Page 244 - Engineering Electromagnetics, 8th Edition

226 ENGINEERING ELECTROMAGNETICS

7.17 A current filament on the z axis carries a current of 7 mA in the a z direction,
and current sheets of 0.5 a z A/m and −0.2 a z A/m are located at ρ = 1cm
and ρ = 0.5 cm, respectively. Calculate H at: (a) ρ = 0.5 cm; (b) ρ =
1.5 cm; (c) ρ = 4 cm. (d) What current sheet should be located at ρ = 4cm
so that H = 0 for all ρ> 4 cm?
7.18 A wire of 3 mm radius is made up of an inner material (0 <ρ < 2 mm) for
7
which σ = 10 S/m, and an outer material (2 mm <ρ < 3 mm) for which
7
σ = 4 × 10 S/m. If the wire carries a total current of 100 mA dc, determine
H everywhere as a function of ρ.
7.19 In spherical coordinates, the surface of a solid conducting cone is described
by θ = π/4 and a conducting plane by θ = π/2. Each carries a total current
I. The current flows as a surface current radially inward on the plane to the
vertex of the cone, and then flows radially outward throughout the cross
section of the conical conductor. (a) Express the surface current density as a
function of r;(b)express the volume current density inside the cone as a
function of r;(c) determine H as a function of r and θ in the region between
the cone and the plane; (d) determine H as a function of r and θ inside the
cone.
7.20 A solid conductor of circular cross section with a radius of 5 mm has a
conductivity that varies with radius. The conductor is 20 m long, and there is
a potential difference of 0.1 V dc between its two ends. Within the conductor,
5 2
H = 10 ρ a φ A/m. (a) Find σ as a function of ρ. (b) What is the resistance
between the two ends?
7.21 Acylindrical wire of radius a is oriented with the z axis down its center line.
The wire carries a nonuniform current down its length of density
2
J = bρ a z A/m where b is a constant. (a) What total current flows in the
wire? (b) Find H in (0 <ρ < a), as a function of ρ;(c) find H out (ρ> a), as a
function of ρ;(d)verify your results of parts (b) and (c)by using ∇× H = J.
7.22 A solid cylinder of radius a and length L, where L
a, contains volume
charge of uniform density ρ 0 C/m . The cylinder rotates about its axis (the
3
z axis) at angular velocity
rad/s. (a) Determine the current density J as a
function of position within the rotating cylinder. (b) Determine H on-axis by
applying the results of Problem 7.6. (c) Determine the magnetic field
intensity H inside and outside. (d) Check your result of part (c)by taking
the curl of H.
2
7.23 Given the field H = 20ρ a φ A/m: (a) Determine the current density J.
(b) Integrate J over the circular surface ρ ≤ 1, 0 <φ < 2π, z = 0, to
determine the total current passing through that surface in the a z direction.
(c) Find the total current once more, this time by a line integral around the
circular path ρ = 1, 0 <φ < 2π, z = 0.
7.24 Infinitely long filamentary conductors are located in the y = 0 plane at x = n
meters where n = 0, ±1, ±2,... Each carries1Ainthe a z direction.