Problem: Find the magnitude of the magnetic field due to a straight current segment of length .02900 m, at a distance of 5.9 meters from the segment, given that the vector from the segment to the point is at angle 46 degrees with the segment, and that a current of 3.5 Amps flows in the segment.
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Solution: In this case, B = k'[(IL)/r ^ 2](sin `theta) = .000000001000 Tesla / Amp meter)[( .1015 Amp m)/( 5.9 m) ^ 2](sin( 46 deg) = .000000000002096 Tesla. If the vector was perpendicular the angle would be 90 deg, the sine would be 1, and the field would be just k'IL/r ^ 2 = .000000000002915 Tesla; the present result reduces this by factor sin( 46 deg) = .7190, giving the answer specified above.
Generalized Response: The magnitude of a magnetic field due to source IL, at distance r from the source such that the angle between I and a line from the source to the point is `theta, is B = k' (IL) / r^2 * sin(`theta). Since sin(`theta) varies from 0 to 1 as `theta varies from 0 to 90 deg, the field has maximum strength k' IL / r^2 when `theta = 90 deg, and is reduced for any other angle, reaching 0 when `theta = 0.
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Figure description: