Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 31

Chapter 17, Problem 31

Two out-of-phase radio antennas at x=±300 m on the x-axis are emitting 3.0 MHz radio waves. Is the point (x, y) =(300 m, 800 m) a point of maximum constructive interference, maximum destructive interference, or something in between?

Verified step by step guidance

Step 1: Calculate the wavelength of the radio waves using the formula \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)) and \( f \) is the frequency (\( 3.0 \times 10^6 \; \text{Hz} \)).

Step 2: Determine the distances from the two antennas to the point \((300 \; \text{m}, 800 \; \text{m})\). Use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for each antenna. For the antenna at \( x = 300 \; \text{m} \), the distance is \( \sqrt{(300 - 300)^2 + (800 - 0)^2} \). For the antenna at \( x = -300 \; \text{m} \), the distance is \( \sqrt{(300 - (-300))^2 + (800 - 0)^2} \).

Step 3: Find the path difference between the distances calculated in Step 2. The path difference is \( \Delta d = |d_1 - d_2| \), where \( d_1 \) and \( d_2 \) are the distances from the two antennas to the point.

Step 4: Compare the path difference \( \Delta d \) to the wavelength \( \lambda \). If \( \Delta d \) is an integer multiple of \( \lambda \) (\( \Delta d = n \lambda \), where \( n \) is an integer), the point is a location of maximum constructive interference. If \( \Delta d \) is an odd multiple of half the wavelength (\( \Delta d = (2n+1) \frac{\lambda}{2} \)), the point is a location of maximum destructive interference. Otherwise, the interference is somewhere in between.

Step 5: Based on the comparison in Step 4, classify the point \((300 \; \text{m}, 800 \; \text{m})\) as a point of maximum constructive interference, maximum destructive interference, or something in between.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Interference of Waves

Interference occurs when two or more waves overlap, resulting in a new wave pattern. Constructive interference happens when waves are in phase, leading to increased amplitude, while destructive interference occurs when waves are out of phase, resulting in reduced amplitude. The type of interference at a point depends on the relative phase difference between the waves at that location.

Path Difference

Path difference refers to the difference in distance traveled by two waves from their sources to a specific point. For constructive interference, the path difference must be an integer multiple of the wavelength, while for destructive interference, it should be a half-integer multiple. Calculating the path difference is essential to determine the interference pattern at a given point.

Wavelength and Frequency

The wavelength is the distance between successive peaks of a wave, and it is inversely related to frequency, which is the number of oscillations per second. For radio waves, the wavelength can be calculated using the speed of light divided by the frequency. Understanding the relationship between wavelength and frequency is crucial for analyzing wave behavior and interference patterns.

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