What area does the flux formula refer to?

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koliko987
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I've been studying em induction and in my book it was explained by considering a metal rod of length l moving through a magnetic field and cutting through the field lines at a constant speed. So in time dt it moves through ds and they showed e=BA/dt, where A = l x ds(the area that the rod "sweeps out".
Then in the next chapter it explains induction when moving a bar magnet through a coil and this time the "A" in the formula referred to the area of the cross section of the coil (the circle area of one of the rings of the wound up wire). Why is this? The magnetic field lines are only cutting through the small area of the wire, the rest of the "circle" is empty space so why are we considering the whole area of the cross section. What's different now that we're not looking for area that the wire "sweeps out"?
Thank you for your help
 

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I presume that this coil is not part of a circuit with a source.
In the case of a coil, there is interest in the magnetic-flux through the coil (that is, an area where the coil marks the boundary of that area). That flux will be related to the electric-field induced along that boundary.