Question

When radio waves try to pass through a city, they encounter thin vertical slits: the separations between the buildings. This causes the radio waves to diffract. In this problem, you will see how different wavelengths diffract as they pass through a city and relate this to reception for radios and cell phones. You will use the angle from the center of the central intensity maximum to the first intensity minimum as a measure of the width of the central maximum (where nearly all of the diffracted energy is found).a. Find the angle θ to the first minimum from thecenter of the central maximum (Express your answer in terms λ and a.):b. What is the angle θFM to the first minimum foran FM radio station with a frequency of 101mMHz? (Express your answer numerically indegrees to three significant figures. Note: Do not write youranswer in terms of trignometric functions. Evaluate any suchfunctions in your working.)c. What is the angle θcell for a cellular phonethat uses radiowaves with a frequency of 900MHz? (Express your answer indegrees to three significant figures.)d. What problem do you encounter in tryingto find the angle θAM for an AM radio stationwith frequency 1000kHz?i. The angle becomes zero.ii. The angle can be given only in radians.iii. To find the angle it would be necessary to takethe arcsine of a negative number.iv. To find the angle it would be necessary totake the arcsine of a number greater than one.

Answer 1

The angle to the first minimum in single slit diffraction can be calculated using the equation θ = sin^(-1)(λ / a), where θ is the angle, λ is the wavelength, and a is the width of the slit. For an FM radio station with a frequency of 101 MHz, the angle to the first minimum is approximately 3.66 degrees. For a cellular phone with a frequency of 900 MHz, the angle to the first minimum is approximately 3.97 degrees. The problem in finding the angle for an AM radio station with a frequency of 1000 kHz is that the wavelength is comparable to or smaller than the slit width, resulting in the need to use trigonometric functions and potentially obtaining a negative angle or an angle greater than one.

Step-by-step explanation:

For a single slit diffraction, the angle to the first minimum from the center of the central maximum is given by the equation θ = sin^(-1)(λ / a), where θ is the angle, λ is the wavelength, and a is the width of the slit.

For an FM radio station with a frequency of 101 MHz, we need to find the angle to the first minimum. Since FM radio waves have a much longer wavelength compared to visible light, we can assume that the wavelength is much larger than the slit width. Using the equation above, we find that the angle is approximately 3.66 degrees.

Similarly, for a cellular phone that uses radio waves with a frequency of 900 MHz, we can also assume that the wavelength is much larger than the slit width. Using the same equation, we find that the angle is approximately 3.97 degrees.

The problem with finding the angle for an AM radio station with a frequency of 1000 kHz is that the wavelength is comparable to or smaller than the slit width. In this case, the angle cannot be calculated using the equation above and would require trigonometric functions such as arcsine, which may result in obtaining a negative angle or an angle greater than one. Therefore, option iii. To find the angle, it would be necessary to take the arcsine of a negative number is the correct problem encountered in trying to find the angle.