FM radio station KRTH in Los Angeles broadcasts on an assigned frequency of 101 MHz with a power of 50,000 W. (a) What is the wavelength of the radio waves produced by this station? (b) Estimate the average intensity of the wave at a distance of 22.3 km from the radio transmitting antenna. Assume for the purpose of this estimate that the antenna radiates equally in all directions, so that the intensity is constant over a hemisphere centered on the antenna. (c) Estimate the amplitude of the electric field at this distance.
FM radio station KRTH in Los Angeles broadcasts on an assigned frequency of 101 MHz with a power of 50,000 W. (a) What is the wavelength of the radio waves produced by this station? (b) Estimate the average intensity of the wave at a distance of 22.3 km from the radio transmitting antenna. Assume for the purpose of this estimate that the antenna radiates equally in all directions, so that the intensity is constant over a hemisphere centered on the antenna. (c) Estimate the amplitude of the electric field at this distance.
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![**FM Radio Station Signal Analysis**
*FM radio station KRTH in Los Angeles broadcasts on an assigned frequency of 101 MHz with a power of 50,000 W.*
**(a) What is the wavelength of the radio waves produced by this station?**
[Space for Answer]
**(b) Estimate the average intensity of the wave at a distance of 22.3 km from the radio transmitting antenna. Assume for the purpose of this estimate that the antenna radiates equally in all directions, so that the intensity is constant over a hemisphere centered on the antenna.**
The average intensity, \(I\), can be calculated using the formula:
\[ I = \frac{P}{A} \]
where:
- \(P\) is the power of the transmitter (50,000 W)
- \(A\) is the area over which the power is distributed.
Since the power is radiated equally in all directions, and we are dealing with a hemisphere (not a full sphere), the area \(A\) is given by the surface area of a hemisphere:
\[ A = 2 \pi r^2 \]
where \(r\) is the distance of 22.3 km.
[Space for Calculation and Answer]
**(c) Estimate the amplitude of the electric field at this distance.**
The relationship between the intensity \(I\) and the amplitude \(E\) of the electric field is:
\[ I = \frac{c \epsilon_0 E^2}{2} \]
where:
- \(c\) is the speed of light in a vacuum (\(3 \times 10^8 \text{ m/s}\))
- \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \text{ F/m}\))
From this relationship, we can solve for \(E\):
\[ E = \sqrt{\frac{2I}{c \epsilon_0}} \]
[Space for Calculation and Answer]
**Diagrams and Graphs:**
There are no diagrams or graphs provided in the problem statement. If there were to be a diagram, it might illustrate the hemispherical distribution of the radiating waves from the antenna and how the power is spread over the surface area at the given distance.
---
**Notes for Students:**
- Ensure to convert all units to standard SI units before performing calculations.
- Be precise with your calculations and retain significant](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2379bb0-49c5-44f7-ac7a-000baa150133%2F9e7ff73d-5c75-45ec-a16a-6f230b67fb57%2Fi2efku_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**FM Radio Station Signal Analysis**
*FM radio station KRTH in Los Angeles broadcasts on an assigned frequency of 101 MHz with a power of 50,000 W.*
**(a) What is the wavelength of the radio waves produced by this station?**
[Space for Answer]
**(b) Estimate the average intensity of the wave at a distance of 22.3 km from the radio transmitting antenna. Assume for the purpose of this estimate that the antenna radiates equally in all directions, so that the intensity is constant over a hemisphere centered on the antenna.**
The average intensity, \(I\), can be calculated using the formula:
\[ I = \frac{P}{A} \]
where:
- \(P\) is the power of the transmitter (50,000 W)
- \(A\) is the area over which the power is distributed.
Since the power is radiated equally in all directions, and we are dealing with a hemisphere (not a full sphere), the area \(A\) is given by the surface area of a hemisphere:
\[ A = 2 \pi r^2 \]
where \(r\) is the distance of 22.3 km.
[Space for Calculation and Answer]
**(c) Estimate the amplitude of the electric field at this distance.**
The relationship between the intensity \(I\) and the amplitude \(E\) of the electric field is:
\[ I = \frac{c \epsilon_0 E^2}{2} \]
where:
- \(c\) is the speed of light in a vacuum (\(3 \times 10^8 \text{ m/s}\))
- \(\epsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \text{ F/m}\))
From this relationship, we can solve for \(E\):
\[ E = \sqrt{\frac{2I}{c \epsilon_0}} \]
[Space for Calculation and Answer]
**Diagrams and Graphs:**
There are no diagrams or graphs provided in the problem statement. If there were to be a diagram, it might illustrate the hemispherical distribution of the radiating waves from the antenna and how the power is spread over the surface area at the given distance.
---
**Notes for Students:**
- Ensure to convert all units to standard SI units before performing calculations.
- Be precise with your calculations and retain significant
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