Page 243 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 225

Figure 7.22 See Problem 7.12.

ρ = 1 cm, −250 mA/m at ρ = 2 cm, and −300 mA/m at ρ = 3 cm. Calculate
H φ at ρ = 0.5, 1.5, 2.5, and 3.5 cm.

7.12 In Figure 7.22, let the regions 0 < z < 0.3m and 0.7 < z < 1.0mbe
2
conducting slabs carrying uniform current densities of 10 A/m in opposite
directions as shown. Find H at z =:(a) −0.2; (b) 0.2; (c) 0.4; (d) 0.75;
(e) 1.2 m.
7.13 A hollow cylindrical shell of radius a is centered on the z axis and carries a
uniform surface current density of K a a φ .(a) Show that H is not a function of
φ or z.(b) Show that H φ and H ρ are everywhere zero. (c) Show that H z = 0
for ρ> a.(d) Show that H z = K a for ρ< a.(e)A second shell, ρ = b,
carries a current K b a φ . Find H everywhere.
7.14 A toroid having a cross section of rectangular shape is defined by the
following surfaces: the cylinders ρ = 2 and ρ = 3 cm, and the planes z = 1
and z = 2.5 cm. The toroid carries a surface current density of −50a z A/m
on the surface ρ = 3 cm. Find H at the point P(ρ, φ, z): (a) P A (1.5 cm, 0,
2 cm); (b) P B (2.1 cm, 0, 2 cm); (c) P C (2.7cm,π/2, 2 cm); (d) P D (3.5cm,
π/2, 2 cm).
7.15 Assume that there is a region with cylindrical symmetry in which the
conductivity is given by σ = 1.5e −150ρ kS/m. An electric field of 30a z V/m
is present. (a) Find J.(b) Find the total current crossing the surface ρ< ρ 0 ,
z = 0, all φ.(c) Make use of Amp`ere’s circuital law to find H.
7.16 A current filament carrying I in the −a z direction lies along the entire
positive z axis. At the origin, it connects to a conducting sheet that forms the
xy plane. (a) Find K in the conducting sheet. (b) Use Ampere’s circuital law
to find H everywhere for z > 0; (c) find H for z < 0.