Four conducting loops, all of the same resistance, are moving with the velocitie

Question

Four conducting loops, all of the same resistance, are moving with the velocities indicated either into (loops 1 and 4) or out of (loops 2 and 3) a region of magnetic field. The field has the same magnitude within the rectangular region, but is directed into the page in the left half of the region and out of the page in the right half. (a) Rank the four loops based on the magnitude of their induced current, from largest to smallest, at the instant shown in the picture. (Use only ">"or "=" symbols. Do not include any parentheses around the letters or symbols.) (b) Now, let's define counterclockwise to be the positive direction for current. Rank the four loops based on the induced current, from most positive to most negative, at the instant shown in the picture. (Use only ">"or "=" symbols. Do not include any parentheses around the letters or symbols.)

Explanation / Answer

a)

induced current is given as

i = BLv/R

B and R are same for each loop , hence current induced is directly propotional to product of "v" and "L"

here L is the length intersecting the magnetic field lines

lets assume side of each box as "a"

for loop 1 : L = 2a ,

hence i1 = B(2a) v = 2 Bav

for loop 2 : L = 2a ,

hence i2 = B(2a) v = 2 Bav

for loop 3 : L = a ,

hence i3 = B(a) (2v) = 2 Bav

for loop 4 : L = 4a ,

hence i4 = B(4a) v = 4 Bav

clearly i4>i1 = i2 = i3

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