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. What is the longest wavelength for which there will be destructive interference at point Q?

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Identify the positions of the antennas and point Q on a coordinate line: let antenna A be at position 0 m, antenna B at 120 m, and point Q at 160 m (since it is 40 m to the right of B).

Calculate the distances from each antenna to point Q: the distance from A to Q is \(d_A = 160\) m, and from B to Q is \(d_B = 40\) m.

Determine the path difference \(\Delta d\) between the waves arriving at Q from antennas A and B: \(\Delta d = |d_A - d_B| = |160 - 40| = 120\) m.

Recall that for destructive interference between two in-phase sources, the path difference must satisfy the condition \(\Delta d = (m + \frac{1}{2}) \lambda\), where \(m\) is an integer (0, 1, 2, ...) and \(\lambda\) is the wavelength.

To find the longest wavelength causing destructive interference, set \(m = 0\) and solve for \(\lambda\): \(\lambda = 2 \Delta d\). This gives the fundamental wavelength for destructive interference at point Q.

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주요 개념

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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, amplifying the signal, while destructive interference occurs when waves are out of phase by half a wavelength, canceling each other out. Understanding interference is key to analyzing the combined effect of waves from antennas at a point.

Path Difference and Phase Difference

The path difference is the difference in distance traveled by waves from two sources to a point. This difference determines the phase difference between the waves at that point. For destructive interference, the path difference must be an odd multiple of half wavelengths, causing the waves to arrive out of phase and cancel.

Relationship Between Wavelength, Frequency, and Speed

Wavelength (λ), frequency (f), and wave speed (v) are related by v = fλ. For electromagnetic waves like radio waves, the speed is constant (speed of light). Varying the frequency changes the wavelength, which affects interference patterns. Finding the longest wavelength for destructive interference involves using this relationship along with path difference conditions.