Page 257 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 239

Figure 8.5 (a) Given a lever arm R extending from an origin O to a point P where
force F is applied, the torque about O is T = R × F.(b)If F 2 =−F 1 , then the torque
T = R 21 × F 1 is independent of the choice of origin for R 1 and R 2 .

This result for uniform fields does not have to be restricted to filamentary circuits
only. The circuit may contain surface currents or volume current density as well. If
the total current is divided into filaments, the force on each one is zero, as we have
shown, and the total force is again zero. Therefore, any real closed circuit carrying
direct currents experiences a total vector force of zero in a uniform magnetic field.
Although the force is zero, the torque is generally not equal to zero.
In defining the torque, or moment, of a force, it is necessary to consider both an
origin at or about which the torque is to be calculated, and the point at which the
force is applied. In Figure 8.5a,we apply a force F at point P, and we establish an
origin at O with a rigid lever arm R extending from O to P. The torque about point
O is a vector whose magnitude is the product of the magnitudes of R,of F, and of
the sine of the angle between these two vectors. The direction of the vector torque T
is normal to both the force F and the lever arm R and is in the direction of progress
of a right-handed screw as the lever arm is rotated into the force vector through the
smaller angle. The torque is expressible as a cross product,

T = R × F

Now assume that two forces, F 1 at P 1 and F 2 at P 2 ,having lever arms R 1 and
R 2 extending from a common origin O,as shown in Figure 8.5b, are applied to an
object of fixed shape and that the object does not undergo any translation. Then the
torque about the origin is

T = R 1 × F 1 + R 2 × F 2
where

F 1 + F 2 = 0