Set 54 Problem number 9

Problem

A magnetic field of magnitude 9.5 Tesla makes an angle of 66.5 degrees with a perpendicular to the plane of circular loop with radius 7 meters.  What is the magnetic flux through the loop?

Solution

The area of the region is `pi ( 7 m) ^ 2 = 153.8 m ^ 2, so the flux of a perpendicular 9.5 T field would be

• flux of perpendicular field:  flux = ( 9.5 T * )( 153.8 m ^ 2) = 1461 T m ^ 2.

The 66.5 degree angle change the flux by factor cos( 66.5 deg) = .3993, so the flux is

• flux at given angle = ( 1461 T m ^ 2)( .3993) = 583.3 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).