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A seemingly simple problem on EMI is as follows:

A circular coil of radius 0.7 m is placed with its plane perpendicular to a 1 T magnetic field. It is rotated about its vertical diameter through 180 degrees in 1 s. Estimate the magnitude of induced emf.

I think that the magnitude of average emf induced must be equal to zero. Faraday's Law says that the average EMF equals the (Change in flux)/(Time), and the flux is the same in both cases. Therefore, the average emf must be zero as well.
However, in the solution stated by the book, the magnetic flux through the coil in the two cases are opposite to each other and hence, the change in magnetic flux does not equal zero, amounting to some non-zero value of EMF.
My question is this:
Aren't the two orientations of the coil identical? Why should the flux be in different directions in that case?
I'd appreciate any help. EDIT: I believe I haven't made myself clear as to my exact question, which reflects in one answer.
The area vector is along the outward normal. For a planar ring, how would you define it? The inward and outward normal, I believe, would be identical(an inward normal from one side would appear to be outward from the other). What now?

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  • $\begingroup$ Are you talking about two flips of the coil in two different fields or one flip in both fields? Either way, what is the orientation of the 1 T field? Also, the problem statement seems to suggest that the earth's field is horizontal. In the U.S. it is not. $\endgroup$
    – R.W. Bird
    Aug 1, 2021 at 15:17
  • $\begingroup$ I am sorry I caused that doubt. I purposely changed the actual value of the Earth's magnetic field to 1 T here for ease of calculation. I shouldn't have done that, its a blunder. I apologise again. $\endgroup$ Aug 1, 2021 at 16:19

2 Answers 2

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You should not that while calculating flux, we take the scalar dot product of magnetic field vector and area vector. When the coil is flipped by 180° , the area vector becomes antiparallel to the initial situation and therefore the flux is same in magnitude but opposite in sign, so the change in flux would be 2BA

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  • $\begingroup$ The area vector is along the outward normal. For a planar ring, how would you define it? The inward and outward normal, I believe, would be identical(an inward normal from one side would appear to be outward from the other). This is precisely what my question was about. $\endgroup$ Aug 1, 2021 at 6:08
  • $\begingroup$ For a planar ring, you take the area vector along it's magnetic moment $\endgroup$ Aug 1, 2021 at 6:10
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When viewed from the top, down along the z axis (of rotation) as the coil is flipped clockwise 180 degrees from the yz plane in a magnetic field which points in the x direction; the part of the coil which is going +y to -y across the field will have its free electrons pushed down (toward – z in a right handed system). In the part of the coil going from -y to +y, the electrons come up. This is true for the entire flip, as the flux goes from positive (with the area vector in the direction of B) to negative (with the area vector reversed.) (Redefine the area vector and you can go from negative flux to positive but the direction of current flow around the circular coil does not change.) In my basic physics labs, we used a ballistic galvanometer to measure the total charge flow on each flip, and used that to calculate the horizontal and vertical components of the earth's field in the lab.

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