field is B=k'(IL)/r ^ 2 = .0000001000 Tesla / Amp meter)( 333100 Amp m)/( .3000 m) ^ 2 = 3.7 Tesla.
Generalized Response: A loop of radius r can be thought of as a series of very short segments, of total length 2 `pi r equal to the circumference of the circle. Each segment `dL is a source I `dL, which contributes `dB = k' I `dL / r^2 to the field at the center. All contributions are in the same direction, as can be easily verified.
When all the magnetic field contributions are added, we obtain
B = `sum(`dB) = `sum( k' I `dL / r^2 ).
Since k', I and r are identical for all contributions, we obtain
B = `sum ( k' I dL / r^2 ) = k' I / r^2 * `sum(`dL).
Since the sum of all the `dL contributions for one loop is 2 `pi r, we finally have
B = k' I / r^2 * 2 `pi r = 2 `pi k' I / r. If there are N loops, then this result is multiplied by N.
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Figure description: