Consider two antennas radiating 6.8-MHz radio waves in phase with each other. They are located at points S₁ and S₂, separated by a distance d = 175 m, Fig. 34–50. Determine the points on the positive y-axis where the signals from the two sources will be out of phase (crests of one meet troughs of the other).

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

Step 1: Understand the problem. The two antennas radiate 6.8-MHz radio waves in phase, and we need to find points on the positive y-axis where the signals are out of phase. This occurs when the path difference between the waves from S₁ and S₂ is an odd multiple of half the wavelength.

Step 2: Calculate the wavelength of the radio waves using the formula λ = c / f, where c is the speed of light (approximately 3 × 10⁸ m/s) and f is the frequency (6.8 MHz). Substitute the values to find λ.

Step 3: Determine the condition for destructive interference. Destructive interference occurs when the path difference ΔL = |L₁ - L₂| is equal to (n + 0.5)λ, where n is an integer (0, 1, 2, ...). Here, L₁ and L₂ are the distances from S₁ and S₂ to a point on the positive y-axis.

Step 4: Express the distances L₁ and L₂ geometrically. For a point on the positive y-axis at a distance y from the origin, L₁ = √(y²) = y and L₂ = √(y² + d²), where d = 175 m is the separation between the antennas.

Step 5: Solve for y by setting the path difference ΔL = |L₁ - L₂| equal to (n + 0.5)λ. Substitute L₁ = y, L₂ = √(y² + d²), and λ from Step 2 into the equation. Rearrange and solve for y for different values of n to find the points on the positive y-axis where destructive interference occurs.

Key Concepts

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

Wave Interference

Wave interference occurs when two or more waves overlap and combine to form a new wave pattern. This can result in constructive interference, where wave crests align, or destructive interference, where a crest meets a trough, leading to cancellation. Understanding this concept is crucial for analyzing how the signals from the two antennas interact at various points in space.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time. For waves to be out of phase, the phase difference must be an odd multiple of π (180 degrees), meaning that the crest of one wave coincides with the trough of another. This concept is essential for determining the specific points along the y-axis where the signals from the antennas will interfere destructively.

Path Length Difference

Path length difference is the difference in distance traveled by two waves from their sources to a common point. In this scenario, the path length difference between the waves from antennas S₁ and S₂ must equal half the wavelength for destructive interference to occur. Calculating this difference helps identify the locations on the y-axis where the signals will be out of phase.

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