Page 246 - Engineering Electromagnetics, 8th Edition

228 ENGINEERING ELECTROMAGNETICS

ρ = 2 cm, z = 1 cm, and z = 4 cm. (b) Find H everywhere. (c) Calculate
the total flux within the toroid.
7.33 Use an expansion in rectangular coordinates to show that the curl of the
gradient of any scalar field G is identically equal to zero.
7.34 A filamentary conductor on the z axis carries a current of 16 A in the a z
direction, a conducting shell at ρ = 6 carries a total current of 12 A in the
−a z direction, and another shell at ρ = 10 carries a total current of4Ain
the −a z direction. (a) Find H for 0 <ρ < 12. (b) Plot H φ versus ρ.
(c) Find the total flux
crossing the surface 1 <ρ < 7, 0 < z < 1, at fixed
φ.
7.35 A current sheet, K = 20 a z A/m, is located at ρ = 2, and a second sheet,
K =−10a z A/m, is located at ρ = 4. (a) Let V m = 0at P(ρ = 3,φ = 0,
z = 5) and place a barrier at φ = π. Find V m (ρ, φ, z) for −π< φ < π.
(b) Let A = 0at P and find A(ρ, φ, z) for 2 <ρ < 4.
7.36 Let A = (3y − z)a x + 2xza y Wb/m in a certain region of free space.
(a) Show that ∇ · A = 0. (b)At P(2, −1, 3), find A, B, H, and J.
7.37 Let N = 1000, I = 0.8A, ρ 0 = 2 cm, and a = 0.8cm for the toroid shown
in Figure 7.12b. Find V m in the interior of the toroid if V m = 0at ρ = 2.5
cm, φ = 0.3π.Keep φ within the range 0 <φ < 2π.
7.38 A square filamentary differential current loop, dL on a side, is centered at the
origin in the z = 0 plane in free space. The current I flows generally in the
a φ direction. (a) Assuming that r >> dL, and following a method similar to
that in Section 4.7, show that

2
µ 0 I(dL) sin θ
dA = a φ
4πr 2
(b) Show that
I(dL) 2
dH = (2 cos θ a r + sin θ a θ )
4πr 3
The square loop is one form of a magnetic dipole.
7.39 Planar current sheets of K = 30a z A/m and −30a z A/m are located in free
space at x = 0.2 and x =−0.2, respectively. For the region −0.2 < x < 0.2
(a) find H;(b) obtain an expression for V m if V m = 0at P(0.1, 0.2, 0.3);
(c) find B;(d) obtain an expression for A if A = 0at P.
7.40 Show that the line integral of the vector potential A about any closed path is
equal to the magnetic flux enclosed by the path, or A · dL =
B · dS.
7.41 Assume that A = 50ρ a z Wb/m in a certain region of free space. (a) Find H
2
and B.(b) Find J.(c) Use J to find the total current crossing the surface
0 ≤ ρ ≤ 1, 0 ≤ φ< 2π, z = 0. (d) Use the value of H φ at ρ = 1to calculate
H · dL for ρ = 1, z = 0.