Problem 4 - University Physics with Modern Physics

Question

Radio Interference. Two radio antennas \(A\) and \(B\) radiate in phase. Antenna \(B\) is 120 \(\mathrm{m}\) to the right of antenna \(A .\) Consider point \(Q\) along the extension of the line connecting the antennas, a horizontal distance of 40 \(\mathrm{m}\) 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 \(Q ?\) (b) What is the longest wave-length for which there will be constructive interference at point \(Q ?\)

Step-by-Step Solution

Answer

(a) 240 m, (b) 120 m.

Step 1: Understand the Problem

We have two radio antennas, A and B, that emit waves in phase. The distance between A and B is 120 meters, and point Q is located 40 meters to the right of antenna B. We are asked to find the longest wavelength for which there is destructive interference and the longest wavelength for constructive interference at point Q.

Step 2: Calculate Path Difference

The path length from antenna A to point Q is the sum of the distances from A to B and from B to Q, which is 160 meters. The path length from antenna B to point Q is 40 meters. The path difference, therefore, is the difference between these two, which is 160 m - 40 m = 120 m.

Step 3: Condition for Destructive Interference

For destructive interference, the path difference should be odd multiples of half the wavelength: \( \Delta d = (m+0.5)\lambda \), where \( m \) is an integer (0, 1, 2,...). The longest wavelength occurs when \( m = 0 \). Thus, \( 120 = 0.5\lambda \) and solving for \( \lambda \), we get \( \lambda = 240 \ m \).

Step 4: Condition for Constructive Interference

For constructive interference, the path difference should be integer multiples of the wavelength: \( \Delta d = m\lambda \), where \( m \) is an integer (0, 1, 2,...). The longest wavelength occurs when \( m = 1 \). Thus, \( 120 = \lambda \) and solving for \( \lambda \), we get \( \lambda = 120 \ m \).

Step 5: Final Answer

Compile the results. For destructive interference, the longest wavelength is 240 meters, and for constructive interference, the longest wavelength is 120 meters.

Key Concepts

Radio Waves

Radio waves are a type of electromagnetic wave used for transmitting information like radio broadcasts, television signals, and even Wi-Fi. These waves are part of the electromagnetic spectrum, which includes other wave types such as microwaves, infrared waves, and visible light.
Radio waves have long wavelengths compared to other forms of waves, which allows them to travel long distances and penetrate through obstacles like buildings and trees. This makes them ideal for communication over vast areas.
When dealing with radio waves, one of the important aspects to consider is their wavelength, especially in situations involving interference, such as in the problem we are discussing.

Path Difference

The concept of path difference is crucial when analyzing interference patterns between waves. It refers to the difference in distance traveled by two waves arriving at the same point.
In the exercise provided, the path difference is the additional distance radio waves from antenna A travel compared to those from antenna B in reaching point Q. The path lengths are 160 meters for antenna A and 40 meters for antenna B, giving a path difference of 120 meters.
Understanding path difference helps determine whether the waves will interfere constructively or destructively when they meet at a point.

Destructive Interference

Destructive interference occurs when two waves meet and cancel each other out, resulting in a lower amplitude or complete nullification of the waves. This phenomenon happens when the waves are out of phase, specifically when their path difference is an odd multiple of half their wavelength. The formula can be expressed as:
\[ \Delta d = (m+0.5)\lambda \]
In this scenario, for the longest wavelength (and hence lowest frequency), where this cancellation occurs most efficiently at point Q, the smallest possible integer value of \( m \) is 0. This gives a longest wavelength of 240 meters, showcasing the condition needed to achieve destructive interference at the farthest possible wavelength.

Constructive Interference

Constructive interference arises when two waves align perfectly with each other, resulting in a combined wave of greater amplitude. This occurs when the waves are in phase and their path difference is a multiple of their wavelength. The formula for this is:
\[ \Delta d = m\lambda \]
In the exercise, at point Q, the path difference of 120 meters for radio waves is exactly equal to one complete wavelength (using the integer \( m = 1 \)). Thus, the longest wavelength for constructive interference in this setup is 120 meters. This means at this wavelength, both radio waves from antennas A and B integrate perfectly to produce a stronger signal at point Q.