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Unformatted text preview: 374 PROBLEMS _‘ 8.10 CHAPTER 3 MAGNETIC FORCES, MATERIALS, AND DEVICES A multilayer coil of 2000 turns of ﬁne wire is 20 mm long and has a thickness 5 mm of
winding. If the coil carries a current of 5 mA, the mmf generated is (a) 10 A ' t (b) 500 A  t (c) 2000 A  t (d) None of the above Answers: 8.1 b,c, 8.2b, 8.3a, 8.4d, 8.5d, 8.6a, 8.7b, 8.80, 8.9c,d, 8.10a. BEETIDN 8.2 FDRBES DUE TE] MAGNETIC FIELDS 8.1 An electron with velocity u = (3ax + 12ay — 4212) X 105 m/s experiences no net force
at a point in a magnetic ﬁeld B = lOax + Zan + 30az me/mz. Find E at that point.
8.2 A charged particle of a mass of 2 kg and a charge of l C starts at the origin with velocity
Sax m/s in a magnetic ﬁeld B = 1.5ay Wb/mz. Determine the location and the kinetic en
ergy of the particle at t = 2 s. "
*8.3 A particle with a mass of 1 kg and a charge of 2 C starts from rest at point (2, 3, —4) in a
region where E = —4ray V/m and B = Sax Wb/mz. Calculate
(a) The location of the particle at t = l s
(b) Its velocity and K.E. at that location
8.4 A —2 mC charge starts at point (0, l, 2) with a velocity of Sax m/s in a magnetic ﬁeld
B = 6ay Wb/mz. Determine the position and velocity of the particle after 10 3, assuming
that the mass of the charge is 1 gram. Describe the motion of the charge.
*8.5 By injecting an electron beam normally to the plane edge of a uniform ﬁeld Boaz, electrons
can be dispersed according to their velocity as in Figure 8.33.
(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 entry point.
y Figure 8.33 For Problem 8.5.
I (D G) ®
I
I
I _.
Ml ‘ 7W“ I
Electron ' x
beam : (9 ® (9 8.6 8.7 8.9 PRDBLEMS 375 Figure 8.34 For Problem 8.6. Given that B = 6xax — 9yay + 3zaz Wb/mz, ﬁnd the total force experienced by the rec
tangular loop (on z = 0 plane) shown in Figure 8.34. A current element of length 2 cm is located at the origin in free space and carries current
12 mA along 3,. A ﬁlamentary current of 153z A is located along x = 3, y = 4. Find the
force on the current ﬁlament. Three inﬁnite lines L1, L2, and L3 deﬁned byx = 0, y = O; x = 0, y = 4; x = 3, y = 4,
respectively, carry ﬁlamentary currents — 100 A, 200 A, and 300 A along az. Find the
force per unit length on (a) L; due to L1 (b) L1 due to [/2 (C) L3 due [0 L] (d) L3 due to L1 and L2. State whether each force is repulsive or attractive. A conductor 2 m long carrying a current of 3 A is placed parallel to the z—axis at distance ,00 = 10 cm as shown in Figure 8.35. If the ﬁeld in the region is cos ((15/3) ap Wb/m2,
how much work is required to rotate the conductor one revolution about the z~axis? Figure 8.35 For Problem 8.9. 376 CHAPTER 8 MAGNETIC FDRCES, MATERIALS, AND DEVICES *8.10 >“8.11 *8.12 8.13 8.14 Figure 8.36 For Problem 8.10. 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. 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, conduc—
tors 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. An inﬁnitely long tube of inner radius a and outer radius 19 is made of a conducting magnetic
material. The tube carries a total current I and is placed along the z—axis. If it is exposed to a
constant magnetic ﬁeld Boap, determine the force per unit length acting on the tube. A current sheet with K = lOax A/m lies in free space in the z = 2 m plane. A ﬁlamentary
conductor on the x—axis carries a current of 2.5 A in the axdirection. Determine the force
per unit length on the conductor. The magnetic ﬁeld in a certain region is B = 40 ax me/mz. 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 form on the conductor. Figure 8.37 For Problem 8.11. P R D B LE M s li'ii':?§ié 377 3’ Figure 8.38 For Problem 8.15. AlI‘ SEBTIDNE 8.3 AND 8.4 MAGNETIC; TBRQUE, MDMENTE, AND DIPULE *8.15 An inﬁnitely long conductor is buried in but insulated from an iron mass (/1 = 2000;40) as
shown in Figure 8.38. Using image theory, estimate the magnetic ﬂux density at point P. 8.16 A galvanometer has a rectangular coil of side 10 m by 30 mm pivoted about the center of
the shorter side. It is mounted in radial magnetic ﬁeld so that a constant magnetic ﬁeld of
0.4 Wb/m2 always acts across the plane of the coil. If the coil has 1000 turns and carries a
current of 2 mA, ﬁnd the torque exerted on it. 8.17 A 60—turn coil carries a current of 2 A and lies in the plane x + 2y ~ 5z = 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. 8.18 A small magnet placed at the origin produces B = —0.5az me/m2 at (10, 0, 0). Find B at (a) (0, 3,0)
(b) (3,4,0)
(0) (1,1, 1) SEGTIDN 8.5 MAGNETIZATICJN IN MATEREALB 8.19 A block of iron (,u. = SOOOMO) is placed in a uniform magnetic ﬁeld with 1.5 Wb/mz. If
iron consists of 8.5 X 1028 atoms/m3, calculate (a) the magnetization M, (b) the average
dipole moment. 8.20 In a certain material, xm = 4.2 and H = 0.2x ay A/m. Determine: (a) u, (b) M, (c) M, (d) B,
(6) J , (f) .112 8.21 In a ferromagnetic material (M = 4.5%),
' B = 4yaz me/m2
calculate: (a) Xm, (b) H, (c) M, (d) J b. 378 CHAPTER B MAGNETIC FORBES, MATERIALS, AND DEVICES 8.22 The magnetic ﬁeld intensity is H = 1200 A/m in a material when B = 2 Wb/mz.
When H is reduced to 400 A/m, B = 1.4 Wb/mZ. Calculate the change in the magneti— zation M. 8.23 An inﬁnitely long cylindrical conductor of radius a and permeability your is placed along
the z—axis. If the conductor carries a uniformly distributed current I along az ﬁnd M and J1, f0r0<p<a. k0 . . _
8.24 If M = Z (—yax + xay) 1n a cube of Size a, ﬁnd J b. Assume k0 is a constant. BEETIDN 8.7 MAGNETIC: BOUNDARY CDNDITIDNE *8.25 (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
M II M 2[ — —— = K and
Xml X1112 X7711 X1112 (b) If the boundary is not current free, show that instead of eq. (8.49), we obtain tan01_ IJ«1[1+ KM ]
tan 62 M2 32 sin 62 8.26 Given that H = 243x — 30ay + 40aZ kA/m in region 1, z > 0 with Mr = 50. If z = O
separates regions l and 2 and carries 63x kA/m, determine the magnetic ﬂux density in re
gion 2, z < 0, with ,u, = 100. 8.27 If ILLI = Zuo for region 1 (0 < (is < 7r) and M = 5M0 for region 2 (7r < ¢ < 271') and
B2 = 10ap + 15:14>  20az me/mz. Calculate (a) B], (b) the energy densities in the
two media. 8.28 The interface 2x + y = 8 between two media carries no current. Medium
1 (2x + y 2 8) is nonmagnetic with H1 = —4ax + 33y  aZ A/m. Find (a) the mag
netic energy density in medium 1, (b) M2 and B2 in medium 2 (2x + y S 8) with
u = 10%, (c) the angles H1 and H2 make with the normal to the interface. 8.29 Let the surface x + y + z = 2 separates air (,u = Mo) and a block of iron (u = 400 M0).
The region x + y + z > 2 is air, where B] = 50ax + 23an  203, Wb/mz. Find 32 and Hz. 8.30 Inside a right circular cylinder, u] = 800 no, while the exterior is free space. Given that
Bl = ,uo (22ap + 4551,75) Wb/mz, determine B2 just outside the cylinder. 8.31 The planez = 0 separates air (2 Z 0, p. = “0) from iron (z E O, p. = 200%). Given that
H = 103,, + 15ay — 3azA/m in air, ﬁnd B in iron and the angle it makes with the interface. P R CI B L E M s 379 Figure 8.39 For Problem 8.32. 8.32 Region 0 E z E 2 m is ﬁlled with an inﬁnite slab of magnetic material (,1, = 2.5 no). If
the surfaces of the slab at z = 0 and z = 2, respectively, carry surface currents 30ax A/m
and ~40aX A/m as in Figure 8.39, calculate H and B for (a)z<0
(b)0<z<2
(c)z>2 SEETIDN 8.8 INDUETURE AND INDUBTANEE *8.33 (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 L ,uoNza [2,0O + a]
= In *W—
27r 2po — a (b) If the toroid has a circular cross section as in Figure 7.15, show that N2 2
L=M
2:00 where p0 >> a. 8.34 When two parallel identical wires are separated by 3 m, the inductance per unit length is
2.5 uH/m. Calculate the diameter of each wire. 8.35 A solenoid with length 10 cm and radius 1 cm has 450 turns. Calculate its inductance. 8.36 A toroidal core is square in cross section with m = 2000. If it has a cross—sectional area
40 cm2 and a mean diameter of 50 cm, determine the number of turns necessary to get a
2 H inductor. ' 8.37 Show that the mutual inductance between the rectangular loop and the inﬁnite line current
of Figure 8.4 is ‘ 1)
M12 : a+po]
27r Po Calculate M12 whena = b = p0 = l m. 380 CHAPTER 8 MAGNETIC FDREES, MATERIALS, AND DEVICES *8.38 Prove that the mutual inductance between the closed wound coaxial solenoids of length 61
and 62 (61 >> 52), turns N1 and N2, and radii r1 and r2 with r1 2 r2 is EEBTIUN 3.9 MAGNETICS ENERGY 8.39 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 (,u = 4M0). If
the cable is 3 m long and carries 25 mA current, calculate the energy stored in the medium. 8.40 In a certain region for which X,” = 19,
H = 5x2yzax + leyzzay — ‘leyzzaz A/m Howmuchenergyis storedinO < x < 1,0 < y < 2, —1 < z < 2? 8.41 The magnetization curve for an iron alloy is approximately given by B = 3H + HZM Wb/m2. Find (a) u, when H = 210 A/m, (b) the energy stored per unit volume in the
alloy as H increases from 0 to 210 A/m. ﬁEﬁBTIDN 8.1m MAGNETIC BIRBUITE 8.42 A cobalt ring (it, = 600) has a mean radius of 30 cm. If a coil wound on the ring carries
12 A, calculate the number of turns required to establish an average magnetic ﬂux density
of 1.5 Wb/m2 in the ring. 8.43 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 u = 500pt0 and that all branches have the same cross—
sectional area of 10 cm2. 8.44 The magnetic circuit of Figure 8.40 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 pt, = 1500. Calculate R, 97, and ‘1’ for (a) The core
(b) The air gap Figure 8.40 For Problem 8.44. PRDBLEMS 381 Figure 8.41 For Problem 8.45. 8.45 Consider the magnetic circuit in Figure 8.41. Assuming that the core (u = 1000pto) has a
uniform cross section of 4 cm2, determine the ﬂux density in the air gap. 8.46 A ferromagnetic core with cross—section 40 X 60 mm experiences a ﬂux ‘1’ = 2.56 me
and an air gap that is 2.5 mm long. Determine the NI drop across the gap. 8.47 A magnetic circuit has a coil with 200 turns. Find the current through the coil to produce
0.1 Wb/m2 ﬂux density in the core (,u = 500 no) of length 40 cm. EEETIDN 8.1 'l FDRBE UN MAENETIB MATERIALS 8.48 An electromagnetic relay is modeled as shown in Figure 8.42. What force is on the arma
ture (moving part) of the relay if the ﬂux in the air gap is 2 me? The area of the gap is
0.3 cm2, and its length 1.5 mm. 8.49 A toroid with air gap, shown in Figure 8.43, has a square cross section. A long conductor
carrying current 12 is inserted in the air gap. If 11 = 200 mA, N = 750, p0 = 10 cm,
a = 5 mm, and Q = 1 mm, calculate (a) The force across the gap when 12 = O and the relative permeability of the toroid
is 300 (b) The force on the conductor when 12 = 2 mA and the permeability of the toroid is
inﬁnite. Neglect fringing in the gap in both cases. Figure 8.42 For Problem 8.48. CHAPTER 8 MAGNETIC FORCES, MATERIALS, AND DEVICES Figure 8.43 For Problem 8.49. Figure 8.44 For Problem 8.50. 8.50 A section of an electromagnet with a plate below it carrying a load is shown in Fig—
ure 8.44. The electromagnet has a contact area of 200 cm2 per pole, and the middle pole
has a winding of 1000 turns with I = 3 A. Calculate the maximum mass that can be lifted.
Assume that the reluctance of the electromagnet and the plate is negligible. 8.51 Figure 8.45 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 NZIZMOS F=— 23x
(a+2x) Figure 8.45 For Problem 8.51. Nonmagnetic sleeve ...
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