Page 419 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 401

From Euler’s identity, we find that e jδ/2 + e − jδ/2 = 2 cos δ/2, and e jδ/2 − e − jδ/2 =
2 j sin δ/2. Using these relations, we obtain

E sT = 2E 0 [cos(δ/2)a x + sin(δ/2)a y ]e − j(βz−δ/2) (102)
We recognize (102) as the electric field of a linearly polarized wave, whose field
vector is oriented at angle δ/2 from the x axis.

Example 11.7 shows that any linearly polarized wave can be expressed as the
sum of two circularly polarized waves of opposite handedness, where the linear po-
larization direction is determined by the relative phase difference between the two
waves. Such a representation is convenient (and necessary) when considering, for
example, the propagation of linearly polarized light through media which contain
organic molecules. These often exhibit spiral structures having left- or right-handed
pitch, and they will thus interact differently with left- or right-hand circular polar-
ization. As a result, the left circular component can propagate at a different speed
than the right circular component, and so the two waves will accumulate a phase
difference as they propagate. As a result, the direction of the linearly polarized field
vector at the output of the material will differ from the direction that it had at the
input. The extent of this rotation can be used as a measurement tool to aid in material
studies.
Polarization issues will become extremely important when we consider wave
reflection in Chapter 12.

REFERENCES
1. Balanis, C. A. Advanced Engineering Electromagnetics.New York: John Wiley & Sons,
1989.
2. International Telephone and Telegraph Co., Inc. Reference Data for Radio Engineers. 7th
ed. Indianapolis, Ind.: Howard W. Sams & Co., 1985. This handbook has some excellent
data on the properties of dielectric and insulating materials.
3. Jackson, J. D. Classical Electrodynamics.3d ed. New York: John Wiley & Sons, 1999.
4. Ramo, S., J. R. Whinnery, and T. Van Duzer. Fields and Waves in Communication
Electronics.3d ed. New York: John Wiley & Sons, 1994.

CHAPTER 11 PROBLEMS
11.1 Show that E xs = Ae j(k 0 z+φ) is a solution of the vector Helmholtz equation,
√
Eq. (30), for k 0 = ω µ 0 0 and any φ and A.
11.2 A10 GHz uniform plane wave propagates in a lossless medium for which
r = 8 and µ r = 2. Find (a) ν p ;(b) β;(c) λ;(d) E s ;(e) H s ;( f ) S .
11.3 An H field in free space is given as H(x, t) = 10 cos(10 t − βx)a y A/m.
8
Find (a) β;(b) λ;(c) E(x, t)at P(0.1, 0.2, 0.3) at t = 1 ns.
11.4 Small antennas have low efficiencies (as will be seen in Chapter 14), and the
efficiency increases with size up to the point at which a critical dimension of