Page 241 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 223

therefore select the φ component of ∇× A = B and integrate to obtain A z .Try it,
you’ll like it!

D7.10. Equation (66) is obviously also applicable to the exterior of any con-
ductor of circular cross section carrying a current I in the a z direction in free
space. The zero reference is arbitrarily set at ρ = b.Now consider two con-
ductors, each of 1 cm radius, parallel to the z axis with their axes lying in
the x = 0 plane. One conductor whose axis is at (0, 4cm, z) carries 12 A
in the a z direction; the other axis is at (0, −4cm, z) and carries 12 A in the
−a z direction. Each current has its zero reference for A located 4 cm from its
axis. Find the total A field at: (a)(0, 0, z); (b)(0, 8cm, z); (c)(4 cm, 4cm, z);
(d)(2 cm, 4cm, z).
Ans. 0; 2.64 µWb/m; 1.93 µWb/m; 3.40 µWb/m

REFERENCES

1. Boast, W. B. (See References for Chapter 2.) The scalar magnetic potential is defined on
p. 220, and its use in mapping magnetic fields is discussed on p. 444.
2. Jordan, E. C., and K. G. Balmain. Electromagnetic Waves and Radiating Systems. 2d ed.
Englewood Cliffs, N.J.: Prentice-Hall, 1968. Vector magnetic potential is discussed on
pp. 90–96.
3. Paul, C. R., K. W. Whites, and S. Y. Nasar. Introduction to Electromagnetic Fields.
3d ed. New York: McGraw-Hill, 1998. The vector magnetic potential is presented on
pp. 216–20.
4. Skilling, H. H. (See References for Chapter 3.) The “paddle wheel” is introduced on
pp. 23–25.

CHAPTER 7 PROBLEMS
7.1 (a) Find H in rectangular components at P(2, 3, 4) if there is a current
filament on the z axis carrying 8 mA in the a z direction. (b) Repeat if the
filament is located at x =−1, y = 2. (c) Find H if both filaments are present.
7.2 A filamentary conductor is formed into an equilateral triangle with sides of
length carrying current I. Find the magnetic field intensity at the center of
the triangle.
7.3 Two semi-infinite filaments on the z axis lie in the regions −∞ < z < −a
and a < z < ∞. Each carries a current I in the a z direction. (a) Calculate H
as a function of ρ and φ at z = 0. (b) What value of a will cause the
magnitude of H at ρ = 1, z = 0, to be one-half the value obtained for an
infinite filament?
7.4 Two circular current loops are centered on the z axis at z =±h. Each loop
has radius a and carries current I in the a φ direction. (a) Find H on the z axis
over the range −h < z < h.Take I = 1A and plot |H| as a function of z/a if