Help! Three conducting loops labeled X, Y, or Z are in the plane of the paper. T

Question

Help! Three conducting loops labeled X, Y, or Z are in the plane of the paper. Two are rectangles and one is a square as shown below. The loops have equal areas and their centers are at equal distances from a very long wire, in the same plane, and carrying a current i = 1.37 t A, where t is in seconds. Rank the magnitudes of the induced Emf's in the loops in order of INCREASING value. Smallest has rank 1, largest has rank 3. (If all are equal, all ranks = 1. If two are equal, ranks can be 1,1,3 or 1,2,2). If A is the same on all three, and emf= BAcos(theta), shouldn't they all be the same? I have the same question with two triangles: A is the same for both and centers are the same d from the wire, but they point in opposite directions. What would be relative emf in this case?

Explanation / Answer

magnetic flux = mu_0*I*a/(2*pi) * ln(1 + b/c) where a is the length of the closest side to the long wire, b is that perpendicular to the wire, c is the distance between long wire and the closest side of the loop. emf = mu_0*(dI/dt)*a/(2*pi) * ln(1 + b/c) let d = distance of their centers from the long wire, A = area, then c = d - b/2, A = a*b emf = mu_0*(dI/dt)*a/(2*pi) * ln[(2da + A)/(2da - A)] induced current = emf/R = mu_0*(dI/dt)*a/(2*pi*R) * ln[(2da + A)/(2da - A)] which depends a ln[(2da + A)/(2da - A)] answer: X is the largest, Y is the smallest

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