A square loop is defined by its four vertices in a 3D coordinate system: P( 2 , 0 , 0 ), Q( 2 , 2 , 0 ), R( 0 , 2 , 0 ), and S( 0 , 0 , 0 ), where all coordinates are in meters. A steady current of 3 A flows through the loop in the sequence P Q R S P. The loop is placed in a uniform magnetic field given by B = (2 i + 3 j + 4 k ) T. The torque vector acting on the loop is :

Step-by-step solution

The area vector of the loop can be found using the cross product of two adjacent edge vectors following the direction of current. The current flows in the sequence P Q R S P. The edge vectors are: PQ = (2 - 2) i + (2 - 0) j + 0 k = 2 j m QR = (0 - 2) i + (2 - 2) j + 0 k = -2 i m The area vector is A = PQ QR = (2 j ) (-2 i ) = -4( j i ) = 4 k m ^2 . The magnetic dipole moment is: M = I A = 3 4 k = 12 k A m ^2 . The torque acting on the loop is given by = M B : = (12 k ) (2 i + 3 j + 4 k ) = 24( k i ) + 36( k j ) + 4