Page 253 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 235

Figure 8.2 A square loop of wire in the xy plane carrying 2 mA
is subjected to a nonuniform B field.

We use the integral form (10),

F =−I B × dL

Let us assume a rigid loop so that the total force is the sum of the forces on the four
sides. Beginning with the left side:

3 a z 2 a z
F =−2 × 10 −3 × 3 × 10 −6 × dx a x + × dy a y
x=1 x y=0 3
1 0
a z a z
+ × dx a x + × dy a y
x=3 x y=2 1

3 2 1 0

y (−a x ) + ln x a y + y (−a x )
=−6 × 10 −9 ln x a y + 1

3
1 0 3 2
2 1

=−6 × 10 −9 (ln 3)a y − a x + ln a y + 2a x
3 3
=−8a x nN
Thus, the net force on the loop is in the −a x direction.
D8.2. The field B =−2a x + 3a y + 4a z mT is present in free space. Find the
vector force exerted on a straight wire carrying 12 A in the a AB direction, given
A(1, 1, 1) and: (a) B(2, 1, 1); (b) B(3, 5, 6).

Ans. −48a y + 36a z mN; 12a x − 216a y + 168a z mN