Page 242 - Engineering Electromagnetics, 8th Edition

224 ENGINEERING ELECTROMAGNETICS

Figure 7.21 See Problem 7.5.

(b) h = a/4; (c) h = a/2; (d) h = a. Which choice for h gives the most
uniform field? These are called Helmholtz coils (of a single turn each in this
case), and are used in providing uniform fields.
7.5 The parallel filamentary conductors shown in Figure 7.21 lie in free space.
Plot |H| versus y, −4 < y < 4, along the line x = 0, z = 2.
7.6 A disk of radius a lies in the xy plane, with the z axis through its center.
Surface charge of uniform density ρ s lies on the disk, which rotates about
the z axis at angular velocity
rad/s. Find H at any point on the z axis.
7.7 A filamentary conductor carrying current I in the a z direction extends along
the entire negative z axis. At z = 0it connects to a copper sheet that fills the
x > 0, y > 0 quadrant of the xy plane. (a) Set up the Biot-Savart law and
find H everywhere on the z axis; (b) repeat part (a), but with the copper sheet
occupying the entire xy plane (Hint: express a φ in terms of a x and a y and
angle φ in the integral).
7.8 For the finite-length current element on the z axis, as shown in Figure 7.5,
use the Biot-Savart law to derive Eq. (9) of Section 7.1.
7.9 A current sheet K = 8a x A/m flows in the region −2 < y < 2in the plane
z = 0. Calculate H at P(0, 0, 3).
7.10 A hollow spherical conducting shell of radius a has filamentary connections
made at the top (r = a,θ = 0) and bottom (r = a,θ = π). A direct current I
flows down the upper filament, down the spherical surface, and out the lower
filament. Find H in spherical coordinates (a) inside and (b) outside the
sphere.
7.11 An infinite filament on the z axis carries 20π mA in the a z direction. Three
a z -directed uniform cylindrical current sheets are also present: 400 mA/m at