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Unformatted text preview: 310 ; : L‘PRLOB‘LLEMS CHAPTER '7 MAGNETDSTATIC FIELDS BEETIDN 7.2 BIDT*‘SAVARTIS LAW 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 10A Figure 7.27 For Problem 7.3. (a) State Biot—Savart’s law. (b) The y and z—axes, respectively, carry ﬁlamentary currents 10 A along ay and 20 A
along ~31. Find H at (—3, 4, 5). An inﬁnitely long conducting ﬁlament is placed along the xaxis and carries current
10 mA in the ax direction. Find H at (—2, 3, 4). Two inﬁnitely long ﬁlaments are placed parallel to the xaxis as shown in Figure 7.27.
(a) Find H at the origin. (b) Determine H at (— l, 2, 2). A current element I d1 = 43x A  m is located at the origin. Determine its contribution to
the magnetic ﬁeld intensity at (a) (l, 0, 0), (b) (0, 1,0), (0) (0, 0, l), (d) (1, l, 1). A conducting ﬁlament carries current I from point A(0, 0, a) to point B(0, 0, b). Show
that at point P(x, y, 0), I ‘ Mm H b a
2 2 2 2 2 2 ad’
Vx +y +19 Vx+y+a Consider AB in Figure 7.28 as part of an electric circuit. Find H at the origin due to AB.
Repeat Problem 7.6 for the conductor AB in Figure 7.29. Line x = 0, y = 0, 0 S 2 S 10 m carries current 2A along 32. Calculate H at points (a) (5, 0,0) (C) (5,15, 0)
(b) (5, 5, 0) (d) (5, —15,0)
y
1 A
6A
B x
10A 0 1 Figure 7.28 For Problem 7.6. P R D B L E M s 3 1 1
y
y
2 A
4 A
5 A
B x . , x
O 1 5 A
Figure 7.29 For Problem 7.7. Figure 7.30 Current ﬁlament for Problem 7.10. *7.9 (a) Find H at (0, 0, 5) due to side 2 of the triangular loop in Figure 7.6(a).
(b) Find H at (0, 0, 5) due to the entire loop. 7.10 An inﬁnitely long conductor is bent into an L shape as shown in Figure 7.30. If a direct current of 5 A ﬂows in the current, ﬁnd the magnetic ﬁeld intensity at (a) (2, 2, 0),
(b) (0, 2, 0), and (C) (0, 0, 2)~ 7.11 Find H at the center C of an equilateral triangular loop of side 4 m carrying 5 A of current
as in Figure 7.31. 7.12 A rectangular loop carrying 10 A of current is placed on z = 0 plane as shown in Fig
ure 7.32. Evaluate H at (a) (2, 2, 0) (c) (4, 8, 0)
(b) (4, 2, 0) (d) (0, 0, 2) 7.13 A square conducting loop of side 2a lies in the 2: = 0 plane and carries a current I in the
counterclockwise direction. Show that at the center of the loop V21 H = a
7m Z
Z
_
—2 0 2 O 4 8
Figure 7.31 Equilateral triangular loop for Figure 7.32 Rectangular loop of Problem 7.11. Problem 7.12. CHAPTER '7 MAGNETDSTATIB FIELDS 312 y 10A Figure 7.33 Filamentary loop of Problem 7.16; not drawn to scale. 7.14 A square conducting loop 3 cm on each side carries a current of 10 A. Calculate the mag
netic ﬁeld intensity at the center of the loop. *7.15 (a) A ﬁlamentary loop carrying current I is bent to assume the shape of a regular polygon
of n sides. Show that at the center of the polygon nI . 7r
H = — sm —
27rr n where r is the radius of the circle circumscribed by the polygon.
(b) Apply this for the cases of n = 3 and n = 4 and see if your results agree with those for
the triangular loop of Problem 7.11 and the square loop of Problem 7.13, respectively. (c) As 11 becomes large, show that the result of part (a) becomes that of the circular loop
of Example 7.3. 7.16 For the ﬁlamentary loop shown in Figure 7.33, ﬁnd the magnetic ﬁeld strength at 0. 7.17 Two identical current loops have their centers at (0, 0, 0) and (0, 0, 4) and their axes the
same as the z—axis (so that the “Helmholtz coil” is formed). If each loop has a radius of 2 m
and carries a current of 5 A in 305, calculate H at (a) (0, 0, 0)
(b) (0, 0, 2) 7.18 A solenoid 3 cm in length carries a current of 400 mA. If the solenoid is to produce a mag
netic ﬂux density of 5 me/mz, how many turns of wire are needed? 7.19 A solenoid of radius 4 mm and length 2 cm has 150 turns/m and carries a current Of
500 mA. Find (a) at the center, (b) at the ends of the solenoid. 7.20 Plane x = 10 carries a current of 100 mA/m along az, while line x = 1, y = —2 carrieS
a ﬁlamentary current of 207r mA along 32. Determine H at (4, 3, 2). 7.21 A wire bent as shown in Figure 7.34 carries a current of 4 A. Find H at (0, 0, 2) due to (a) the
segment at —4 < x < — 1, (b) the segment at 1 < x < 4, (c) the semicircular portion F’RDBLEMS 313 Figure 7.34 For Problem 7.21. SEBTIDN 7.3 AMPERE’E CIRCUIT LAW 7.22 (a) State Ampere’s circuit law. (b) A hollow conducting cylinder has inner radius a and outer radius 19 and carries current
I along the positive z—direction. Find H everywhere. 7.23 A current sheet K1 = Sax A/m ﬂows on y = 10, while K2 = —lOax A/m ﬂows on
y = —4. Find H at the origin. 7.24 The 2 = 0 plane carries current K = 103x A/m, while current ﬁlament situated at
y = 0, z = 6 carries current I along ax. Find I such that H(O, 0, 3) = 0. 7.25 (a) An inﬁnitely long solid conductor of radius a is placed along the z—axis. If the con—
ductor carries current I in the +z direction, show that
I
H = p 395 21rd2 within the conductor. Find the corresponding current density.
(b) HI = 3 A and a = 2 cm in part (a), ﬁnd H at (0, 1 cm, 0) and (0, 4 cm, 0). 7.26 If H = yax — xay A/m on plane z = 0, (a) determine the current density and (b) verify
Ampere’s law by taking the circulation of H around the edge of the rectangle
z=0,0<x<3,1<y<4. 7.27 Let H = ko<£>a¢, p < a, where k0 is a constant. (a) Find J for p < a. (b) Find H for
a
p > a. 7.28 An inﬁnitely long ﬁlamentary wire carries a current of 2 A along the zaxis in the
+z—direction. Calculate the following: (a) B at (—3, 4, 7)
(b) The ﬂux through the square loop described by 2 S p S 6, 0 S z S 4. ¢ = 90° 314 CHAPTER l7 MAGNETDSTATIE FIELDS Figure 7.35 Electric motor pole of Problem 7.29.
Pole Armature 7.29 The electric motor shown in Figure 7.35 has ﬁeld 106
H = Tsin 2(1) ap A/m Calculate the ﬂux per pole passing through the air gap if the axial length of the pole is
20 cm. 7.30 Consider the twowire transmission line whose cross section is illustrated in Figure 7.36.
Each wire is of radius 2 cm, and the wires are separated 10 cm. The wire centered at (0, 0)
carries a current of S A while the other centered at (10 cm, 0) carries the return current. Find H at (a) (5 cm, 0)
(b) (10 cm, 5 cm) BEETIDN “7.5 MAGNETIC F‘Lux DENSITY 7.31 Determine the magnetic ﬂux through a rectangular loop (a X 19) due to an inﬁnitely long
conductor carrying current I as shown in Figure 7.37. The loop and the straight conductors are separated by distance d. Figure 7.36 Twowire line of Problem
7.30. 4cm PROBLEMS 315 Brass ring Figure 7.38 Cross section of a brass ring enclosing a long
Figure 7.37 For Problem 731 straight wire; for Problem 7.34. 7.32 Let B = 4 cos<:—y)e_3z ax Wb/mz. Calculate the total magnetic ﬂux crossing the surface x=0,0<y<1,z>0. 7.33 In free space, the magnetic ﬂux density is B = yzax + zzay + xzaz Wb/m2 (a) Show that B is a magnetic ﬁeld
(b) Find the magnetic ﬂux through x = l, 0 < y < l, l < z < 4.
(c) Calculate J. *7.34 A brass ring with triangular cross section encircles a very long straight wire concentrically
as in Figure 7.38. If the wire carries a current I, show that the total number of magnetic
ﬂux lines in the ring is +b
[b—alna ] Calculate ‘Pifa = 30 cm, 19 = 10cm, h = 5 cm, and] = 10A. SECTIDN 7.6 MAXWELL’S EQUATIDNE 7.35 Consider the following arbitrary ﬁelds. Find out which of them can possibly represent an
electrostatic or magnetostatic ﬁeld in free space. (a) A = y cos axax + (y + e"‘)aZ
20 (b) B = 721‘, (c) C = r2 sin63¢ 316 CHAPTER '7 MAGNETDSTATIB FIELDS 7.36 Reconsider Problem 7.35 for the following ﬁelds. (a) D = yzzax + 2(x + 1)yzay — (x + 1)z2az + 1 sin
(Z ) cos gb a9 + (is
p p (b) E = az 1
(c) F=—2(20030a,+ sindag)
r SECTION 7.7 MAGNETIC] SGALAR AND VEBTUR PDTENTIALE 7.37 Find the magnetic ﬂux density B for each of these vector magnetic potentials. (a) A = e” sin yax + (1 + cos y)ay %
bA= a
() ﬁ+1z
r6 '0
(c) A: CO: ar—I— sm a9
r I” 7.38 For a current distribution in free space, A = (2x2); + yz)ax + (xy2 — xz3)ay — (6xyz  2362y2)az Wb/m (a) Calculate B.
(b) Find the magnetic ﬂux through a loop described by x = 1, 0 < y, z < 2.
(c) ShowthatVA = OandVB = 0. 7.39 The magnetic vector potential of a current distribution in free space is given by
A = 156” sin (15 az Wb/m
Find H at (3, 7r/4, — 10). Calculate the ﬂux through p = 5, O S (j) S 7r/2, 0 S z S 10. 7.40 An inﬁnitely long conductor of radius a carries a uniform current with J = JO az. Show
that the magnetic vector potential for p < a is 1
A = _Z MoJopzaz 7.41 The current sheet y = 0 has a uniform current density K = ko ax A/m. Determine the
vector potential. 7.42 Find the B ﬁeld corresponding to the magnetic vector potential , 7rx 7ry
A = s1n—cos—az
2 2 7.43 The magnetic vector potential of two parallel inﬁnite straight current ﬁlaments in fr65 _
space carrying equal current I in the opposite direction is
M dp A=—l
27l'n p az 7.44 7.45 7.46 *7.47 7.48 7.49 7.50 7.51 PROBLEMS 317 where d is the separation distance between the ﬁlaments (with one ﬁlament placed along
the z—axis). Find the corresponding magnetic ﬂux density B. The magnetic ﬁeld intensity in a certain conducting medium is H = xyzax + xzzay  yzzaz A/m (a) Calculate the current density at point P(2,  1, 3). a
(b) What is p”
a: at P? Find the current density J due to 0
2 a2 Wb/m in free space. Prove that the magnetic scalar potential at (0, O, z) due to a circular loop of radius a shown
in Figure 7.8(a) is I z
Vm = — 1— ——~————
2 I: [Z2 + a2]1/2] A coaxial transmission line is constructed such that the radius of the inner conductor is a
and the outer conductor has radii 3a and 4a. Find the vector magnetic potential within the
outer conductor. Assume AZ = 0 for p = 3a. The zaxis carries a ﬁlamentary current 12 A along az. Calculate Vm at (4, 30°, —2) if
V," = 0 at (10, 60°, 7). Plane z = * 2 carries a current of 50:1y A/m. If Vm = 0 at the origin, ﬁnd Vm at (a) (—2, 0, 5)
(b) (10,3,1) Prove in cylindrical coordinates that (a)V><(VV)=0
(b) V(VXA)=0 HR = r — r’ andR = IR,show that R
_—1l3— where V and V’ are del operators with respect to (x, y, z) and (x’, y’, 2’), respectively. ...
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 Fall '08
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 Magnetic Field, Vector potential, R D B L E M

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