Eight identical rectangular wire loops are in a region in which there is a unifo

Question

Eight identical rectangular wire loops are in a region in which there is a uniform magnetic field coming out of the page. Six of loops are being pushed to the right at various constant speeds and two of the loops are not moving as indicated. The loops have different locations with various portions (25, 50%, 75%, or 100%) within the field as shown. Rank the induced magnitude field at the center of the rectangular loops at the positions shown. Assume fields out of the page to be "positive" and fields into the page to be "negative". Assume there is no interaction between the loops. Greatest out of the page 1 _______ 2 _____ 3 _______ 4 ______ 5 ________ 6 ___________ 7 ___________ 8 _________ Greatest into the page OR, all the loops have the same (but not zero) induced magnetic field at the center of the loops. __________ OR, the induced magnetic field at all these points is zero._______ OR, the ranking for the induced magnetic field cannot be determined. _______ Carefully explain your reasoning.

Explanation / Answer

Let the area of the loops be A. B is the uniform magnetic field, v is the speed of loop and l is the length of the side of the loop perpendicular to its velocity.

For loop A: magnetic field induced in the loop would be proporitonal to the current induced in the loop which is prportional to the emf induced in the loop. Let the emf induced be Ea

Then Ea = -Blv = -Bl(2cm/s) , as the flux through A is increasing.

So induced current would also be negative and so the inudced magnetic field would be negative which would be proprotional to E.

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For loop B: The flux through the loop B is constant. So induced emf would be zero, and induced current would be zero and thus the induced magenitc field would be zero.

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For loop C: magnetic field induced in the loop would be proporitonal to the current induced in the loop which is prportional to the emf induced in the loop. Let the emf induced be Ec

Then Ec = Blv = Bl(4cm/s) , as the flux through C is decreasing.

So induced current would also be positive and so the inudced magnetic field would be positive which would be proprotional to Ec. The field induced would be twice in strength compared to the field induced in A as emf induced in C is twice the emef induced in A.

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For loop D: The flux through the loop D is constant. So induced emf would be zero, and induced current would be zero and thus the induced magenitc field would be zero.

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For loop E: The flux through the loop E is constant. So induced emf would be zero, and induced current would be zero and thus the induced magenitc field would be zero.

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For loop F: The flux through the loop F is constant as it's speed is zero. So induced emf would be zero, and induced current would be zero and thus the induced magenitc field would be zero.

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For loop G: magnetic field induced in the loop would be proporitonal to the current induced in the loop which is prportional to the emf induced in the loop. Let the emf induced be Eg

Then Ea = -Blv = -Bl(4cm/s) , as the flux through G is increasing.

So induced current would also be negative and so the inudced magnetic field would be negative which would be proprotional to E.

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For loop H: The flux through the loop H is constant as it's speed is zero. So induced emf would be zero, and induced current would be zero and thus the induced magenitc field would be zero.

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So from above analysis we can arrange the induced fields as:

1.C > 2.B = 3.D = 4.E = 5.F = 6.H > 7.A > 8.G

The field induced in C is greatest out of page, field induced in B, D, E, F, and H is zero, and field induced in G is greatest into the page.

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This concludes the answers. Check the answer and let me know if it's correct. If you need anymore clarification or correction I will be happy to oblige....

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