Questions and Answers of Elements Of Electromagnetics

(a) If A = x²yax + xay + 2yzz, find ∇ · A at point ∇ · B; (-3, 4, 2).(b) Given that B = 3ρ sin Φaρ – 5p²zaΦ + 8z cos² Φaz, find ∇ · B at point (5,30°,1)
Find the potential V(x, y) due to the two-dimensional systems of Figure 6.35.a.b.c. V=0 y a V=V₂ V=V₁ V=0 X
In a source-free region, show that JE dt VE – μετ με Ε of² = 0
(a) Show that E(x, t) = cos(x + ωt) + cos(x – ωt) satisfies the scalar wave equation.(b) Determine the velocity of wave propagation.
A uniform plane wave has a wavelength of 6.4 cm in free space and 2.8 cm in a dielectric (σ = 0, ε = εo εr, μ = μo). Find εr.
For a uniform plane wave at 4 GHz, the intrinsic impedance and phase velocity of an unknown material are measured as 105 Ω and 7.6 x 107 m/s, respectively. Find εr and μr of the material.
The electric field intensity in a dielectric medium (μ = μo, ε = εoεr) is given by E = 150 cos(109t + 8x)az V/m Calculate(a) The dielectric constant εr(b) The intrinsic impedance(c) The
A coaxial cable consists of two conducting cylinders of radii a and b. The electric and magnetic fields in the cable arewhere Vo and Io are constants. (a) Determine the time-average
Human exposure to the electromagnetic radiation in air is regarded as safe if the power density is less than 10 mW/m2. What is the corresponding electric field intensity?
At the bottom of a microwave oven, E = 2.4 kV/m. If this value is found uniformly over the entire area of the oven, which is 450 cm2, determine the power delivered by the oven. Assume μ = μo, ε =
Identify the configuration in Figure 7.24 that is not a correct representation of I and H.(a)(b)(c)(d) (e) 1 H
A hollow cylinder of radius a = 2 cm is 10 m long. Find the inductance of the cylinder. 1. Wire L = 2. Hollow cylinder нов 2π l» a L = Hol 877 L = In 20 3. Parallel wires Hol d L == In
Given the total electromagnetic energy show from Maxwell’s equations that w=f W (E-D+ H B) dv 2
Retrieve Faraday’s law in differential form from E = -VV - JA at
A semicircular loop of radius a in free space carries a current I. Determine the magnetic flux density at the center of the loop.
In a certain homogeneous medium, ε = 81 εo, and μ = μo,If ω = 2π 109 rad/m, find b and Ho. E = 10e(t + B²), V/m B₂), H, = Hejt+B₂), A/m
Assume an electric field intensity of 20 kV/m and a magnetic flux density of 5 μ Wb/m2 exist in a region. Find the ratio of the magnitudes of electric and magnetic forces on an electron that
Region 1 is defined by x – y + 2z > 5 with μ1 = 2μo, while region 2 is defined by x – y + 2z < 5 with μ2 = 5μo. If H1 = 40ax + 20ay – 30az A/m, find (a)
In free space, the electric field is given by E = Eo cosz costazFind the charge density ρv that will produce this field.
Use Maxwell’s equations to derive the continuity equation.
Show that the phase constant in eq. (10.24) can be approximated as β = wV με 1 + . ων σ for = 1. ωε σ δωτε? 2
In air, E = cos(12πx)sin(1011t – αy)az V/m. Find H and a.
In a certain medium, E = 10 cos(108t – 3y)ax V/m. What type of medium is it?(a) Free space(b) Lossy dielectric(c) Lossless dielectric(d) Perfect conductor
Electromagnetic waves travel faster in conductors than in dielectrics.(a) True(b) False
In a nonmagnetic medium,Find E. H = 50e-100x cos(2π X 10°t - 200x)a, mA/m
Determine the loss tangent for each of the following nonmagnetic media at 12 MHz.(a) Wet earth (ε = 10εo, σ = 10–2 S/m)(b) Dry earth (ε = 4εo, σ = 10–4 S/m)(c) Seawater (ε = 81εo ,
In a good conductor, E and H are in time phase.(a) True(b) False
Alumina is a ceramic material used in making printed circuit boards. At 15 GHz, ε = 9.6eo, μ = μo, tan θ = 3 x 3 10–4. Calculate(a) The penetration depth, (b) The total attenuation over a
The Poynting vector physically denotes the power density leaving or entering a given volume in a time-varying field.(a) True(b) False
The properties of a medium are σ = εω, ε = 4εo, μ = μo. If the wavelength in free space is 12 cm, find α, β, and u in the medium.
At f = 100 MHz, show that silver (σ = 6.1 x 107 S/m, μr = 1, εr = 1) is a good conductor, while rubber (σ = 10–15 S/m, μr = 1, εr = 3.1) is a good insulator.
A certain medium has σ = 1 S/m, ε = 4εo, and μ = 9 μo at a frequency of 1 GHz.Determine the (a) Attenuation constant, (b) Phase constant, (c) Intrinsic impedance, and(d) Wave
The electric field of a TV broadcast signal progagating in air is given by E (z,t) = 0.2 cos(ωt – 6.5z)ax V/m(a) Determine the wave frequency ω and the wavelength λ.(b) Sketch Ex as a
The electric field intensity of a uniform plane wave in air is given by E = 50 sin(108 πt – βx)az mV/m(a) Calculate β.(b) Determine the location(s) where E vanishes at t = 50 ns.(c)
In a lossless medium (εr = 4.5, μr = 1), a uniform plane wave E = 8 cos(ωt – βz)ax – 6 sin(ωt – βz)ay V/m propagates at 40 MHz. (a) Find H.(b) Determine β, λ, η,
The magnet field intensity of a uniform plane wave in a good conductor (ε = εo,μ = μo) is H = 20e–12z cos(2π x 106t + 12z)ay mA/mFind the conductivity and the corresponding E field.
A uniform plane wave impinges normally on a conducting medium. If the frequency s 100 MHz and the skin depth is 0.02 mm, determine the velocity of the wave in the conducting medium.
A 10 GHz wave passes through a medium made of copper (ε = εo, μ = μo, σ = 5.8 x 107 S/m); find: (a) Attenuation constant,(b) The skin depth,(c) The intrinsic impedance.
For silver, σ = 6.1 x 107 S/m, μr = 1, εr = 1, determine the frequency at which the penetration depth is 2 mm.
Compute the penetration depth of copper at the power frequency of 60 Hz.
By measurements conducted at 12 MHz on a certain material, it is found that the intrinsic impedance is 24.6∠45° Ω with μ = μ . Find α, β, λ, and u.
How deep does a radar wave at 2 GHz travel in seawater before its amplitude is reduced to 10–5 of its amplitude just below the surface? Assume that μ = μo, ε = 24 εo, σ = 4 S/m.
Fat tissue at 2.42 GHz has the following properties: σ = 0.12 S/m, ε = 5.5 εo, and μ = μo. Find the penetration depth.
Brass waveguides are often silver plated to reduce losses. If the thickness of silver (μ = μo, ε = εo, σ = 6.1 x 107 S/m) must be 5d, find the minimum thickness required for a waveguide
The electric field intensity of a uniform plane wave in a medium(σ = 0, μ = μo, ε = εo εr) is E = 12 sin(2π x 107 t – 3y)az V/m(a) Determine the polarization of the wave.(b)
An antenna is located at the origin of a spherical coordinate system. The fields produced by the antenna in free space arewhereDetermine the time-average power radiated by the antenna. E E =
Show that eqs. (10.77) and (10.78) are equivalent. Pave (2) T To P(z, t) dt
Let E = 2 sin(ωt – βx)ay – 5 sin(ωt – βx)az V/m. What is the wave polarization?
Determine the wave polarization of each of the following waves:(a) Eo cos(ωt + βy)ax + Eo sin(ωt + βy)az V/m(b) Eo cos(ωt – βy)ax – 3Eo sin(ωt + βy)az V/m
The electric field intensity of a uniform plane wave in free space is given by E = 40 cos(ωt – βz)ax + 60 sin(ωt – βz)ay V/m(a) What is the wave polarization?(b) Determine the
Show that a linearly polarized plane wave of the form Es = Eoe–jbz ax can be expressed as the sum of two circularly polarized waves.
Determine the polarization of the following waves:(a) Es = 40e j10zax + 60e j10zay V/m(b) Es = 12ejπ/3e–j10xay + 5e–jπ/3e–j10xaz V/m
Suppose E(y,t) = Eo1 cos(ωt – βy)ax + Eo2 cos(ωt – βy + Φ)az V/m. Determine the polarization when (a) Φ = 0, (b) Φ = π/2, (c) Φ = π.
In free space, E = 40 cos(ωt – 10z)ay V/m. Find the total average power passing through a circular disk of radius 1.5 m in the z = 0 plane.
The electric field component of a uniform plane wave traveling in seawater (σ = 4 S/m, ε = 81 εo, μ = μo ) is E = 8e–0.1z cos(ωt – 0.3z)ax V/m(a) Determine the average power
A signal in air (z ≥ 0) with the electric field component E = 10 sin(ωt + 3z)ax V/m hits normally the ocean surface at z = 0 as in Figure 10.24. Assuming that the ocean surface is smooth and
A plane wave in free space has H (x, t) = (10ay – 20az) sin(ωt – 40x) A/m Find ω, E, and Pave.
Given that E = cos(ωt – βz)ax + sin(ωt – βz)ay V/m, show that the Poynting vector is constant everywhere.
Electromagnetic radiation can be used to heat cancerous tumors. If a plane wave is normally incident on the tissue surface at 1.2 GHz as shown in Figure. 10.26, determine the refection coefficient.
A plane wave in a lossless medium (σ = 0, μ = 2 μo, ε = 8εo) is given as E = 60sin(ωt – 10z)ax + 30sin(ωt – 10z + π/6)ay V/m.This wave is incident on a lossless medium (σ = 0,
A uniform plane wave in free space impinges perpendicularly on a lossless nonmagnetic material with ε = 9εo. Calculate the fraction of the incident average power that is transmitted.
A uniform plane wave propagates in a medium for which σ = 0, ε = 16 εo μ = μo. The electric field in the medium is E = 60cos(ωt – βz)ax V/m where ω = 90 Mrad/s. If the wave hits
A 100 MHz plane wave is normally incident from air to the sea surface, which may be assumed to be calm and smooth. If σ = 4 S/m,μr = 1, and εr = 81 for seawater, calculate the fractions of
A uniform plane wave in a certain medium (μ = μ0, ε4εo) is given by E = 12 cos(tω – 40πx)az V/m(a) Find ω.(b) If the wave is normally incident on a dielectric (μ = μo, ε = 3.2εo),
Show that the field Es = 20 sin(kx x) cos(kyy)az where k2x + k2y = ω2 μoεo, can be represented as the superposition of four propagating plane waves. Find the
An EM plane wave in a lossless medium impinges normally on a lossy medium.(a) Determine the ratio of transmitted to incident power in terms of the standing wave ratio s.(b) Express the ratio of
A plane wave in free space has a propagation vector k = 124ax + 124ay 263azFind the wavelength, frequency, and angles k makes with the x-, y-, and z-axes.
In free space, Es = [Eoax + ay + (3 + j4)az ]e – j(3.4x – 4.2y) V/m Determine Eo, Hs, and frequency.
If region 1 is in free space, while region 2 is a nonmagnetic dielectric medium (σ2 = 0, εr2 = 6.4), compute Ero/Eio and Eto/Eio for oblique incidence at θi = 12º. Assume parallel
In a dielectric medium (ε = 9εo, μ = μo), a plane wave with H = 0.2 cos(109t – kx – k√8z)ay A/m is incident on an air boundary at z = 0. Find(a) θr and θt(b) k(c) The wavelength
A plane wave in free space has Ei = (4ax + 5ay – 3az)cos(ωt – 0.5πx – 0.866πz)Determine: (a) The perpendicular- and parallel-polarized components of the
A parallel-polarized wave in free space impinges on a dielectric medium (σ = 0, ε = εo εr, μ = μo ). If the Brewster angle is 68°, find εr.
If u is the phase velocity of an EM wave in a given medium, the index of refraction of the medium is n = c/u, where c is the speed of light in vacuum.(a) Paraffin has μr = 1, εr = 2.1.
A coaxial cable has inner radius a and outer radius b. If the inner and outer conductors are separated by a material with conductivity σ, show that the conductance per unit length is G
In free space, a microwave signal has a frequency of 8.4 GHz. Calculate the wavelength of the signal.
Which of the following statements are not true of the line parameters R, L, G, and C?(a) R and L are series elements.(b) G and C are shunt elements.(c) G = 1 / R.(d) LC = με and RG = σε.(e) Both
For a lossy transmission line, the characteristic impedance does not depend on(a) The operating frequency of the line(b) The length of the line(c) The load terminating the line(d) The conductivity of
A coaxial cable has an inner conductor of radius a = 0.8 mm and an outer conductor of radius b = 2.6 mm. The conductors have σc = 5.28 x 107 S/m, μc = μo, and ∈c = ∈o; they
Match the following normalized impedances with points A, B, C, D, and E on the Smith chart of Figure 11.44.(i) 0 + j0 (ii) 1+ j0(iii) 0 – j1 (iv) 0 + j1(v) ∞ + j∞(vi) [Zin /
Which of the following conditions will not guarantee a distortionless transmission line?(a) R = 0 = G(b) RC = GL(c) Very low frequency range (R >> ωL, G >> ωC)(d) Very high frequency
A small section Δz of a transmission line may be represented by the equivalent circuit in Figure 11.47. Determine the voltage–current relationship for the section. I(z.t)
A planar transmission line is made of copper strips of width 30 mm and are separated by a dielectric of thickness 2 mm, σ = 10–3 S/m, ε = 4 εo, μ = μo. The conductivity of copper is 5.8 x
(a) Show that at high frequencies (R << ωL, G << ωL),(b) Obtain a similar formula for Zo. RC *-(√²+√9+ L Y + jw VLC
Which of these is not true of a lossless line?(a) Zin = 2jZo for a shorted line with ℓ = λ /8.(b) Zin = j∞ for a shorted line with ℓ = λ /4.(c) Zin = jZo for an open line with ℓ =
A lossless transmission line of length 50 cm with L = 10 mH/m, C = 40 pF/m is operated at 30 MHz. Its electrical length is(a) 20λ(b) 0.2λ(c) 108°(d) 40π(e) None of the above
A TV twin-lead is made of two parallel copper wires with a = 1.2 mm. The wires are separated by 1.5 cm of a dielectric material with εr = 4. Calculate L, C, and Zo.
A transmission line operates at 12 MHz and has the following parameters:R = 0.2 Ω/m, G = 4 mS/m, L = 40 μH/m, C = 25 μH/m(a) Compute γ using eq. (11.11).(b) Compute γ using the result of
A 500 m lossless transmission line is terminated by a load that is located at P on the Smith chart of Figure 11.45. If λ = 150 m, how many voltage maxima exist on the line?(a) 7 (b) 6(c)
Write true (T) or false (F) for each of the following statements.(a) All r- and x-circles pass through point (Γr , Γi) = (1, 0).(b) Any impedance repeats itself every λ/4 on the
The voltage on a line is given bywhere ℓ is the distance from the load. Calculate γ and u. V(C) = 80e¹0-3 cos(2 X 10t+ 0.01€) + 60e-10-³ cos(27 X 10¹t+ 0.01€) V
In an air line, adjacent maxima are found at 12.5 cm and 37.5 cm. The operating frequency is(a) 1.5 GHz (b) 600 MHz(c) 300 MHz (d) 1.2 GHz
The current along a transmission line is given byI(z,t) = I1(ωt – βz) + I2(ωt + βz)(a) Determine which of the two components represents a wave traveling from source to load.(b) Find the
Two identical pulses each of magnitude 12 V and width 2 μs are incident at t = 0 on a lossless transmission line of length 400 m terminated with a load. If the two pulses are separated 3 μs and μ
At 60 MHz, the following characteristics of a lossy line are measured:Zo = 50Ω, α = 0.04 dB/m, β = 2.5 rad/mCalculate R, L, C, and G of the line.
A telephone line operating at 1 kHz has R = 6.8 Ω/mi, L = 3.4 mH/mi, C = 8.4 nF/mi, and G = 0.42 μS/mi. Find (a) Z and γ, (b) Phase velocity, (c) Wavelength.
A TV antenna lead-in wire 10 cm long has a characteristic impedance of 250 Ω and is open-circuited at its end. If the line operates at 400 MHz, determine its input impedance.
A coaxial cable has its conductors made of copper (σc = 5.8 x 107 S/m) and its dielectric made of polyethylene (εr = 2.25, μr = 1). If the radius of the outer conductor is 3
A distortionless cable is 4 m long and has a characteristic impedance of 60 Ω. An attenuation of 0.24 dB is observed at the receiving end. Also, a signal applied to the cable is delayed by 80 μs
Two lines are cascaded as shown in Figure 11.51. Determine:(a) The input impedance(b) The standing wave ratio for sections XY and YZ(c) The reflection coefficient at Z X 75 Ω W/4 Y 50 Ω W/2 Z 150 Ω
A 60 Ω lossless line is connected to a source with and Zg = 50 –j40 Ω and terminated with a load of j40Ω. If the line is 100 m long and β = 0.25 rad/m, calculate Zin and V at(a) The

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