Problems
Chapter 1
P1.1 For a simple lossless medium, derive the wave equations for E and H. P1.2 Obtain the wave Equations 1.18 and 1.19 subject to the Lorentz condition given by Equation 1.21. P1.3 Obtain Equation 1.27. P1.4 Show that for a time-harmonic case, Equation 1.22 reduces to Equation 1.50. P1.5 a. Obtain the expression for the time-harmonic retarded potential A for a current element of small length h located on the z-axis as shown in the figure.
P
z
h
R
b. Hence obtain the electric and magnetic fields at P. Assume that the current in the filament is I0(A). c. In the far zone, that is, for R >> λ, h << λ, obtain E, H, E × H, and <S>. P1.6 Lecture module-week 1 is devoted to reviewing Maxwell’s equations. In the undergraduate prerequisite course(s), considerable amount of time is spent in arriving at these equations based on experimental laws. In the process, vector calculus and coordinate systems are explained at length. The questions for the week make you revise the undergraduate background. You can look into the textbook you used in your undergraduate prerequisite course or search the internet in answering these two questions. The questions are: a. Equations 1.1 through 1.4 are in differential form. Write down the corresponding equations in the integral form. b. The integral form of Equation 1.1, Faraday’s law, allows you to compute the voltage induced (emf) in a circuit. There are two components to it: (i) transformer emf and (ii) motional emf. Give an example where only (i) is induced; give an example where only (ii) is induced and give a third example where both (i) and (ii) are induced. In the third case, if your interest is in computing the total induced voltage and not in separating them into the two components, is there a simpler way of computing the total induced voltage? Illustrate the simpler way through an example. P1.7 This question is to make you think about the concept of retarded potentials. Equations 1.22 and 1.24 are useful in answering this question. Note that Equation 1.22 is written when the source is a volume current element. For a filamentary current source, you will replace J dV by I dL. Suppose the source is a differential filamentary current element located at the origin along the y-axis, where
I dL = 2t yˆ dy. 387
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Obtain the differential vector potential dAP at the point P (5, 10, 15). Note the following. You can use a bold letter or a letter with an arrow on the top to denote a vector. The hat on the top of y denotes a unit vector in the y-direction in the Cartesian system of coordinates (rectangular coordinate system). P1.8 Consider a cylindrical capacitor of length . The radius of the inner cylinder is a and the radius of the outer cylinder is b. The dielectric constant of the dielectric inside the capacitor varies as (K is a constant) ε r (ρ) = Kρ2 .
Show that the conduction current I(t) in the wire is the same as the displacement current in the capacitor.
b I(t)
a
V0 sin(ωt)
Dielectric εr(ρ) = Kρ2
P1.9 A parallel plate capacitor is of cross-sectional area A and is filled with a dielectric material whose permittivity varies linearly from ε = ε1 at one plate (y = 0) to ε = ε2 at the other plate (y = d). Neglecting fringing effects, determine the capacitance. +Q ε(y) –Q
Plate area a
y=d
E y=0 Area a
P1.10 Show that the displacement current between two concentric cylindrical conduc ting shells of radii r1 and r2, r2 > r1 is exactly the same as the conduction current in the external circuit. The applied voltage is V = V0 sin ωt.
r2
r1
V0 sin ωt
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P1.11 Derive (2.42) through (2.44), based on Section 1.6.
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Chapter 2
P2.1 Derive Equations 2.1 through 2.3. P2.2 Assume that the time-harmonic fields in a perfect conductor are zero. Show that boundary conditions at the interface of a perfect conductor are given by Equations 2.23 or 2.24 or 2.25 or 2.26. P2.3 The principle of induction heating is explored in this problem. A sheet of metal (good conductor) of conductivity σc is inserted into AC magnetic field whose ˆ 1. value at the surface z = ± d is assumed to be H1 = − yH
z H1
d
μ, σc x
H1
a. Show that the AC magnetic field in the conductor is given by H(z) = H1 (cosh τz/cosh τd) where τ = (1 + j)/δ and δ = 1/ πf µσ c . Assume that δ ~ d. b. Show that J = Jxˆ , where J = τH1(sinh τz/cosh τd). c. Find the power consumed by the sheet of length (along x-axis) 1 m and width (along y-axis) 1 m. You do not have to show it for the homework, but a little further work will show that the power consumed per unit volume is maximized if d ≈ 1.125 δ. P2.4 Figure P2.4a shows the circuit equivalent of a section Δz of a transmission line. a. Find the differential equation for the instantaneous current I(z, t), b. Assume that the current varies harmonically in space and time, that is,
I ( z , t) = I exp[ j(ωt − kz)].
–d
Find the relation between ω and k. c. A sketch of ω and k is given in Figure P2.4b. Find the value of the cutoff frequency ωc and the slope of the asymptotes (dotted lines). (a) I(z, t )
C′0/Δz
I(z + Δz, t )
(b) ω
L′Δz V(z, t )
C′/Δz
V(z + Δz, t )
ωc k
Figure P2.4
P2.5 The electric and magnetic fields of a coaxial transmission line whose crosssection is shown in Figure P2.5 are given as follows: The line connects a source to a load.
At ρ = 1: E = 100a ρ + 0.05a z V/m , H = 2a φ A/m At ρ = 2: E = Ca ρ + 0.02a z V/m , H = Da φ A/m
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a. Determine C and D. b. Let the length of the line be 30 m. Find the source power, power lost in the line, and the power received by the load. c. There could be a misprint in the statement: “At ρ = 2: E = Ca ρ + 0.02a z V/m.” Find the misprint and correct it. Give one or two reasons as to why you think there is a misprint. This misprint perhaps has not affected your answers to parts (a) and (b).
y
Dielectric 1
x
2
Inner conductor current direction out of the paper
Outer conductor current direction into the paper
P2.6 A loss-free nonuniform transmission line has
L′ = L′( z),
C ′ = C ′( z),
where L′ and C′ are per meter values of the series inductance and parallel capacitance of the transmission line, respectively. a. Determine the partial differential equation for the instantaneous voltage V(z, t). b. For an exponential transmission line
L′( z) = L0 exp(qz),
C ′( z) = C0 exp(− qz),
assuming V(z, t) = V0 exp[j(ωt − kz)], determine the relation between ω and k. P2.7 A distortionless line satisfies the condition
R′ G ′ = . L′ C′
Find α, β, and Z0 for such a line. P2.8 The magnetic field vector-phasor of a plane wave is given by
H( x) = 2[ y + jz ]exp( jkx).
(P2.1)
a. Use the properties of a plane wave to determine E( x , t).
b. You probably recognized from the expression for E( x , t) that it is circularly polarized. Is it L wave or R wave?
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c. Had you used Equation P2.1 to determine the polarization, you would get the same answer as that for (b). Check this out by drawing the diagram like Figure 2.6b or c. d. Determine the instantaneous power density S( x , t) and active power density < SR ( x) > . P2.9 A conducting film of impedance 377 Ω/square is placed a quarter-wavelength in air from a plane conductor (PEC) to eliminate wave reflection at 9 GHz. Assume negligible displacement currents in the film. Plot a curve showing the fraction of incident power reflected versus frequency for frequencies 6–18 GHz.
377 Ω/sq. PEC
Free space d = λ/4
P2.10 Calculate the reflection coefficient and percent of incident energy reflected when a uniform plane wave is normally incident on a Plexiglas radome (dielectric window) of thickness 3/8″, relative permittivity εr = 2.8, with free space on both sides. λ0 = 20 cm. Repeat for λ0 = 10 cm. Repeat for λ0 = 3 cm. Comment on the results obtained. P2.11 A green ion laser beam, operating at λ0 = 5.45 μm, is generated in vacuum, and then passes through a glass window of refractive index 1.5 into water with n = 1.34. Design a window to give zero reflection at the two surfaces for a wave polarized with E in the plane of incidence, that is, find θB2 − θB1.
θB2 θB1 n1 = 1
n3 = 1.34
n2 = 1.5
P2.12 The transmitting antenna of a ground-to-air communication system is placed at a height of 10 m above the water, as shown in the figure. For a separation of 10 km between the transmitter and the receiver, which is placed on an airborne platform, find the height h2 above water of the receiving system so that the wave reflected by the water does not possess a parallel-polarized component. Assume that the water surface is flat and lossless. [Reference 2 of Chapter 2]
h2 ε0, μ0
10 m
Water (εr = 81) 104 m
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P2.13 A source Vgs = 60∠0° with internal impedance 200 Ω is connected to a resistive load ZL1 = 200 Ω by a two-wire lossless transmission line in air (velocity of propagation c = 3 × 108 m/s) of length l1 = 10 cm and characteristic impedance
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