Page 289 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 271

current of 6 mA, flowing in the a z direction from B to C.A filamentary
current of 15 A flows along the entire z axis in the a z direction. (a) Find F on
side BC.(b) Find F on side AB.(c) Find F total on the loop.
8.6 Show that the differential work in moving a current element IdL through a
distance dl in a magetic field B is the negative of that done in moving the
element Idl through a distance dL in the same field.
8.7 Uniform current sheets are located in free space as follows: 8a z A/m at
y = 0, −4a z A/m at y = 1, and −4a z A/m at y =−1. Find the vector force
per meter length exerted on a current filament carrying 7 mA in the a L
direction if the filament is located at (a) x = 0, y = 0.5, and a L = a z ;
(b) y = 0.5, z = 0, and a L = a x ;(c) x = 0, y = 1.5, and a L = a z .
8.8 Two conducting strips, having infinite length in the z direction, lie in the xz
plane. One occupies the region d/2 < x < b + d/2 and carries surface
current density K = K 0 a z ; the other is situated at −(b + d/2) < x < −d/2
and carries surface current density −K 0 a z .(a) Find the force per unit length
in z that tends to separate the two strips. (b) Let b approach zero while
maintaining constant current, I = K 0 b, and show that the force per unit
length approaches µ 0 I /(2πd) N/m.
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8.9 A current of −100a z A/m flows on the conducting cylinder ρ = 5 mm, and
+500a z A/m is present on the conducting cylinder ρ = 1 mm. Find the
magnitude of the total force per meter length that is acting to split the outer
cylinder apart along its length.
8.10 A planar transmission line consists of two conducting planes of width b
separated d min air, carrying equal and opposite currents of I A. If b d,
find the force of repulsion per meter of length between the two conductors.
8.11 (a) Use Eq. (14), Section 8.3, to show that the force of attraction per unit
length between two filamentary conductors in free space with currents I 1 a z
at x = 0, y = d/2, and I 2 a z at x = 0, y =−d/2, is µ 0 I 1 I 2 /(2πd). (b) Show
how a simpler method can be used to check your result.
8.12 Two circular wire rings are parallel to each other, share the same axis, are of
radius a, and are separated by distance d, where d << a. Each ring carries
current I. Find the approximate force of attraction and indicate the relative
orientations of the currents.
8.13 A current of6Aflows from M(2, 0, 5) to N(5, 0, 5) in a straight, solid
conductor in free space. An infinite current filament lies along the z axis
and carries 50 A in the a z direction. Compute the vector torque on the wire
segment using an origin at: (a) (0, 0, 5); (b) (0, 0, 0); (c) (3, 0, 0).
8.14 A solenoid is 25 cm long, 3 cm in diameter, and carries4Adcinits400
turns. Its axis is perpendicular to a uniform magnetic field of 0.8 Wb/m in
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air. Using an origin at the center of the solenoid, calculate the torque acting
on it.