Problem: What is the flux of a magnetic field of magnitude 4.9 Tesla through a circular loop with radius .7000 meters, if the field makes an angle of 39.5 degrees with a perpendicular to the plane of the loop?
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Solution: The area of the region is `pi ( .7000 m) ^ 2 = 1.538 m ^ 2, so the flux of a perpendicular 4.9 T field would be ( 4.9 T)( 1.538 m ^ 2) = 7.536 Tm ^ 2. The 39.5 degree angle entails an additional factor of cos( 39.5 deg) = .7718, so the flux is ( 7.536 N m ^ 2/C)( .7718) = 5.816 N m ^ 2/C.
Generalized Response: The flux of any vector field through an area A is equal to the product of the field strength and the area, as long as the vector field is directed perpendicular to the area. If the field is directed at angle `theta from perpendicular, the flux will be reduced by factor cos(`theta). In this case we have a magnetic field B directed at angle `theta from a perpendicular to an area A, so the flux is just flux = B A * A * cos(`theta).
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Figure description: