Page 261 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 243

Figure 8.7 Arectangular loop is located in a uniform
magnetic flux density B 0 .

On side 3 we obtain the negative of this result,

F 3 = 3.2a y + 2.4a z mN

Next, we attack side 2:

−3
F 2 = IL 2 × B 0 = 4 × 10 (2a y ) × (−0.6a y + 0.8a z )
= 6.4a x mN

with side 4 again providing the negative of this result,
F 4 =−6.4a x mN

Because these forces are distributed uniformly along each of the sides, we treat
each force as if it were applied at the center of the side. The origin for the torque may
be established anywhere since the sum of the forces is zero, and we choose the center
of the loop. Thus,

T = T 1 + T 2 + T 3 + T 4 = R 1 × F 1 + R 2 × F 2 + R 3 × F 3 + R 4 × F 4
= (−1a y ) × (−3.2a y − 2.4a z ) + (0.5a x ) × (6.4a x )
+ (1a y ) × (3.2a y + 2.4a z ) + (−0.5a x ) × (−6.4a x )
= 2.4a x + 2.4a x = 4.8a x mN · m

Crossing the loop moment with the magnetic flux density is certainly easier.