Page 252 - Engineering Electromagnetics, 8th Edition

234 ENGINEERING ELECTROMAGNETICS

and thus the Lorentz force equation may be applied to surface current density,

dF = K × B dS (6)

or to a differential current filament,

dF = IdL × B (7)

Integrating (5), (6), or (7) over a volume, a surface which may be either open or
closed (why?), or a closed path, respectively, leads to the integral formulations

F = J × B dν (8)
vol

K × B dS (9)
F =
S
and

F = IdL × B =−I B × dL (10)

One simple result is obtained by applying (7) or (10) to a straight conductor in a
uniform magnetic field,

F = IL × B (11)

The magnitude of the force is given by the familiar equation

F = BIL sin θ (12)
where θ is the angle between the vectors representing the direction of the current flow
and the direction of the magnetic flux density. Equation (11) or (12) applies only to
a portion of the closed circuit, and the remainder of the circuit must be considered in
any practical problem.

EXAMPLE 8.1
As a numerical example of these equations, consider Figure 8.2. We have a square
loop of wire in the z = 0 plane carrying 2 mA in the field of an infinite filament on
the y axis, as shown. We desire the total force on the loop.
Solution. The field produced in the plane of the loop by the straight filament is
I 15
H = a z = a z A/m
2πx 2πx
Therefore,
3 × 10 −6
−7
B = µ 0 H = 4π × 10 H = a z T
x