1. Uniform plane wave. The electric field of a uniform plane wave in
air is given by...(a) Find the phase constant β and the wavelength λ.
(b) Sketch Ex (z , t ) as a function of t at z = 0 and z = λ/4. (c)
Sketch Ex (z , t ) as a function of z at t = 0 and t = π/ω. Get solution
2. Uniform plane wave. The electric field phasor of an 18 GHz uniform plane wave propagating in free space is given by...(a) Find the phase constant, β, and wavelength, λ. (b) Find the corresponding magnetic field phasor H(y). Get solution
3. Uniform plane wave. A uniform plane wave is traveling in the x direction in air with its magnetic field oriented in the z direction. At the instant t = 0, the wave magnetic field has two adjacent zero values, observed at locations x = 2.5 cm and x = 7.5 cm, with a maximum value of 70 mA-m−1 at x = 5 cm. (a) Find the wave magnetic field H(x, t ) and its phasor H(x). (b) Find the corresponding wave electric field E(x, t ) and its phasor E(x). Get solution
4. Broadcast signal. The magnetic field of a TV broadcast signal propagating in air is given as...(a) Find the wave frequency f = ω/(2π). (b) Find the corresponding E(y, t ). Get solution
5. Uniform plane wave. A NASA spacecraft orbiting around Mars receives a radio signal transmitted by the UHF antenna of Curiosity Rover, with an electric field given by...(a) Determine the frequency f and the wavelength λ of this radio signal. (b) Find the corresponding magnetic field H(z , t ). Get solution
6. Lossless nonmagnetic medium. The magnetic field component of a uniform plane wave propagating in a lossless simple nonmagnetic medium (μ = μ0) is given by...(a) Find the frequency, wavelength, and the phase velocity. (b) Find the relative permittivity, єr , and the intrinsic impedance, η, of the medium. (c) Find the corresponding E. (d) Find the time-average power density carried by this wave. Get solution
7. A wireless communication signal. The electric field of a wireless communication signal traveling in air is given in phasor form as...(a) Find the frequency f and wavelength λ. (b) Find the corresponding phasor-form magnetic field H(x). (c) Find the total time-average power density carried by this wave. Get solution
8. Uniform plane wave. An 8 GHz uniform plane wave traveling in air is represented by a magnetic field vector given in phasor form as follows:...(a) Find β and frequency f . (b) Find the corresponding electric field vector in phasor form. (c) Find the total time-average power density carried by this wave. Get solution
9. Cellular phones. The electric field component of a uniform plane wave in air emitted by a mobile communication system is given by...(a) Find the frequency f and wavelength λ. (b) Find θ if E(x, z , t ) ≃ −ŷ25 mV-m−1 at t = 0 and at x = 3 m, z = 2 m. (c) Find the corresponding magnetic field H(x, z , t ). (d) Find the time-average power density carried by this wave. Get solution
10. Superposition of two waves. The sum of the electric fields of two time-harmonic (sinusoidal) electromagnetic waves propagating in opposite directions in air is given as...(a) Find the constant β. (b) Find the corresponding H. (c) Assuming that this wave may be regarded as a sum of two uniform plane waves, determine the direction of propagation of the two component waves. Get solution
11. Unknown material. The intrinsic impedance and the wavelength of a uniform plane wave traveling in an unknown dielectric at 900 MHz are measured to be ∼42Ω and ∼3.7 cm, respectively. Determine the constitutive parameters (i.e., єr and μr) of the material. Get solution
12. Uniform plane wave. A 10 GHz uniform plane wave with maximum electric field of 100 V-m−1 propagates in air in the direction of a unit vector given by û= 0.8x − 0.6ẑ. The wave magnetic field has only a y component, which is approximately equal to 40 mA-m−1 at x = z = t = 0. Find E and H. Get solution
13. Standing waves. The electric field phasor of an electromagnetic wave in air is given by...(a) Find the wavelength, λ. (b) Find the corresponding magnetic field H(x). (c) Is this a traveling wave? Get solution
14. Propagation through wet versus dry earth. Assume the conductivities of wet and dry earth to be σwet = 10−2 S-m−1 and σdry = 10−4 S-m−1, respectively, and the corresponding permittivities to be єwet = 10є0 and єdry = 3є0. Both media are known to be nonmagnetic (i.e., μr = 1). Determine the attenuation constant, phase constant, wavelength, phase velocity, penetration depth, and intrinsic impedance for a uniform plane wave of 20 MHz propagating in (a) wet earth, (b) dry earth. Use approximate expressions whenever possible. Get solution
15. 15 Propagation in lossy media. (a) Show that the penetration depth (i.e., the depth at which the field amplitude drops to 1/e of its value at the surface) in a lossy medium with μ = μ0 is approximately given by...where tan δc is the loss tangent.70 (b) For tan δc≪1, show that the above equation can be further approximated as d ≃ [0.318(c/f )]/[tan δcr...]. (c) Assuming the properties of fat tissue at 2.45 GHz to be σ = 0.12 S-m−1, єr = 5.5, and μr = 1, find the penetration depth of a 2.45 GHz plane wave in fat tissue using both expressions, and compare the results. Get solution
16. Glacier ice. A glacier is a huge mass of ice that, unlike sea ice, sits over land. Glaciers are formed in the cold polar regions and in high mountains. They spawn billions of tons of icebergs each year from tongues that reach the sea. The icebergs drift over an area of 70,000,000 km2, which is more than 20% of the ocean area, and pose a serious threat to navigation and offshore activity in many areas of the world. Glacier ice is a low-loss dielectric material that permits significant microwave penetration.71 The depth of penetration of an electromagnetic wave into glacier ice with loss tangent of tan δc≃ 0.001 at X-band (assume 3 cm air wavelength) is found to be ∼5.41 m. (a) Calculate the dielectric constant and the effective conductivity of the glacier ice. (Note that μr = 1.) (b) Calculate the attenuation of the signal measured in dB-m−1. Get solution
17. Good dielectric. Alumina (Al2O3) is a low-loss ceramic material that is commonly used as a substrate for printed circuit boards. At 10 GHz, the relative permittivity and loss tangent of alumina are approximately equal to єr = 9.7 and tan δc = 2 × 10−4. Assume μr = 1. For a 10 GHz uniform plane wave propagating in a sufficiently large sample of alumina, determine the following: (a) attenuation constant, α, in np-m−1; (b) penetration depth, d; (c) total attenuation in dB over thicknesses of 1 cm and 1 m. Get solution
18. Concrete wall. The effective complex dielectric constant of walls in buildings are investigated for wireless communication applications.72 The relative dielectric constant of the reinforced concrete wall of a building is found to be єr = 6.7 − j 1.2 at 900 MHz and єr = 6.2 − j 0.69 at 1.8 GHz, respectively. (a) Find the appropriate thickness of the concrete wall to cause a 10 dB attenuation in the field strength of the 900 MHz signal traveling over its thickness. Assume μr = 1 and neglect the reflections from the surfaces of the wall. (b) Repeat the same calculations at 1.8 GHz. Get solution
19. Unknown medium. The magnetic field phasor of a 100 MHz uniform plane wave in a nonmagnetic medium is given by...(a) Find the conductivity σ and relative permittivity єr of the medium. (b) Find the corresponding time-domain electric field E(z , t ). Get solution
20. Unknown biological tissue. The electric field component of a uniform plane wave propagating in a biological tissue with relative dielectric constant is given by...(a) Find ...and ... Assume σ = 0 and μr = 1. (b) Write the corresponding expression for the wave magnetic field. (c) Write the mathematical expression for the time-average Poynting vector Sav and sketch its magnitude from y = 0 to 5 cm. Get solution
21. Propagation in seawater. Transmission of electromagnetic energy through the ocean is practically impossible at high frequencies because of the high attenuation rates encountered. For seawater, take σ = 4 S-m−1, єr = 81, and μr = 1, respectively. (a) Show that seawater is a good conductor for frequencies much less than ∼890 MHz. (b) For frequencies less than 100 MHz, calculate, as a function of frequency (in Hz), the approximate distance over which the amplitude of the electric field is reduced by a factor of 10. Get solution
22. Wavelength in seawater. Find and sketch the wavelength in seawater as a function of frequency. Calculate λsw at the following frequencies: 1 Hz, 1 kHz, 1 MHz, and 1 GHz. Sketch log λsw vs. log f . Use the following properties for seawater: σ = 4 S-m−1, єr = 81, and μr = 1. Get solution
23. Communication in seawater. ELF communication signals (f ≤ 3 kHz) can more effectively penetrate seawater than VLF signals (3 kHz ≤ f ≤ 30 kHz). In practice, an ELF signal used for communication can penetrate and be received at a depth of up to 80 m below the ocean surface.73 (a) Find the ELF frequency at which the skin depth in seawater is equal to 80 m. For seawater, use σ = 4 S-m−1, єr = 81, and μr = 1. (b) Find the ELF frequency at which the skin depth is equal to half of 80 m. (c) At 100 Hz, find the depth at which the peak value of the electric field propagating vertically downward in seawater is 40 dB less than its value immediately below the surface of the sea. (d) A surface vehicle-based transmitter operating at 1 kHz generates an electromagnetic signal of peak value 1 V-m−1 immediately below the sea surface. If the antenna and the receiver system of the submerged vehicle can measure a signal with a peak value of as low as 1 μV-m−1, calculate the maximum depth beyond which the two vehicles cannot communicate. Get solution
24. Submarine communication near a river delta. A submarine submerged in the sea (σ = 4 S-m−1, єr = 81, μr = 1) wants to receive the signal from a VLF transmitter operating at 20 kHz. (a) How close must the submarine be to the surface in order to receive 0.1% of the signal amplitude immediately below the sea surface? (b) Repeat part (a) if the submarine is submerged near a river delta, where the average conductivity of seawater is ten times smaller. Get solution
25. Human brain tissue. Consider a 1.9 GHz electromagnetic wave produced by a wireless communication telephone inside a human brain tissue74 (єr = 43.2, μr = 1, and σ = 1.29 S-m−1) such that the peak electric field magnitude at the point of entry (z = 0) inside the tissue is about 100 V-m−1. Assuming plane wave approximation, do the following: (a) Calculate the electric field magnitudes at points z = 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm inside the brain tissue and sketch it with respect to z . (b) Calculate the time-average power density at the same points and sketch it with respect to z . (c) Calculate the time-average power absorbed in the first 1 cm thickness of a tissue sample having a cross-sectional area of 1 cm2. Get solution
26. Dispersion in sea water. A uniform plane electromagnetic wave in free space propagates with the speed of light, namely, c ≃ 3 × 108 m-s−1. In a conducting medium, however, the velocity of propagation of a uniform plane wave depends on the signal frequency, leading to the “dispersion” of a signal consisting of a band of frequencies. (a) For sea water (σ = 4 S-m−1, єr = 81, and μr = 1), show that for frequencies much less than ∼890 MHz, the velocity of propagation is approximately given by vp ≃ k1√f, where k1 is a constant. What is the value of k1? (b) Consider two different frequency components of a signal, one at 1 kHz, the other at 2 kHz. If these two signals propagate in the same direction in seawater and are in phase at z = 0, what is the phase delay (in degrees) between them (e.g., between their peak values) at a position 100 m away? Get solution
27. Electromagnetic earthquake precursor. A group of Stanford scientists measured75 mysterious electromagnetic waves varying with ultralow frequencies in the range of 0.01–10 Hz during two different earthquakes which occurred in Santa Cruz, California, in 1989 and in Parkfield, California, in 1994. A member of the group speculates that these waves may result from a local disturbance in the earth’s magnetic field caused by charged particles carried by water streams that flow along the fault lines deep in the earth’s crust as a result of the shifts that led to the quake. These low-frequency waves can penetrate rock much more easily than those of higher frequencies but can still travel only about 15 km through the ground. Since this low-frequency electromagnetic activity was recorded close to a month before the quake and lasted about a month after, this phenomenon has a potential use as an earthquake predictor. Consider three plane waves of equal amplitudes with frequencies of 0.1 Hz, 1 Hz, and 10 Hz, all produced at a depth of 15 km below the earth’s surface during an earthquake. Assuming each of these waves to be propagating vertically up toward the surface of the earth, (a) calculate the percentage time-average power of each wave that reaches the surface of the earth and (b) using the results of part (a), comment on which one of the three signals is more likely to be picked up by a receiver located on the earth’s surface, based on their signal strengths. For simplicity, assume the earth’s crust to be homogeneous, isotropic, and nonmagnetic with properties σ = 10−3 S-m−1 and єr = 10, respectively. Get solution
28. Phantom muscle tissue. In order to develop radiofrequency (RF) heating techniques for treating tumors at various locations and depths in patients, it is necessary to carry out experiments to determine the energy absorbed by an object exposed to electromagnetic fields over a wide range of RF frequencies. An artificial muscle tissue (“muscle phantom”) is designed to be used in these experiments to simulate actual muscle tissue for applications in the frequency range used for RF hyperthermia.76 (a) Given the relative dielectric constant and the conductivity of the muscle phantom at 915 MHz and 22◦C to be єr ≃ 51.1 and σ≃1.27 S-m−1, calculate the depth of penetration in the phantom. Note that μr = 1. (b) Repeat part (a) at 2.45 GHz when єr ≃ 47.4 and σ ≃ 2.17 S-m−1. Which frequency can penetrate deeper into the muscle phantom? (c) Calculate the total dB attenuation over a muscle phantom of 1.5 cm thickness at both frequencies. Get solution
29. Unknown medium. The skin depth and the loss tangent of a nonmagnetic conducting medium at 21.4 kHz are approximately equal to 1.72 m and 4.15 × 104, respectively. (a) Find the conductivity σ and the relative dielectric constant єr of the medium. What medium is this? (b) Write the mathematical expressions for the electric and magnetic field components of a 21.4 kHz uniform plane wave propagating in this medium, assuming the maximum peak value of the electric field to be 10 V-m−1. (c) Repeat part (b) at 2.14 MHz. Assume the properties of the medium to be the same at both frequencies. Get solution
30. Unknown medium. A uniform plane wave propagates in the x direction in a certain type of material with unknown properties. At t = 0, the wave electric field is measured to vary with x as shown in Figure At x = 40 m, the temporal variation of the wave electric field is measured to be in the form shown in Figure Using the data in these two Figure, find (a) σ and єr (assume nonmagnetic case), (b) the depth of penetration and the attenuation in dB-m−1, and (c) the total dB attenuation and the phase shift over a distance of 100 m through this medium.Figure Unknown medium. Problem.... Get solution
31. Uniform plane waves. The electric field of a 1 GHz uniform plane wave propagating in a low-loss dielectric is given by...(a) Stating all assumptions, determine the conductivity σ and permittivity є of this dielectric and calculate the depth of penetration d. (b) Write a complete (i.e., with all quantities specified in terms of numerical values) expression for the vector magnetic field intensity H(x, y, t ) of this wave. Get solution
32. Thickness of beef products. Microwave heating is generally uniform over the entire body of the product being heated if the thickness of the product does not exceed about 1–1.5 times its penetration depth.77 (a) Consider a beef product to be heated in a microwave oven operating at 2.45 GHz. The dielectric properties of raw beef at 2.45 GHz and 25◦C are ...= 52.4, μr = 1, and tan δc = 0.33.78 What is the maximum thickness of this beef product for it to be heated uniformly? (b) Microwave ovens operating at 915 MHz are evidently more appropriate for cooking products with large cross sections and high dielectric loss factors. The dielectric properties of raw beef at 915 MHz and 25◦C are ...= 54.5, μr = 1, and tan δc = 0.411. Find the maximum thickness of the beef product at 915 MHz and compare it with the results of part (a). Get solution
33. Beef versus bacon. The dielectric properties of cooked beef and smoked bacon at 25◦C are given by єr ≃ 31.1 − j 10.3 at 2.45 GHz and єr ≃ 2.5 − j 0.125 at 3 GHz, respectively (see the references in the preceding problem). Assuming μr = 1, calculate the loss tangent and the penetration depth for each and explain the differences. Get solution
34. Aluminum foil. A sheet of aluminum foil of thickness ∼25 μm is used to shield an electronic instrument at 100 MHz. Find the dB attenuation of a plane wave that travels from one side to the other side of the aluminum foil. (Neglect the effects from the boundaries.) For aluminum, σ = 3.54 × 107 S-m−1 and єr = μr = 1. Get solution
35. Unknown material. Using the results of a reflection measurement technique, the intrinsic impedance of a material at 200 MHz is found to be approximately given by...Assuming that the material is nonmagnetic, determine its conductivity σ and the relative dielectric constant ... Get solution
36. Poynting flux. The electric and magnetic field expressions for a uniform plane wave propagating in a lossy medium are as follows:...The frequency of operation is f = 108 Hz, and the electrical parameters of the medium are є = 18.5є0, μ0, and σ. (a) Find the time-average electromagnetic power density entering a rectangular box-shaped surface like that shown in Figure assuming a = d = 1 m and b = 0.5 m. (b) Determine the power density exiting this region and compare with (a). (c) The difference between your results in (a) and (b) should represent electromagnetic power lost in the region enclosed by the square-box region. Can you calculate this dissipated power by any other method (i.e., without using the Poynting vector)? If yes, carry out this calculation. Hint: You may first need to find σ.Figure Poynting flux. Problem.... Get solution
37. Laser beams. The electric field component of a laser beam propagating in the z direction is approximated by...where E0 is the amplitude on the axis and a is the effective beam radius, where the electric field amplitude is a factor of e−1 lower than E0. (a) Find the corresponding expression for the magnetic field E. (b) Show that the time-average power density at the center of the laser beam is given by...where η = 377. (c) Find the total power of the laser beam. Consider a typical laboratory helium-neon laser with a total power of 5 mW and an effective radius of a = 400 μm. What is the power density at the center of the beam? (d) The power density of solar electromagnetic radiation at the Earth’s surface is 1400 W-m−2. At what distance from the Sun would its power density be equal to that for the helium-neon laser in part (c)? (e) One of the highest-power lasers built for fusion experiments operates at λ = 1.6 μm, produces 10.2 kJ for 0.2 ns and is designed for focusing on targets of 0.5 mm diameter. Estimate the electric field strength at the center of the beam. Is the field large enough to break down air? What is the radiation pressure of the laser beam? How much weight can be lifted with the pressure of this beam? Get solution
38. Maxwell’s equations. Consider a parallel-plate transmission line with perfectly conducting plates of large extent, separated by a distance of d meters. As shown in Figure, an alternating surface current density Js in the z direction flows on the conductor surface:...Figure Surface current. Problem....(a) Find an expression for the electric field, and determine the voltage between the plates, for d = 0.1 m and J0 = 1 A-m−1. (b) Use the continuity equation to find an expression for the surface charge density ρs (z , t ). Get solution
39. Uniform plane wave. A uniform plane electromagnetic wave propagates in free space with electric and magnetic field components as shown in Figure:...The wave frequency is 300 MHz and the electric field amplitude E0 = 1 V-m−1. A square loop antenna with side length a = 10 cm is placed at z = 2 m as shown. (a) Find the voltage Vind(t ) induced at the terminals of the loop. (b) Repeat (a) for the loop located at a distance of d = 3 m from the x axis instead of 2 m as shown. Compare your answers in (a) and (b).Figure Uniform plane wave. Problem.... Get solution
40. FM radio. An FM radio station operating at 100 MHz radiates a circularly polarized plane wave with a total isotropically radiated power of 200 kW. The transmitter antenna is located on a tower 500 m above the ground. (a) Find the rms value of the electric field 1 km away from the base of the antenna tower. Neglect the effects of reflections from the ground and other boundaries. (b) If the primary coverage radius of this station is ∼100 km, find the approximate time-average power density of the FM wave at this distance. Get solution
41. Mobile phones. Cellular phone antennas installed on cars have a maximum output power of 3 W, set by the Federal Communications Commission (FCC) standards. The incident electromagnetic energy to which the passengers in the car are exposed does not pose any health threats, both because they are some distance away from the antenna and also because the body of the car and glass window shield them from much of the radiation.79 (a) For a car with a synthetic roof, the maximum localized power density in the passenger seat is about 0.3 mW-(cm)−2. For cars with metal roofs, this value reduces to 0.02 mW-(cm)−2 or less. Antennas mounted on the trunk or in the glass of the rear windshield deliver power densities of about 0.35–0.07 mW-(cm)−2 to passengers in the back seat. Compare these values with the IEEE safety limit (IEEE Standard C95.1-1991) in the cellular phone frequency range (which is typically 800–900 MHz) and comment on the safety of the passengers. Note that from 300 MHz to 15 GHz, IEEE safety limits 80 specify a maximum allowable power density that increases linearly with frequency as |Sav|max = f /1500 in mW-(cm)−2, where the frequency f is in MHz. For reference, the maximum allowable power density is 1 mW-(cm)−2 at 1.5 GHz. (b) Calculate the maximum output power of a cellular phone antenna installed on the metal roof of a car such that the localized power density in the passenger compartment is equal to the IEEE safety limit at 850 MHz. Get solution
42. Radar aboard a Navy ship. Some shipboard personnel work daily in an environment where the radio frequency (RF) power density only a few feet above their heads may exceed safe levels. Some areas on the deck of the ship are not even allowed to personnel due to high power densities. Pilots of aircrafts routinely fly through the ship’s radar beams during takeoff and landing operations. On one of the Navy aircraft carriers, the average power density along the axis of the main beam (where the field intensity is the greatest) 100 ft away from a 6 kW missile control radar operating in the C-band is measured 81 to be about 300 mW-(cm)−2. (a) If it is assumed that human exposures in such environments are limited to less than 10 mW-(cm)−2, calculate the approximate distance along the beam axis that can be considered as the hazardous zone. (b) Assuming the operation frequency of the C-band radar to be 5 GHz, recalculate the hazardous zone along the radar’s main beam based on the IEEE safety limit 82 for the average power density given by |Sav|max = f /1500 in mW-(cm)−2 (valid over the frequency range 300 MHz–15 GHz), where f is in MHz. Get solution
43. Radio frequency exposure time. In 1965, the U.S. Army and Air Force amended their use of the prevailing 10 mW-(cm)−2 exposure guideline to include a time limit for exposures, given by the formula...where |Sav| is the average power density [in mW-(cm)−2] of exposure and tmax is the maximum recommended exposure duration, in minutes.83 Consider a person standing on the deck of a Navy ship where the peak rms electric field strength due to a microwave radar transmitter is measured to be 140 V-m−1. Using the above formula, calculate the maximum exposure time allowed (in hours) for this person to stay at that location. Get solution
44. Microwave cataracts in humans. Over 50 cases of human cataract induction have been attributed to microwave exposures, primarily encountered in occupational situations involving acute exposure to presumably relatively high-intensity fields.84 The following are three reported incidents of cataracts caused by microwave radiation: (1) A 22-year-old technician exposed approximately five times to 3 GHz radiation at an estimated average power density of 300 mW-(cm)−2 for 3 min/exposure developed bilateral cataracts. (2) A person was exposed to microwaves for durations of approximately 50 hour/month over a 4-year period at average power densities of less than 10 mW-(cm)−2 in most instances, but with a period of 6 months or more during which the average power density was approximately 1 W-(cm)−2. (3) A 50-year-old woman was intermittently exposed to leakage radiation from a 2.45 GHz microwave oven of approximately 1 mW-(cm)−2 during oven operation, with levels of up to 90 mW-(cm)−2 when the oven door was open, presumably over a period of approximately 6 years prior to developing cataract. For each of these above cases, compare the power densities with the IEEE standards (see Problem ) and comment.Mobile phones. Cellular phone antennas installed on cars have a maximum output power of 3 W, set by the Federal Communications Commission (FCC) standards. The incident electromagnetic energy to which the passengers in the car are exposed does not pose any health threats, both because they are some distance away from the antenna and also because the body of the car and glass window shield them from much of the radiation.79 (a) For a car with a synthetic roof, the maximum localized power density in the passenger seat is about 0.3 mW-(cm)−2. For cars with metal roofs, this value reduces to 0.02 mW-(cm)−2 or less. Antennas mounted on the trunk or in the glass of the rear windshield deliver power densities of about 0.35–0.07 mW-(cm)−2 to passengers in the back seat. Compare these values with the IEEE safety limit (IEEE Standard C95.1-1991) in the cellular phone frequency range (which is typically 800–900 MHz) and comment on the safety of the passengers. Note that from 300 MHz to 15 GHz, IEEE safety limits 80 specify a maximum allowable power density that increases linearly with frequency as |Sav|max = f /1500 in mW-(cm)−2, where the frequency f is in MHz. For reference, the maximum allowable power density is 1 mW-(cm)−2 at 1.5 GHz. (b) Calculate the maximum output power of a cellular phone antenna installed on the metal roof of a car such that the localized power density in the passenger compartment is equal to the IEEE safety limit at 850 MHz. Get solution
45. VHF TV signal. The magnetic field component of a 10 μW-m−2, 200 MHz TV signal in air is given by...(a) What are the values of H0 and a? (b) Find the corresponding electric field E(x, y, t ). What is the polarization of the wave? (c) An observer at z = 0 is equipped with a wire antenna capable of detecting the component of the electric field along its length. Find the maximum value of the measured electric field if the antenna wire is oriented along the (i) x direction, (ii) y direction, (iii) 45◦ line between the x and y directions. Get solution
46. FM polarization. Find the type (linear, circular, elliptical) and sense (right- or left-handed) of the polarization of the FM broadcast signal given in ExampleFM broadcast signal. An FM radio broadcast signal traveling in the y direction in air has a magnetic field given by the phasor...(a) Determine the frequency (in MHz) and wavelength (in m). (b) Find the corresponding E(y).(c) Write the instantaneous expression for E(y, t ) and H(y, t ). Get solution
47. Unknown wave polarization. The magnetic field component of a uniform plane wave in air is given by...where a is a real constant. (a) Find the wavelength λ and frequency f . (b) Find the total time-average power density carried by this wave. (c) Determine the type (linear, circular, elliptical) and sense (right- or left-handed) of the polarization of this wave when a = 1. (d) Repeat part (c) when a = 3. Get solution
48. Linear and circularly polarized waves. Two electromagnetic waves operating at the same frequency and propagating in the same direction (y direction) in air are such that one of them is linearly polarized in the x direction, whereas the other is left-hand circularly polarized (LHCP). However, the electric and magnetic field components of the two waves appear identical at one instant within every 50 ps time interval. The linearly polarized wave carries a time-average power density of 1.4 W-m−2. An observer located at y = 0 uses a receiving antenna to measure the x component of the total electric field only and records a maximum field magnitude of about 65 V-m−1 over every time interval of 0.5 ns. (a) Write the mathematical expressions for the electric field components of each wave, using numerical values of various quantities whenever possible. (b) Find the ratio of the time-average power densities of the LHCP and the linearly polarized waves. Get solution
49. Two circularly polarized waves. Consider two circularly polarized waves traveling in the same direction transmitted by two different satellites operating at the same frequency given by...where E01 and E02 are real constants. (a) If the total time-average power densities of these two waves are equal, find the polarization of the total wave. (b) Repeat part (a) for the case when the total time-average power density of the first wave is four times the total time-average power density of the second wave. Get solution
50. Wave polarization. Consider the following complex phasor expression for a time-harmonic magnetic field in free space:...(a) Is this a uniform plane wave? What is its frequency? (b) What is the direction of propagation and the state of polarization (specify both the type and sense of polarization) of this electromagnetic field? (c) Find the associated electric field phasor and the total time-average power density in the direction of propagation. Get solution
51. Wave polarization. The electric field component of a communication satellite signal travelling in free space is given by...(a) Find the corresponding H(z ). (b) Find the total time-average power density carried by this wave. (c) Determine the polarization (both type and sense) of the wave. Get solution
52. Wave polarization. A fellow engineer makes the following two measurements of the electric field vector of a uniform plane wave propagating in the x direction in a simple, lossless and nonmagnetic (μ = μ0) medium:...(a) Are these two measurements enough to determine the polarization type and sense of the wave? What about the wavelength of the wave? (b) Now the engineer offers two other measurements made at the same time t = 0:...Furthermore, you are informed that the electric field amplitude at t = 0 does not exceed 5 V-m−1 at any point between x = 0 and x = 0.75 m. Given this information, determine the type and sense of polarization of the wave. Get solution
53. Superposition of two waves. The electric field components of two electromagnetic waves at the same frequency and propagating in free space are represented by...Find and sketch the locus of the total electric field measured at the origin (x = y = z = 0) if E0 is equal to (a) 10 V-m−1, (b) 20 V-m−1, (c) 40 V-m−1, respectively. Get solution
2. Uniform plane wave. The electric field phasor of an 18 GHz uniform plane wave propagating in free space is given by...(a) Find the phase constant, β, and wavelength, λ. (b) Find the corresponding magnetic field phasor H(y). Get solution
3. Uniform plane wave. A uniform plane wave is traveling in the x direction in air with its magnetic field oriented in the z direction. At the instant t = 0, the wave magnetic field has two adjacent zero values, observed at locations x = 2.5 cm and x = 7.5 cm, with a maximum value of 70 mA-m−1 at x = 5 cm. (a) Find the wave magnetic field H(x, t ) and its phasor H(x). (b) Find the corresponding wave electric field E(x, t ) and its phasor E(x). Get solution
4. Broadcast signal. The magnetic field of a TV broadcast signal propagating in air is given as...(a) Find the wave frequency f = ω/(2π). (b) Find the corresponding E(y, t ). Get solution
5. Uniform plane wave. A NASA spacecraft orbiting around Mars receives a radio signal transmitted by the UHF antenna of Curiosity Rover, with an electric field given by...(a) Determine the frequency f and the wavelength λ of this radio signal. (b) Find the corresponding magnetic field H(z , t ). Get solution
6. Lossless nonmagnetic medium. The magnetic field component of a uniform plane wave propagating in a lossless simple nonmagnetic medium (μ = μ0) is given by...(a) Find the frequency, wavelength, and the phase velocity. (b) Find the relative permittivity, єr , and the intrinsic impedance, η, of the medium. (c) Find the corresponding E. (d) Find the time-average power density carried by this wave. Get solution
7. A wireless communication signal. The electric field of a wireless communication signal traveling in air is given in phasor form as...(a) Find the frequency f and wavelength λ. (b) Find the corresponding phasor-form magnetic field H(x). (c) Find the total time-average power density carried by this wave. Get solution
8. Uniform plane wave. An 8 GHz uniform plane wave traveling in air is represented by a magnetic field vector given in phasor form as follows:...(a) Find β and frequency f . (b) Find the corresponding electric field vector in phasor form. (c) Find the total time-average power density carried by this wave. Get solution
9. Cellular phones. The electric field component of a uniform plane wave in air emitted by a mobile communication system is given by...(a) Find the frequency f and wavelength λ. (b) Find θ if E(x, z , t ) ≃ −ŷ25 mV-m−1 at t = 0 and at x = 3 m, z = 2 m. (c) Find the corresponding magnetic field H(x, z , t ). (d) Find the time-average power density carried by this wave. Get solution
10. Superposition of two waves. The sum of the electric fields of two time-harmonic (sinusoidal) electromagnetic waves propagating in opposite directions in air is given as...(a) Find the constant β. (b) Find the corresponding H. (c) Assuming that this wave may be regarded as a sum of two uniform plane waves, determine the direction of propagation of the two component waves. Get solution
11. Unknown material. The intrinsic impedance and the wavelength of a uniform plane wave traveling in an unknown dielectric at 900 MHz are measured to be ∼42Ω and ∼3.7 cm, respectively. Determine the constitutive parameters (i.e., єr and μr) of the material. Get solution
12. Uniform plane wave. A 10 GHz uniform plane wave with maximum electric field of 100 V-m−1 propagates in air in the direction of a unit vector given by û= 0.8x − 0.6ẑ. The wave magnetic field has only a y component, which is approximately equal to 40 mA-m−1 at x = z = t = 0. Find E and H. Get solution
13. Standing waves. The electric field phasor of an electromagnetic wave in air is given by...(a) Find the wavelength, λ. (b) Find the corresponding magnetic field H(x). (c) Is this a traveling wave? Get solution
14. Propagation through wet versus dry earth. Assume the conductivities of wet and dry earth to be σwet = 10−2 S-m−1 and σdry = 10−4 S-m−1, respectively, and the corresponding permittivities to be єwet = 10є0 and єdry = 3є0. Both media are known to be nonmagnetic (i.e., μr = 1). Determine the attenuation constant, phase constant, wavelength, phase velocity, penetration depth, and intrinsic impedance for a uniform plane wave of 20 MHz propagating in (a) wet earth, (b) dry earth. Use approximate expressions whenever possible. Get solution
15. 15 Propagation in lossy media. (a) Show that the penetration depth (i.e., the depth at which the field amplitude drops to 1/e of its value at the surface) in a lossy medium with μ = μ0 is approximately given by...where tan δc is the loss tangent.70 (b) For tan δc≪1, show that the above equation can be further approximated as d ≃ [0.318(c/f )]/[tan δcr...]. (c) Assuming the properties of fat tissue at 2.45 GHz to be σ = 0.12 S-m−1, єr = 5.5, and μr = 1, find the penetration depth of a 2.45 GHz plane wave in fat tissue using both expressions, and compare the results. Get solution
16. Glacier ice. A glacier is a huge mass of ice that, unlike sea ice, sits over land. Glaciers are formed in the cold polar regions and in high mountains. They spawn billions of tons of icebergs each year from tongues that reach the sea. The icebergs drift over an area of 70,000,000 km2, which is more than 20% of the ocean area, and pose a serious threat to navigation and offshore activity in many areas of the world. Glacier ice is a low-loss dielectric material that permits significant microwave penetration.71 The depth of penetration of an electromagnetic wave into glacier ice with loss tangent of tan δc≃ 0.001 at X-band (assume 3 cm air wavelength) is found to be ∼5.41 m. (a) Calculate the dielectric constant and the effective conductivity of the glacier ice. (Note that μr = 1.) (b) Calculate the attenuation of the signal measured in dB-m−1. Get solution
17. Good dielectric. Alumina (Al2O3) is a low-loss ceramic material that is commonly used as a substrate for printed circuit boards. At 10 GHz, the relative permittivity and loss tangent of alumina are approximately equal to єr = 9.7 and tan δc = 2 × 10−4. Assume μr = 1. For a 10 GHz uniform plane wave propagating in a sufficiently large sample of alumina, determine the following: (a) attenuation constant, α, in np-m−1; (b) penetration depth, d; (c) total attenuation in dB over thicknesses of 1 cm and 1 m. Get solution
18. Concrete wall. The effective complex dielectric constant of walls in buildings are investigated for wireless communication applications.72 The relative dielectric constant of the reinforced concrete wall of a building is found to be єr = 6.7 − j 1.2 at 900 MHz and єr = 6.2 − j 0.69 at 1.8 GHz, respectively. (a) Find the appropriate thickness of the concrete wall to cause a 10 dB attenuation in the field strength of the 900 MHz signal traveling over its thickness. Assume μr = 1 and neglect the reflections from the surfaces of the wall. (b) Repeat the same calculations at 1.8 GHz. Get solution
19. Unknown medium. The magnetic field phasor of a 100 MHz uniform plane wave in a nonmagnetic medium is given by...(a) Find the conductivity σ and relative permittivity єr of the medium. (b) Find the corresponding time-domain electric field E(z , t ). Get solution
20. Unknown biological tissue. The electric field component of a uniform plane wave propagating in a biological tissue with relative dielectric constant is given by...(a) Find ...and ... Assume σ = 0 and μr = 1. (b) Write the corresponding expression for the wave magnetic field. (c) Write the mathematical expression for the time-average Poynting vector Sav and sketch its magnitude from y = 0 to 5 cm. Get solution
21. Propagation in seawater. Transmission of electromagnetic energy through the ocean is practically impossible at high frequencies because of the high attenuation rates encountered. For seawater, take σ = 4 S-m−1, єr = 81, and μr = 1, respectively. (a) Show that seawater is a good conductor for frequencies much less than ∼890 MHz. (b) For frequencies less than 100 MHz, calculate, as a function of frequency (in Hz), the approximate distance over which the amplitude of the electric field is reduced by a factor of 10. Get solution
22. Wavelength in seawater. Find and sketch the wavelength in seawater as a function of frequency. Calculate λsw at the following frequencies: 1 Hz, 1 kHz, 1 MHz, and 1 GHz. Sketch log λsw vs. log f . Use the following properties for seawater: σ = 4 S-m−1, єr = 81, and μr = 1. Get solution
23. Communication in seawater. ELF communication signals (f ≤ 3 kHz) can more effectively penetrate seawater than VLF signals (3 kHz ≤ f ≤ 30 kHz). In practice, an ELF signal used for communication can penetrate and be received at a depth of up to 80 m below the ocean surface.73 (a) Find the ELF frequency at which the skin depth in seawater is equal to 80 m. For seawater, use σ = 4 S-m−1, єr = 81, and μr = 1. (b) Find the ELF frequency at which the skin depth is equal to half of 80 m. (c) At 100 Hz, find the depth at which the peak value of the electric field propagating vertically downward in seawater is 40 dB less than its value immediately below the surface of the sea. (d) A surface vehicle-based transmitter operating at 1 kHz generates an electromagnetic signal of peak value 1 V-m−1 immediately below the sea surface. If the antenna and the receiver system of the submerged vehicle can measure a signal with a peak value of as low as 1 μV-m−1, calculate the maximum depth beyond which the two vehicles cannot communicate. Get solution
24. Submarine communication near a river delta. A submarine submerged in the sea (σ = 4 S-m−1, єr = 81, μr = 1) wants to receive the signal from a VLF transmitter operating at 20 kHz. (a) How close must the submarine be to the surface in order to receive 0.1% of the signal amplitude immediately below the sea surface? (b) Repeat part (a) if the submarine is submerged near a river delta, where the average conductivity of seawater is ten times smaller. Get solution
25. Human brain tissue. Consider a 1.9 GHz electromagnetic wave produced by a wireless communication telephone inside a human brain tissue74 (єr = 43.2, μr = 1, and σ = 1.29 S-m−1) such that the peak electric field magnitude at the point of entry (z = 0) inside the tissue is about 100 V-m−1. Assuming plane wave approximation, do the following: (a) Calculate the electric field magnitudes at points z = 1 cm, 2 cm, 3 cm, 4 cm, and 5 cm inside the brain tissue and sketch it with respect to z . (b) Calculate the time-average power density at the same points and sketch it with respect to z . (c) Calculate the time-average power absorbed in the first 1 cm thickness of a tissue sample having a cross-sectional area of 1 cm2. Get solution
26. Dispersion in sea water. A uniform plane electromagnetic wave in free space propagates with the speed of light, namely, c ≃ 3 × 108 m-s−1. In a conducting medium, however, the velocity of propagation of a uniform plane wave depends on the signal frequency, leading to the “dispersion” of a signal consisting of a band of frequencies. (a) For sea water (σ = 4 S-m−1, єr = 81, and μr = 1), show that for frequencies much less than ∼890 MHz, the velocity of propagation is approximately given by vp ≃ k1√f, where k1 is a constant. What is the value of k1? (b) Consider two different frequency components of a signal, one at 1 kHz, the other at 2 kHz. If these two signals propagate in the same direction in seawater and are in phase at z = 0, what is the phase delay (in degrees) between them (e.g., between their peak values) at a position 100 m away? Get solution
27. Electromagnetic earthquake precursor. A group of Stanford scientists measured75 mysterious electromagnetic waves varying with ultralow frequencies in the range of 0.01–10 Hz during two different earthquakes which occurred in Santa Cruz, California, in 1989 and in Parkfield, California, in 1994. A member of the group speculates that these waves may result from a local disturbance in the earth’s magnetic field caused by charged particles carried by water streams that flow along the fault lines deep in the earth’s crust as a result of the shifts that led to the quake. These low-frequency waves can penetrate rock much more easily than those of higher frequencies but can still travel only about 15 km through the ground. Since this low-frequency electromagnetic activity was recorded close to a month before the quake and lasted about a month after, this phenomenon has a potential use as an earthquake predictor. Consider three plane waves of equal amplitudes with frequencies of 0.1 Hz, 1 Hz, and 10 Hz, all produced at a depth of 15 km below the earth’s surface during an earthquake. Assuming each of these waves to be propagating vertically up toward the surface of the earth, (a) calculate the percentage time-average power of each wave that reaches the surface of the earth and (b) using the results of part (a), comment on which one of the three signals is more likely to be picked up by a receiver located on the earth’s surface, based on their signal strengths. For simplicity, assume the earth’s crust to be homogeneous, isotropic, and nonmagnetic with properties σ = 10−3 S-m−1 and єr = 10, respectively. Get solution
28. Phantom muscle tissue. In order to develop radiofrequency (RF) heating techniques for treating tumors at various locations and depths in patients, it is necessary to carry out experiments to determine the energy absorbed by an object exposed to electromagnetic fields over a wide range of RF frequencies. An artificial muscle tissue (“muscle phantom”) is designed to be used in these experiments to simulate actual muscle tissue for applications in the frequency range used for RF hyperthermia.76 (a) Given the relative dielectric constant and the conductivity of the muscle phantom at 915 MHz and 22◦C to be єr ≃ 51.1 and σ≃1.27 S-m−1, calculate the depth of penetration in the phantom. Note that μr = 1. (b) Repeat part (a) at 2.45 GHz when єr ≃ 47.4 and σ ≃ 2.17 S-m−1. Which frequency can penetrate deeper into the muscle phantom? (c) Calculate the total dB attenuation over a muscle phantom of 1.5 cm thickness at both frequencies. Get solution
29. Unknown medium. The skin depth and the loss tangent of a nonmagnetic conducting medium at 21.4 kHz are approximately equal to 1.72 m and 4.15 × 104, respectively. (a) Find the conductivity σ and the relative dielectric constant єr of the medium. What medium is this? (b) Write the mathematical expressions for the electric and magnetic field components of a 21.4 kHz uniform plane wave propagating in this medium, assuming the maximum peak value of the electric field to be 10 V-m−1. (c) Repeat part (b) at 2.14 MHz. Assume the properties of the medium to be the same at both frequencies. Get solution
30. Unknown medium. A uniform plane wave propagates in the x direction in a certain type of material with unknown properties. At t = 0, the wave electric field is measured to vary with x as shown in Figure At x = 40 m, the temporal variation of the wave electric field is measured to be in the form shown in Figure Using the data in these two Figure, find (a) σ and єr (assume nonmagnetic case), (b) the depth of penetration and the attenuation in dB-m−1, and (c) the total dB attenuation and the phase shift over a distance of 100 m through this medium.Figure Unknown medium. Problem.... Get solution
31. Uniform plane waves. The electric field of a 1 GHz uniform plane wave propagating in a low-loss dielectric is given by...(a) Stating all assumptions, determine the conductivity σ and permittivity є of this dielectric and calculate the depth of penetration d. (b) Write a complete (i.e., with all quantities specified in terms of numerical values) expression for the vector magnetic field intensity H(x, y, t ) of this wave. Get solution
32. Thickness of beef products. Microwave heating is generally uniform over the entire body of the product being heated if the thickness of the product does not exceed about 1–1.5 times its penetration depth.77 (a) Consider a beef product to be heated in a microwave oven operating at 2.45 GHz. The dielectric properties of raw beef at 2.45 GHz and 25◦C are ...= 52.4, μr = 1, and tan δc = 0.33.78 What is the maximum thickness of this beef product for it to be heated uniformly? (b) Microwave ovens operating at 915 MHz are evidently more appropriate for cooking products with large cross sections and high dielectric loss factors. The dielectric properties of raw beef at 915 MHz and 25◦C are ...= 54.5, μr = 1, and tan δc = 0.411. Find the maximum thickness of the beef product at 915 MHz and compare it with the results of part (a). Get solution
33. Beef versus bacon. The dielectric properties of cooked beef and smoked bacon at 25◦C are given by єr ≃ 31.1 − j 10.3 at 2.45 GHz and єr ≃ 2.5 − j 0.125 at 3 GHz, respectively (see the references in the preceding problem). Assuming μr = 1, calculate the loss tangent and the penetration depth for each and explain the differences. Get solution
34. Aluminum foil. A sheet of aluminum foil of thickness ∼25 μm is used to shield an electronic instrument at 100 MHz. Find the dB attenuation of a plane wave that travels from one side to the other side of the aluminum foil. (Neglect the effects from the boundaries.) For aluminum, σ = 3.54 × 107 S-m−1 and єr = μr = 1. Get solution
35. Unknown material. Using the results of a reflection measurement technique, the intrinsic impedance of a material at 200 MHz is found to be approximately given by...Assuming that the material is nonmagnetic, determine its conductivity σ and the relative dielectric constant ... Get solution
36. Poynting flux. The electric and magnetic field expressions for a uniform plane wave propagating in a lossy medium are as follows:...The frequency of operation is f = 108 Hz, and the electrical parameters of the medium are є = 18.5є0, μ0, and σ. (a) Find the time-average electromagnetic power density entering a rectangular box-shaped surface like that shown in Figure assuming a = d = 1 m and b = 0.5 m. (b) Determine the power density exiting this region and compare with (a). (c) The difference between your results in (a) and (b) should represent electromagnetic power lost in the region enclosed by the square-box region. Can you calculate this dissipated power by any other method (i.e., without using the Poynting vector)? If yes, carry out this calculation. Hint: You may first need to find σ.Figure Poynting flux. Problem.... Get solution
37. Laser beams. The electric field component of a laser beam propagating in the z direction is approximated by...where E0 is the amplitude on the axis and a is the effective beam radius, where the electric field amplitude is a factor of e−1 lower than E0. (a) Find the corresponding expression for the magnetic field E. (b) Show that the time-average power density at the center of the laser beam is given by...where η = 377. (c) Find the total power of the laser beam. Consider a typical laboratory helium-neon laser with a total power of 5 mW and an effective radius of a = 400 μm. What is the power density at the center of the beam? (d) The power density of solar electromagnetic radiation at the Earth’s surface is 1400 W-m−2. At what distance from the Sun would its power density be equal to that for the helium-neon laser in part (c)? (e) One of the highest-power lasers built for fusion experiments operates at λ = 1.6 μm, produces 10.2 kJ for 0.2 ns and is designed for focusing on targets of 0.5 mm diameter. Estimate the electric field strength at the center of the beam. Is the field large enough to break down air? What is the radiation pressure of the laser beam? How much weight can be lifted with the pressure of this beam? Get solution
38. Maxwell’s equations. Consider a parallel-plate transmission line with perfectly conducting plates of large extent, separated by a distance of d meters. As shown in Figure, an alternating surface current density Js in the z direction flows on the conductor surface:...Figure Surface current. Problem....(a) Find an expression for the electric field, and determine the voltage between the plates, for d = 0.1 m and J0 = 1 A-m−1. (b) Use the continuity equation to find an expression for the surface charge density ρs (z , t ). Get solution
39. Uniform plane wave. A uniform plane electromagnetic wave propagates in free space with electric and magnetic field components as shown in Figure:...The wave frequency is 300 MHz and the electric field amplitude E0 = 1 V-m−1. A square loop antenna with side length a = 10 cm is placed at z = 2 m as shown. (a) Find the voltage Vind(t ) induced at the terminals of the loop. (b) Repeat (a) for the loop located at a distance of d = 3 m from the x axis instead of 2 m as shown. Compare your answers in (a) and (b).Figure Uniform plane wave. Problem.... Get solution
40. FM radio. An FM radio station operating at 100 MHz radiates a circularly polarized plane wave with a total isotropically radiated power of 200 kW. The transmitter antenna is located on a tower 500 m above the ground. (a) Find the rms value of the electric field 1 km away from the base of the antenna tower. Neglect the effects of reflections from the ground and other boundaries. (b) If the primary coverage radius of this station is ∼100 km, find the approximate time-average power density of the FM wave at this distance. Get solution
41. Mobile phones. Cellular phone antennas installed on cars have a maximum output power of 3 W, set by the Federal Communications Commission (FCC) standards. The incident electromagnetic energy to which the passengers in the car are exposed does not pose any health threats, both because they are some distance away from the antenna and also because the body of the car and glass window shield them from much of the radiation.79 (a) For a car with a synthetic roof, the maximum localized power density in the passenger seat is about 0.3 mW-(cm)−2. For cars with metal roofs, this value reduces to 0.02 mW-(cm)−2 or less. Antennas mounted on the trunk or in the glass of the rear windshield deliver power densities of about 0.35–0.07 mW-(cm)−2 to passengers in the back seat. Compare these values with the IEEE safety limit (IEEE Standard C95.1-1991) in the cellular phone frequency range (which is typically 800–900 MHz) and comment on the safety of the passengers. Note that from 300 MHz to 15 GHz, IEEE safety limits 80 specify a maximum allowable power density that increases linearly with frequency as |Sav|max = f /1500 in mW-(cm)−2, where the frequency f is in MHz. For reference, the maximum allowable power density is 1 mW-(cm)−2 at 1.5 GHz. (b) Calculate the maximum output power of a cellular phone antenna installed on the metal roof of a car such that the localized power density in the passenger compartment is equal to the IEEE safety limit at 850 MHz. Get solution
42. Radar aboard a Navy ship. Some shipboard personnel work daily in an environment where the radio frequency (RF) power density only a few feet above their heads may exceed safe levels. Some areas on the deck of the ship are not even allowed to personnel due to high power densities. Pilots of aircrafts routinely fly through the ship’s radar beams during takeoff and landing operations. On one of the Navy aircraft carriers, the average power density along the axis of the main beam (where the field intensity is the greatest) 100 ft away from a 6 kW missile control radar operating in the C-band is measured 81 to be about 300 mW-(cm)−2. (a) If it is assumed that human exposures in such environments are limited to less than 10 mW-(cm)−2, calculate the approximate distance along the beam axis that can be considered as the hazardous zone. (b) Assuming the operation frequency of the C-band radar to be 5 GHz, recalculate the hazardous zone along the radar’s main beam based on the IEEE safety limit 82 for the average power density given by |Sav|max = f /1500 in mW-(cm)−2 (valid over the frequency range 300 MHz–15 GHz), where f is in MHz. Get solution
43. Radio frequency exposure time. In 1965, the U.S. Army and Air Force amended their use of the prevailing 10 mW-(cm)−2 exposure guideline to include a time limit for exposures, given by the formula...where |Sav| is the average power density [in mW-(cm)−2] of exposure and tmax is the maximum recommended exposure duration, in minutes.83 Consider a person standing on the deck of a Navy ship where the peak rms electric field strength due to a microwave radar transmitter is measured to be 140 V-m−1. Using the above formula, calculate the maximum exposure time allowed (in hours) for this person to stay at that location. Get solution
44. Microwave cataracts in humans. Over 50 cases of human cataract induction have been attributed to microwave exposures, primarily encountered in occupational situations involving acute exposure to presumably relatively high-intensity fields.84 The following are three reported incidents of cataracts caused by microwave radiation: (1) A 22-year-old technician exposed approximately five times to 3 GHz radiation at an estimated average power density of 300 mW-(cm)−2 for 3 min/exposure developed bilateral cataracts. (2) A person was exposed to microwaves for durations of approximately 50 hour/month over a 4-year period at average power densities of less than 10 mW-(cm)−2 in most instances, but with a period of 6 months or more during which the average power density was approximately 1 W-(cm)−2. (3) A 50-year-old woman was intermittently exposed to leakage radiation from a 2.45 GHz microwave oven of approximately 1 mW-(cm)−2 during oven operation, with levels of up to 90 mW-(cm)−2 when the oven door was open, presumably over a period of approximately 6 years prior to developing cataract. For each of these above cases, compare the power densities with the IEEE standards (see Problem ) and comment.Mobile phones. Cellular phone antennas installed on cars have a maximum output power of 3 W, set by the Federal Communications Commission (FCC) standards. The incident electromagnetic energy to which the passengers in the car are exposed does not pose any health threats, both because they are some distance away from the antenna and also because the body of the car and glass window shield them from much of the radiation.79 (a) For a car with a synthetic roof, the maximum localized power density in the passenger seat is about 0.3 mW-(cm)−2. For cars with metal roofs, this value reduces to 0.02 mW-(cm)−2 or less. Antennas mounted on the trunk or in the glass of the rear windshield deliver power densities of about 0.35–0.07 mW-(cm)−2 to passengers in the back seat. Compare these values with the IEEE safety limit (IEEE Standard C95.1-1991) in the cellular phone frequency range (which is typically 800–900 MHz) and comment on the safety of the passengers. Note that from 300 MHz to 15 GHz, IEEE safety limits 80 specify a maximum allowable power density that increases linearly with frequency as |Sav|max = f /1500 in mW-(cm)−2, where the frequency f is in MHz. For reference, the maximum allowable power density is 1 mW-(cm)−2 at 1.5 GHz. (b) Calculate the maximum output power of a cellular phone antenna installed on the metal roof of a car such that the localized power density in the passenger compartment is equal to the IEEE safety limit at 850 MHz. Get solution
45. VHF TV signal. The magnetic field component of a 10 μW-m−2, 200 MHz TV signal in air is given by...(a) What are the values of H0 and a? (b) Find the corresponding electric field E(x, y, t ). What is the polarization of the wave? (c) An observer at z = 0 is equipped with a wire antenna capable of detecting the component of the electric field along its length. Find the maximum value of the measured electric field if the antenna wire is oriented along the (i) x direction, (ii) y direction, (iii) 45◦ line between the x and y directions. Get solution
46. FM polarization. Find the type (linear, circular, elliptical) and sense (right- or left-handed) of the polarization of the FM broadcast signal given in ExampleFM broadcast signal. An FM radio broadcast signal traveling in the y direction in air has a magnetic field given by the phasor...(a) Determine the frequency (in MHz) and wavelength (in m). (b) Find the corresponding E(y).(c) Write the instantaneous expression for E(y, t ) and H(y, t ). Get solution
47. Unknown wave polarization. The magnetic field component of a uniform plane wave in air is given by...where a is a real constant. (a) Find the wavelength λ and frequency f . (b) Find the total time-average power density carried by this wave. (c) Determine the type (linear, circular, elliptical) and sense (right- or left-handed) of the polarization of this wave when a = 1. (d) Repeat part (c) when a = 3. Get solution
48. Linear and circularly polarized waves. Two electromagnetic waves operating at the same frequency and propagating in the same direction (y direction) in air are such that one of them is linearly polarized in the x direction, whereas the other is left-hand circularly polarized (LHCP). However, the electric and magnetic field components of the two waves appear identical at one instant within every 50 ps time interval. The linearly polarized wave carries a time-average power density of 1.4 W-m−2. An observer located at y = 0 uses a receiving antenna to measure the x component of the total electric field only and records a maximum field magnitude of about 65 V-m−1 over every time interval of 0.5 ns. (a) Write the mathematical expressions for the electric field components of each wave, using numerical values of various quantities whenever possible. (b) Find the ratio of the time-average power densities of the LHCP and the linearly polarized waves. Get solution
49. Two circularly polarized waves. Consider two circularly polarized waves traveling in the same direction transmitted by two different satellites operating at the same frequency given by...where E01 and E02 are real constants. (a) If the total time-average power densities of these two waves are equal, find the polarization of the total wave. (b) Repeat part (a) for the case when the total time-average power density of the first wave is four times the total time-average power density of the second wave. Get solution
50. Wave polarization. Consider the following complex phasor expression for a time-harmonic magnetic field in free space:...(a) Is this a uniform plane wave? What is its frequency? (b) What is the direction of propagation and the state of polarization (specify both the type and sense of polarization) of this electromagnetic field? (c) Find the associated electric field phasor and the total time-average power density in the direction of propagation. Get solution
51. Wave polarization. The electric field component of a communication satellite signal travelling in free space is given by...(a) Find the corresponding H(z ). (b) Find the total time-average power density carried by this wave. (c) Determine the polarization (both type and sense) of the wave. Get solution
52. Wave polarization. A fellow engineer makes the following two measurements of the electric field vector of a uniform plane wave propagating in the x direction in a simple, lossless and nonmagnetic (μ = μ0) medium:...(a) Are these two measurements enough to determine the polarization type and sense of the wave? What about the wavelength of the wave? (b) Now the engineer offers two other measurements made at the same time t = 0:...Furthermore, you are informed that the electric field amplitude at t = 0 does not exceed 5 V-m−1 at any point between x = 0 and x = 0.75 m. Given this information, determine the type and sense of polarization of the wave. Get solution
53. Superposition of two waves. The electric field components of two electromagnetic waves at the same frequency and propagating in free space are represented by...Find and sketch the locus of the total electric field measured at the origin (x = y = z = 0) if E0 is equal to (a) 10 V-m−1, (b) 20 V-m−1, (c) 40 V-m−1, respectively. Get solution
