At one instant, the electric and magnetic fields at one point of an electromagnetic wave are and . What are the values of and ?
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32. Electromagnetic Waves
Intensity of EM Waves
Problem 75cd
Textbook Question
Suppose a 25-kW radio station emits EM waves uniformly in all directions. What is the rms voltage induced in a 1.0-m-long vertical car antenna (c) 1.0 km away, (d) 50 km away?

1
Step 1: Understand the problem. The radio station emits electromagnetic (EM) waves uniformly in all directions with a power of 25 kW. We need to calculate the root mean square (rms) voltage induced in a 1.0-m-long car antenna at two distances: 1.0 km and 50 km. The induced voltage depends on the electric field strength of the EM wave at the given distances.
Step 2: Calculate the intensity of the EM wave at a given distance. The intensity (I) of the wave is related to the power (P) emitted and the distance (r) from the source. Use the formula: , where is the power (25 kW) and is the distance (1.0 km or 50 km).
Step 3: Relate the intensity to the electric field strength. The intensity of an EM wave is related to the electric field strength (E) by the formula: , where is the speed of light in a vacuum and is the electric field strength. Rearrange this formula to solve for : .
Step 4: Calculate the rms voltage induced in the antenna. The rms voltage (Vrms) induced in the antenna is related to the electric field strength (E) and the length of the antenna (L) by the formula: , where is the length of the antenna (1.0 m).
Step 5: Perform the calculations for both distances. First, calculate the intensity at 1.0 km and 50 km using Step 2. Then, use Step 3 to find the electric field strength at each distance. Finally, use Step 4 to calculate the rms voltage induced in the antenna at both distances. Ensure units are consistent throughout the calculations.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Electromagnetic Waves
Electromagnetic (EM) waves are oscillations of electric and magnetic fields that propagate through space. They travel at the speed of light and can carry energy across distances. Understanding EM waves is crucial for analyzing how radio waves emitted by the station interact with objects like antennas.
Recommended video:
Guided course
Introduction to Electromagnetic (EM) Waves
Power and Intensity
The power of a radio station, measured in watts, indicates the total energy emitted per second. Intensity, defined as power per unit area, decreases with distance from the source due to the spreading of the wavefront. This relationship is essential for calculating the voltage induced in the antenna based on its distance from the transmitter.
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Wave Intensity
Induced Voltage in Antennas
The induced voltage in an antenna is related to the electric field strength of the incoming EM waves. The root mean square (rms) voltage can be calculated using the relationship between the electric field and the antenna's length. This concept is vital for determining how effectively the antenna can convert EM wave energy into an electrical signal.
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RMS Current and Voltage
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