Page 260 - Engineering Electromagnetics, 8th Edition

242 ENGINEERING ELECTROMAGNETICS

and

dT = dm × B (17)

If we extend the results we obtained in Section 4.7 for the differential electric
dipole by determining the torque produced on it by an electric field, we see a similar
result,
dT = dp × E
Equations (15) and (17) are general results that hold for differential loops of any
shape, not just rectangular ones. The torque on a circular or triangular loop is also
given in terms of the vector surface or the moment by (15) or (17).
Because we selected a differential current loop so that we might assume B was
constant throughout it, it follows that the torque on a planar loop of any size or shape
in a uniform magnetic field is given by the same expression,

T = IS × B = m × B (18)

We should note that the torque on the current loop always tends to turn the loop
so as to align the magnetic field produced by the loop with the applied magnetic field
that is causing the torque. This is perhaps the easiest way to determine the direction
of the torque.

EXAMPLE 8.3
To illustrate some force and torque calculations, consider the rectangular loop shown
in Figure 8.7. Calculate the torque by using T = IS × B.
Solution. The loop has dimensions of1mby2mand lies in the uniform field
B 0 =−0.6a y + 0.8a z T. The loop current is 4 mA, a value that is sufficiently small to
avoid causing any magnetic field that might affect B 0 .
We have
−3
T = 4 × 10 [(1)(2)a z ] × (−0.6a y + 0.8a z ) = 4.8a x mN · m
Thus, the loop tends to rotate about an axis parallel to the positive x axis. The small
magnetic field produced by the 4 mA loop current tends to line up with B 0 .

EXAMPLE 8.4
Now let us find the torque once more, this time by calculating the total force and
torque contribution for each side.
Solution. On side 1 we have
−3
F 1 = IL 1 × B 0 = 4 × 10 (1a x ) × (−0.6a y + 0.8a z )
=−3.2a y − 2.4a z mN