Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point Q?

Verified step by step guidance

1. Identify the path difference between the waves reaching point Q from antennas A and B. Since antenna B is 120 m to the right of A and point Q is 40 m to the right of B, the total distance from A to Q is 120 m + 40 m = 160 m. The distance from B to Q is 40 m. Therefore, the path difference is 160 m - 40 m = 120 m.
2. Recall the condition for destructive interference, which occurs when the path difference is an odd multiple of half the wavelength, i.e., \((2n+1)\frac{\lambda}{2}\) where \(n\) is an integer (0, 1, 2, ...).
3. Set up the equation for destructive interference using the path difference: \(120 m = (2n+1)\frac{\lambda}{2}\).
4. Solve for \(\lambda\) in terms of \(n\) to find the possible wavelengths that satisfy the condition for destructive interference: \(\lambda = \frac{240 m}{2n+1}\).
5. Determine the longest wavelength by finding the smallest integer value of \(n\) that satisfies the equation. For the longest wavelength, use \(n = 0\) which gives \(\lambda = \frac{240 m}{1} = 240 m\).

Key Concepts

Destructive Interference

Destructive interference occurs when two waves meet in such a way that their crests and troughs align oppositely, resulting in a reduction or cancellation of the overall wave amplitude. This typically happens when the path difference between the two waves is an odd multiple of half the wavelength (λ/2, 3λ/2, etc.). In this scenario, understanding the conditions for destructive interference is crucial to determine the longest wavelength that will lead to this effect at point Q.

Path Difference

The path difference is the difference in distance traveled by two waves from their respective sources to a given point. For destructive interference to occur, this path difference must equal an odd multiple of half the wavelength. In this problem, calculating the path difference between the waves from antennas A and B at point Q is essential to find the longest wavelength that results in destructive interference.

Wavelength

Wavelength (λ) is the distance between successive crests (or troughs) of a wave and is inversely related to frequency. In the context of this problem, varying the wavelength affects the conditions for interference. The longest wavelength that results in destructive interference at point Q can be determined by analyzing the path difference and applying the principles of wave interference.