Problems 409

PROBLEMS
Section 8.2—Forces due to Magnetic Fields

8.1 A 4 mC charge has velocity u 5 1.4ax 2 3.2ay 2 az m/s at point P(2, 5, 23) in the
presence of E 5 2xyzax 1 x2zay 1 x2yaz V/m and B 5 y2ax 1 z2ay 1 x2az Wb/m2. Find
the force on the charge at P.
8.2 An electron (m  9.11  10231 kg) moves in a circular orbit of radius 0.4  10210 m with
an angular velocity of 2  1016 rad/s. Find the centripetal force required to hold the electron.
8.3 A 1 mC charge with velocity 10ax 2 2ay 1 6az m/s enters a region where the magnetic
flux density is 25az Wb/m2. (a) Calculate the force on the charge. (b) Determine the
electric field intensity necessary to make the velocity of the charge constant.
8.4 Assume an electric field intensity of 20 kV/m and a magnetic flux density of
5 mWb/m2 exist in a region. Find the ratio of the magnitudes of electric and magnetic
forces on an electron that has attained a velocity of 0.5  108 m/s.
8.5 A 22 mC charge starts at point 1 0, 1, 2 2 with a velocity of 5ax m/s in a magnetic field
B 5 6ay Wb/m2. Determine the position and velocity of the particle after 10 s, assuming
that the mass of the charge is 1 gram. Describe the motion of the charge.

*8.6 By injecting an electron beam normally to the plane edge of a uniform field Boaz, elec-
trons can be dispersed according to their velocity as in Figure 8.34.
(a) Show that the electrons would be ejected out of the field in paths parallel to the
input beam as shown.
(b) Derive an expression for the exit distance d above the entry point.
8.7 Two large conducting plates are 8 cm apart and have a potential difference 12 kV. A drop of
oil with mass 0.4 g is suspended in space between the plates. Find the charge on the drop.
8.8 A straight conductor 0.2 m long carries a current 4.5 A along ax. If the conductor lies in
the magnetic field B 5 2.5(ay 1 az) mWb/m2, calculate the force on the conductor.
8.9 Determine |B| that will produce the same force on a charged particle moving at 140 m/s
that an electric field of 12 kV/m produces.
*8.10 Three infinite lines L1, L2, and L3 defined by x 5 0, y 5 0; x 5 0, y 5 4; x 5 3, y 5 4,
respectively, carry filamentary currents 2100 A, 200 A, and 300 A along az. Find the
force per unit length on
(a) L2 due to L1
(b) L1 due to L2
(c) L3 due to L1
(d) L3 due to L1 and L2.
State whether each force is repulsive or attractive.
    8.11 Two infinitely long parallel wires are separated by a distance of 20 cm. If the wires carry
current of 10 A in opposite directions, calculate the force on the wires.
    8.12 A conductor 2 m long carrying a current of 3 A is placed parallel to the z-axis at distance
ro 5 10 cm as shown in Figure 8.35. If the field in the region is cos 1 f/3 2 ar Wb/m2, how
much work is required to rotate the conductor one revolution about the z-axis?

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 410 CHAPTER 8 MAGNETIC FORCES, MATERIALS, AND DEVICES

3A

y
ρo
Electron
x

FIGURE 8.34 For Problem 8.6. FIGURE 8.35 For Problem 8.12.

*8.13 A conducting triangular loop carrying a current of 2 A is located close to an infinitely

long, straight conductor with a current of 5 A, as shown in Figure 8.36. Calculate (a) the
force on side 1 of the triangular loop and (b) the total force on the loop.
*8.14 A three-phase transmission line consists of three conductors that are supported at points
A, B, and C to form an equilateral triangle as shown in Figure 8.37. At one instant, con-
ductors A and B both carry a current of 75 A while conductor C carries a return current
of 150 A. Find the force per meter on conductor C at that instant.
    8.15 A current sheet with K 5 10ax A/m lies in free space in the z 5 2 m plane. A filamentary
conductor on the x-axis carries a current of 2.5 A in the ax-direction. Determine the force
per unit length on the conductor.
    8.16 The magnetic field in a certain region is B 5 40 ax mWb/m2. A conductor that is 2 m
in length lies in the z-axis and carries a current of 5 A in the az-direction. Calculate the
force on the conductor.

y
5A
A 75 A
2 3
2m
2A 1

150 A
2m 4m 2m x
C

B 75 A

FIGURE 8.36 For Problem 8.13. FIGURE 8.37 For Problem 8.14.

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 Problems 411

Sections 8.3 and 8.4—Magnetic Torque, Moments, and Dipole

*8.17 A rectangular loop shown in Figure 8.38 carries current I 5 10 A and is situated in the field
B 5 4.5(ay 2 az) Wb/m2. Find the torque on the loop.
8.18 A 60-turn coil carries a current of 2 A and lies in the plane x 1 2y 2 5z 5 12 such that
the magnetic moment m of the coil is directed away from the origin. Calculate m, assum-
ing that the area of the coil is 8 cm2.
8.19 The earth has a magnetic moment of about 8 3 1022 A ? m2 and its radius is 6370 km.
Imagine that there is a loop around the equator and determine how much current in the loop
would result in the same magnetic moment.
8.20 A triangular loop is placed in the x-z plane, as shown in Figure 8.39. Assume that a dc
current I  2 A flows in the loop and that B  30az m Wb/m exists in the region. Find
the forces and torque on the loop.
8.21 A loop with 50 turns and surface area of 12 cm2 carries a current of 4 A. If the loop rotates
in a uniform magnetic field of 100 mWb/m2, find the torque exerted on the loop.
8.22 High-current circuit breakers typically consist of coils that generate a magnetic field to blow
out the arc formed when the contacts open. An arc 30 mm long carries a current of 520
A in a direction perpendicular to a magnetic flux density of 0.4 mWb/m2. Determine the
magnetic force on the arc.

Section 8.5—Magnetization in Materials

xm
8.23 For a linear, isotropic, and homogeneous magnetic medium, show that M 5 B.
mo 1 1 1 xm 2
8.24 A block of iron 1 m 5 5000mo 2 is placed in a uniform magnetic field with 1.5 Wb/m2. If
iron consists of 8.5 3 1028 atoms/m3, calculate (a) the magnetization M, (b) the average
magnetic moment.

6
y

I = 10A
B

y 5
0 2
I

x
5
x
FIGURE 8.38 For Problem 8.16. FIGURE 8.39 For Problem 8.20.

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 412 CHAPTER 8 MAGNETIC FORCES, MATERIALS, AND DEVICES

8.25 In a magnetic material, with xm = 6.5, the magnetization is M 5 24y2az A/m. Find r, H,
and J at y 5 2 cm.
8.26 In a ferromagnetic material (m 5 80mo), B = 20xay mWb/m2. Determine: (a) mr, (b) xm,
(c) H, (d) M, (e) Jb.
8.27 An electromagnet is made of a ferromagnetic material whose magnetization curve can be
approximated by

B(H) 5 BoH/(Ho 1 H) mWb/m2

where Bo 5 2 Wb/m2 and Ho 5 100 A/m

Find r when H = 250 A/m.
8.28 An infinitely long cylindrical conductor of radius a and permeability or is placed along
the z-axis. If the conductor carries a uniformly distributed current I along az , find M and
Jb for 0 , r , a.

Section 8.7—Magnetic Boundary Conditions

*8.29 (a) For the boundary between two magnetic media such as is shown in Figure 8.16, show
that the boundary conditions on the magnetization vector are

M1t M2t m1 m2
2 5 K and M1n 5 M
xm1 xm2 xm1 xm2 2n
(b) If the boundary is not current free, show that instead of eq. (8.49), we obtain

tan u 1 m1 Km2
5 c1 1 d
tan u 2 m2 B2 sin u 2

   8.30 Region 1, for which µ1 5 2.5µo, is defined by z < 0, while region 2, for which µ2 5 4µo, is
defined by z > 0. If B1 5 6ax 2 4.2ay 1 1.8az mWb/m2, find H2 and the angle H2 makes
with the interface.
   8.31 In medium 1 (z , 0) µ1 5 5µo, while in medium 2 (z . 0) µ2 5 2µo. If B1 5 4ax 2 10ay
1 12az mWb/m2, find B2 and the energy density in medium 2.
   8.32 In region x , 0, µ 5 µo, a uniform magnetic field makes angle 42 with the normal to the
interface. Calculate the angle the field makes with the normal in region x . 0, µ 5 6.5µo.
   8.33 A current sheet with K 5 12ay A/m is placed at x 5 0, which separates region 1, x , 0,
µ 5 2µo and region 2, x . 0, µ 5 4µo. If H1 5 10ax 1 6az, A/m, find H2.
   8.34 Suppose space is divided into region 1 (y  0, 1  or1) and region 2 (y  0, 2 
or2). If H1  ax  ay az A/m, find H2.
   8.35 If m1 5 2mo for region 1 1 0 , f , p 2 and m2 5 5mo for region 2 1 p , f , 2p 2 and
B2 5 10ar 1 15af 2 20az mWb/m2. Calculate (a) B1, (b) the energy densities in the two
media.
*8.36 Region 1 is defined by x 2 y 1 2z  5 with µ1 5 2µo, while region 2 is defined by x 2 y 1
2z  5 with µ2 5 5µo. If H1 5 40ax 1 20ay 2 30az A/m, find (a) H1n, (b) H2t, (c ) B2.

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 Problems 413

   8.37 Inside a right circular cylinder, m1 5 800 mo, while the exterior is free space. Given that
B1 5 mo 1 22ar 1 45af 2 Wb/m2, determine B2 just outside the cylinder.
   8.38 The plane z 5 0 separates air 1 z $ 0, m 5 mo 2 from iron 1 z # 0, m 5 200mo 2 . Given that

H 5 10ax 1 15ay 2 3az A/m

in air, find B in iron and the angle it makes with the interface.
   8.39 Region 0 # z # 2 m is filled with an infinite slab of magnetic material 1 m 5 2.5mo 2 . If
the surfaces of the slab at z 5 0 and z 5 2, respectively, carry surface currents 30ax A/m
and 240ax A/m as in Figure 8.40, calculate H and B for
(a) z , 0
(b) 0 , z , 2
(c) z . 2
   8.40 Medium 1 is free space and is defined by r  a, while medium 2 is a magnetic material
with permeability µ2 and defined by r  a. The magnetic flux densities in the media are:

1.6a3 0.8a3
B1 5 Bo1 c a1 1 3 bcos uar 2 a1 2 bsin uau d
r r3

B2 5 Bo2 1 cos uar 2 sin uau 2

Find µ2.

Section 8.8—Inductors and Inductance

*8.41 (a) If the cross section of the toroid of Figure 7.15 is a square of side a, show that the
self-inductance of the toroid is
moN2a 2ro 1 a
L5 ln c d
2p 2ro 2 a
(b) If the toroid has a circular cross section as in Figure 7.15, show that

moN2a2
L5
2ro
where ro W a.
FIGURE 8.40 For Problem 8.39.

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 414 CHAPTER 8 MAGNETIC FORCES, MATERIALS, AND DEVICES

   8.42 An air-filled toroid of square cross section has inner radius 3 cm, outer radius 5 cm, and
height 2 cm. How many turns are required to produce an inductance of 45 H?
   8.43 A wire of radius 2 mm is 40 m long. Calculate its inductance. Assume µ 5 µo.
   8.44 A coaxial cable has an internal inductance that is twice the external inductance. If the
inner radius is 6.5 mm, calculate the outer radius.
   8.45 A hollow cylinder of radius a 5 2 cm is 10 m long. Find the inductance of the cylinder.
(See Table 8.3.)
   8.46 Show that the mutual inductance between the rectangular loop and the infinite line
­current of Figure 8.4 is

mb a 1 ro
M12 5 ln c d
2p ro

Calculate M12 when a 5 b 5 ro 5 1 m.

*8.47 Prove that the mutual inductance between the close-wound coaxial solenoids of length
1 and 2 1 ,1 W ,2 2 , turns N1 and N2, and radii r1 and r2 with r1 . r2 is

mN1N2 2
M12 5 pr1
,1
*8.48 A loop resides outside the region between two parallel long wires carrying currents in
opposite directions as shown in Figure 8.41. Find the total flux linking the loop.

Section 8.9—Magnetic Energy

   8.49 A coaxial cable consists of an inner conductor of radius 1.2 cm and an outer conductor
of radius 1.8 cm. The two conductors are separated by an insulating medium 1 m 5 4mo 2 .
If the cable is 3 m long and carries 25 mA current, calculate the energy stored in the
medium.

w
I I

b FIGURE 8.41 For Problem 8.48.

a

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 Problems 415

0.2 A 500 turns

L = 42 cm

Ig = 0.1 cm

FIGURE 8.42 For Problem 8.54. FIGURE 8.43 For Problem 8.55.

   8.50 In a certain region for which xm 5 19,

H 5 5x2yzax 1 10xy2zay 2 15xyz2az A/m

How much energy is stored in 0 , x , 1, 0 , y , 2, 21 , z , 2?

   8.51 The magnetic field in a material space (µ 5 15µo) is given by

B 5 4ax 1 12ay mWb/m2

Calculate the energy stored in region 0 x 2, 0  y  3, 0  z  4.

Section 8.10—Magnetic Circuits

   8.52 A cobalt ring 1 mr 5 600 2 has a mean radius of 30 cm. If a coil wound on the ring car-
ries 12 A, calculate the number of turns required to establish an average magnetic flux
density of 1.5 Wb/m2 in the ring.
   8.53 Refer to Figure 8.27. If the current in the coil is 0.5 A, find the mmf and the magnetic
field intensity in the air gap. Assume that m 5 500mo and that all branches have the same
cross-sectional area of 10 cm2.
   8.54 The magnetic circuit of Figure 8.42 has a current of 10 A in the coil of 2000 turns.
Assume that all branches have the same cross section of 2 cm2 and that the material of
the core is iron with mr 5 1500. Calculate R, , and  for
(a) The core
(b) The air gap

I2
I1

N1 N2

FIGURE 8.44 For Problem 8.56.

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 416 CHAPTER 8 MAGNETIC FORCES, MATERIALS, AND DEVICES

FIGURE 8.45 For Problem 8.58. FIGURE 8.46 For Problem 8.59.

FIGURE 8.47 For Problem 8.60. FIGURE 8.48 For Problem 8.61.

8.55 Consider the magnetic circuit in Figure 8.42. Assuming that the core 1 m 5 1000mo 2 has a
uniform cross section of 4 cm2, determine the flux density in the air gap.
8.56 For the magnetic circuit shown in Figure 8.44, draw the equivalent electric circuit.
Assume that all the sections have constant cross-sectional areas.
8.57 An air gap in an electric machine has length 4.4 mm and area 4.82 3 1022 m2. Find the
reluctance of the gap.

Section 8.11—Force on Magnetic Materials

8.58 An electromagnetic relay is modeled as shown in Figure 8.45. What force is on the arma-
ture (moving part) of the relay if the flux in the air gap is 2 mWb? The area of the gap is
0.3 cm2, and its length 1.5 mm.
8.59 A toroid with air gap, shown in Figure 8.46, has a square cross section. A long conduc-
tor carrying current I2 is inserted in the air gap. If I1 5 200 mA, N 5 750, ro 5 10 cm,
a 5 5 mm, and ,a 5 1 mm, calculate
(a) The force across the gap when I2 5 0 and the relative permeability of the toroid is 300.
(b) The force on the conductor when I2 5 2 mA and the permeability of the toroid is
­infinite. Neglect fringing in the gap in both cases.

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 Problems 417

8.60 A section of an electromagnet with a plate below it carrying a load as shown in Figure 8.47.
The electromagnet has a contact area of 200 cm2 per pole, and the middle pole has a wind-
ing of 1000 turns with I 5 3 A. Calculate the maximum mass that can be lifted. ­Assume
that the reluctance of the electromagnet and the plate is negligible.
8.61 Figure 8.48 shows the cross section of an electromechanical system in which the plunger
moves freely between two nonmagnetic sleeves. Assuming that all legs have the same
cross-sectional area S, show that

2 N2I2moS
F52 a
1 a 1 2x 2 2 x

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