Set 54 Problem number 9

Problem

Find the magnetic flux through a loop with radius 6.1 meters if a uniform magnetic field of magnitude 1.2 Tesla makes an angle of 88.1 degrees with a perpendicular to the plane of loop.

Solution

The area of the region is `pi ( 6.1 m) ^ 2 = 116.8 m ^ 2, so the flux of a perpendicular 1.2 T field would be

• flux of perpendicular field:  flux = ( 1.2 T * )( 116.8 m ^ 2) = 140.1 T m ^ 2.

The 88.1 degree angle change the flux by factor cos( 88.1 deg) = .03478, so the flux is

• flux at given angle = ( 140.1 T m ^ 2)( .03478) = 4.872 T m ^ 2.

Generalized Solution

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 * cos(`theta).