Textbook Question
(II) How much energy is transported across a 1.00-cm² area per hour by an EM wave whose E field has an rms strength of 25.8 mV/m?
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32. Electromagnetic Waves
Intro to Electromagnetic (EM) Waves
5:00 minutesProblem 20
Textbook Question
(II) How long would it take a message sent as radio waves from Earth to reach Mars when Mars is (a) nearest Earth, (b) farthest from Earth? Assume that Mars and Earth are in the same plane and that their orbits around the Sun are circles (Mars is 230 x 106 km from the Sun).
Verified step by step guidance1
Step 1: Understand the problem. The goal is to calculate the time it takes for radio waves (which travel at the speed of light) to travel from Earth to Mars when Mars is at its nearest and farthest distances from Earth. Radio waves are electromagnetic waves, and their speed in a vacuum is approximately 3.00 × 10⁸ m/s.
Step 2: Convert the distances from kilometers to meters for consistency with the speed of light. Mars is 230 × 10⁶ km from the Sun, which is equivalent to 230 × 10⁶ × 10³ m. Similarly, Earth's distance from the Sun is approximately 150 × 10⁶ km, or 150 × 10⁶ × 10³ m.
Step 3: Calculate the nearest distance between Earth and Mars. When Mars is nearest to Earth, the distance is the difference between their distances from the Sun: (230 × 10⁶ × 10³ m) - (150 × 10⁶ × 10³ m).
Step 4: Calculate the farthest distance between Earth and Mars. When Mars is farthest from Earth, the distance is the sum of their distances from the Sun: (230 × 10⁶ × 10³ m) + (150 × 10⁶ × 10³ m).
Step 5: Use the formula for time, \( t = \frac{d}{v} \), where \( d \) is the distance and \( v \) is the speed of light (3.00 × 10⁸ m/s). Substitute the nearest and farthest distances calculated in steps 3 and 4 into this formula to find the respective times.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Speed of Light
The speed of light in a vacuum is approximately 299,792 kilometers per second. This constant is crucial for calculating the time it takes for electromagnetic waves, such as radio waves, to travel between two points in space. Since radio waves travel at the speed of light, understanding this speed allows us to determine how long it takes for a message to reach Mars from Earth.
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Distance in Space
The distance between Earth and Mars varies significantly due to their elliptical orbits around the Sun. When Mars is at its closest approach (approximately 54.6 million kilometers), the travel time for radio waves is shorter than when Mars is at its farthest point (about 401 million kilometers). Knowing these distances is essential for accurately calculating the time it takes for a message to be transmitted.
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Orbital Mechanics
Orbital mechanics is the study of the motion of celestial bodies in space, governed by gravitational forces. Understanding the orbits of Earth and Mars helps in predicting their relative positions over time. This knowledge is vital for determining the specific distances at which the two planets are located during different times of the year, impacting the time it takes for signals to travel between them.
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