FM radio waves from a distant tower pass through a relatively small gap (width b
ID: 1700507 • Letter: F
Question
FM radio waves from a distant tower pass through a relatively small gap (width b) between two very large buildings that are opaque to radio waves. (Assume that the radio waves are plane-waves by the time they reach the gap, and their direction of travel is exactly perpendicular to b.) You walk along a street (shown here as the x-axis) at distance s from the gap, perpendicular to the direction the waves are traveling.
a. Suppose that b = 12.5 m, s = 20.0 m, and the wavelength of the radio waves is 2.897 m. As you walk along the x-axis from –xmax = –30.0 m to +xmax = 30.0 m while carrying your portable radio, at what positions will you discover total destructive interference of the radio waves? Find them all. Show your work. (Use the above diagram if needed. You may use the symmetry of the situation to save some work, but explain your reasoning as necessary.)
b. Calculate the frequency of the FM station. Show your work.
c. Suppose that your portable radio uses a series L-R-C circuit to tune in to the desired radio station. As you
rotate the tuner dial, you adjust the value of a variable capacitor, C. If L is fixed at a value of 4.80 µH and R is fixed at 2.2 , find the value of capacitor C when the radio is tuned to the station in part (b). Show your work.
Explanation / Answer
at position of x, the part different ?S ?S=sqrt((x+6.25)^2+20^2)-sqrt((x-6.25)^2+20^2) its destructive interference. so ?S=(0.5+k)*2.897. ------------- so plug k=0,1,2,... in and then find x from the equation above. ?S+sqrt(x^2-12.5x+439)=sqrt(x^2+12.5x+439). 2*?S*sqrt(x^2-12.5x+439)=25x. 4?S^2*(x^2-12.5x+439)=25^2*x^2. solve this quadratic function by plugging ?S=(0.5+k)*2.897 to find x ------------------------------------------- the FM wave is mag-elec wave to its velocity is c. c*T=2.897 so f=c/2.897=0.1e9(Hz) -------------- the frequence. w=sqrt(w0^2-b^2) where w0^2=1/CL b=R/2L. so w=sqrt(1/C^2L^2 - R^2/4L^2) w=2pif so 4.23e17=(1/C^2-R^2/4)/L^2 C=3.2e-4(F)
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