Page 245 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 227

(a) Find H on the y axis. As a help,

y 1
∞
π π
= − +
y + n 2 2 2y e 2πy − 1
2
n=1
(b) Compare your result of part (a)to that obtained if the filaments are
replaced by a current sheet in the y = 0 plane that carries surface current
density K = 1a z A/m.
7.25 When x, y, and z are positive and less than 5, a certain magnetic field
2 2 2
intensity may be expressed as H = [x yz/(y + 1)]a x + 3x z a y −
2
[xyz /(y + 1)]a z . Find the total current in the a x direction that crosses the
strip x = 2, 1 ≤ y ≤ 4, 3 ≤ z ≤ 4, by a method utilizing: (a)a surface
integral; (b)a closed line integral.
7.26 Consider a sphere of radius r = 4 centered at (0, 0, 3). Let S 1 be that portion
of the spherical surface that lies above the xy plane. Find
(∇× H) · dS if
H = 3ρ a φ in cylindrical coordinates. S 1

7.27 The magnetic field intensity is given in a certain region of space as H =
2
[(x + 2y)/z ]a y + (2/z)a z A/m. (a) Find ∇× H. (b) Find J.(c) Use J to find
the total current passing through the surface z = 4, 1 ≤ x ≤ 2, 3 ≤ z ≤ 5,
in the a z direction. (d) Show that the same result is obtained using the other
side of Stokes’ theorem.
2
7.28 Given H = (3r / sin θ)a θ + 54r cos θa φ A/m in free space: (a) Find the total
current in the a θ direction through the conical surface θ = 20 ,0 ≤ φ ≤ 2π,
◦
0 ≤ r ≤ 5, by whatever side of Stokes’ theorem you like the best. (b) Check
the result by using the other side of Stokes’ theorem.
7.29 A long, straight, nonmagnetic conductor of 0.2 mm radius carries a
uniformly distributed current of2Adc.(a) Find J within the conductor.
(b) Use Amp`ere’s circuital law to find H and B within the conductor.
(c) Show that ∇× H = J within the conductor. (d) Find H and B outside the
conductor. (e) Show that ∇× H = J outside the conductor.
7.30 (An inversion of Problem 7.20.) A solid, nonmagnetic conductor of circular
cross section has a radius of 2 mm. The conductor is inhomogeneous, with
6 2
6
σ = 10 (1 + 10 ρ ) S/m. If the conductor is1min length and has a voltage
of 1 mV between its ends, find: (a) H inside; (b) the total magnetic flux
inside the conductor.
7.31 The cylindrical shell defined by 1 cm <ρ < 1.4cm consists of a
nonmagnetic conducting material and carries a total current of 50 A in the a z
direction. Find the total magnetic flux crossing the plane φ = 0, 0 < z < 1:
(a)0 <ρ < 1.2 cm; (b) 1.0 cm <ρ < 1.4 cm; (c) 1.4 cm <ρ < 20 cm.
7.32 The free space region defined by 1 < z < 4cm and 2 <ρ < 3cmisa toroid
of rectangular cross section. Let the surface at ρ = 3cm carry a surface
current K = 2a z kA/m. (a) Specify the current densities on the surfaces at