Instructions, notes, and hints: Provide the details of all solutions, including important intermediate steps. You will not receive credit if you do not show your work. Questions: 1. An airplane is flying 1 km above the ocean radiating 9,000 Watts at 10 kHz in order to communicate with a submarine submerged in the ocean. Let's perform a back-of-the- envelope calculation to determine the how deep the submarine can be and still receive the signal from the airplane. The ocean is characterized by $sigma$ = 4 (S/m), $epsilon_r$ = 80, and $mu_r$ = 1. The antenna on the airplane radiates uniformly in all directions (spherically) from the airplane. (After our discussion about antennas in the next part of this course, we could improve our antenna design, radiation pattern, and calculations for this problem). Airplane Surface of the ocean $E_{incident}$ Outgoing isotropic wave a. Find the amplitude of the electric field incident on the surface of the ocean directly under the airplane: $|E_{incident}|$. Since the wave radiates uniformly outward from the airplane, you can assume the 9,000 Watts of power spreads out equally over a spherical surface until it reaches the ocean surface. b. The wave directly below the airplane can be approximated locally as a plane wave incident on the ocean. Determine how much of the wave is transmitted into the ocean (i.e. find the transmission coefficient). c. Write an expression for the electric field phasor as a function of depth ($z$) into the ocean (straight under the airplane). For simplicity, assume $z$ = 0 is at the surface of the ocean, and the +$z$-direction is downward, into the ocean. d. Write an expression for the magnetic field phasor as a function of depth ($z$) into the ocean (straight under the airplane). e. Write a corresponding expression for the Poynting vector as a function of depth ($z$) into the ocean. f. The receiver on the submarine requires $|E| ge 10^{-10}$ (V/m) in order to detect the radio signal from the airplane. What is the maximum depth that the submarine can still detect the signal?
Added by Juan Antonio A.
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Plugging in the values, we get: Eincitem = sqrt(9,000/(4*pi*(1,000)^2)) = 0.003 V/m b. The transmission coefficient (T) is the ratio of the transmitted electric field to the incident electric field. For a plane wave incident on a dielectric interface, the Show more…
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Adi S.
PROBLEMS & EXERCISES 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed 1. Verify that the correct value for the speed of light c is obtained when numerical values for the permeability and permittivity of free space (μ₀ and ε₀) are entered into the equation c = 1 / √(μ₀ε₀). 2. Show that, when SI units for μ₀ and ε₀ are entered, the units given by the right-hand side of the equation in the problem above are m/s. 24.2 Production of Electromagnetic Waves 3. What is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of 5.00 × 10⁻⁴ T (about 10 times the Earth’s)? 4. The maximum magnetic field strength of an electromagnetic field is 5 × 10⁻⁶ T. Calculate the maximum electric field strength if the wave is traveling in a medium in which the speed of the wave is 0.75c. 5. Verify the units obtained for magnetic field strength B in Example 24.1 (using the equation B = E/c) are in fact teslas (T). 24.3 The Electromagnetic Spectrum 6. (a) Two microwave frequencies are authorized for use in microwave ovens: 900 and 2560 MHz. Calculate the wavelength of each. (b) Which frequency would produce smaller hot spots in foods due to interference effects? 7. (a) Calculate the range of wavelengths for AM radio given its frequency range is 540 to 1600 kHz. (b) Do the same for the FM frequency range of 88.0 to 108 MHz. 8. A radio station utilizes frequencies between commercial AM and FM. What is the frequency of a 11.12-m-wavelength channel? 9. Find the frequency range of visible light, given that it encompasses wavelengths from 380 to 760 nm. 10. Combing your hair leads to excess electrons on the comb. How fast would you have to move the comb up and down to produce red light? 11. Electromagnetic radiation having a 15.0 – μm wavelength is classified as infrared radiation. What is its frequency? 12. Approximately what is the smallest detail observable with a microscope that uses ultraviolet light of frequency 1.20 × 10¹⁵ Hz? 13. A radar used to detect the presence of aircraft receives a pulse that has reflected off an object 6 × 10⁻⁵ s after it was transmitted. What is the distance from the radar station to the reflecting object? 14. Some radar systems detect the size and shape of objects such as aircraft and geological terrain. Approximately what is the smallest observable detail utilizing 500-MHz radar? 15. Determine the amount of time it takes for X-rays of frequency 3 × 10¹⁸ Hz to travel (a) 1 mm and (b) 1 cm. 16. If you wish to detect details of the size of atoms (about 1 × 10⁻¹⁰ m) with electromagnetic radiation, it must have a wavelength of about this size. (a) What is its frequency? (b) What type of electromagnetic radiation might this be? 17. If the Sun suddenly turned off, we would not know it until its light stopped coming. How long would that be, given that the Sun is 1.50 × 10¹¹ m away?
Sri K.
When the Federal Aviation Administration (FAA) allocates numerous frequencies for an airport radio transmitter quite often nearby transmitters use the same frequencies. As a consequence, the FAA would like to minimize the interference between these transmitters. In FIGURE 4.8.50, the point $\left(x_{t}, y_{t}\right)$ represents the location of a transmitter whose radio jurisdiction is indicated by the circle $C$ of radius with center at the origin. A second transmitter is located at $\left(x_{i}, 0\right)$ as shown in the figure. In this problem you will develop and analyze a function to find the interference between two transmitters. (a) The strength of the signal from a transmitter to a point is inversely proportional to the square of the distance between them. Assume that a point $(x, y)$ is located on the upper portion of the circle $C$ as shown in Figure 4.8.50. Express the primary strength of the signal at $(x, y)$ from a transmitter at $\left(x_{t}, y_{t}\right)$ as a function of $x$ Express the secondary strength at $(x, y)$ from the transmitter at $\left(x_{i}, 0\right)$ as a function of $x$. Now define a function $R(x)$ as a quotient of the primary signal strength to the secondary signal strength. $R(x)$ can be thought of as a signal to noise ratio. To guarantee that the interference remains small we need to show that the minimum signal to noise ratio is greater than the FAA's minimum threshold of -0.7 . (b) Suppose that $x_{t}=760 \mathrm{~m}, y_{t}=-560 \mathrm{~m}, r=1.1 \mathrm{~km}$ and $x_{i}=12 \mathrm{~km} .$ Use a CAS to simplify and then plot the graph of $R(x)$. Use the graph to estimate the domain and range of $R(x)$. (c) Use the graph in part (b) to estimate the value of $x$ where the minimum ratio $R$ occurs. Estimate the value of $R$ at that point. Does this value of $R$ exceed the FAA's minimum threshold? (d) Use a CAS to differentiate $R(x)$. Use a CAS to find the root of $R^{\prime}(x)=0$ and to compute the corresponding value of $R(x)$. Compare your answers here with the estimates in part (c). (e) What is the point $(x, y)$ on circle $C ?$ (f) We assumed that the point $(x, y)$ was in the top half plane when $\left(x_{t}, y_{t}\right)$ was in the lower half plane. Explain why this assumption is correct. (g) Use a CAS to find the value of $x$ where the minimum interference occurs in terms of the symbols $x_{t}, y_{t}, x_{i}$ and $r$. (h) Where is that point that minimizes the signal to noise ratio when the transmitter at $\left(x_{t}, y_{t}\right)$ is on the $x$ -axis? Give a convincing argument justifying your answer.
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