Problem 47 - Physics

Question

The transmitting antenna for a radio station is \(7.00 \mathrm{~km}\) from your house. The frequency of the electromagnetic wave broad cast by this station is \(536 \mathrm{kHz}\). The station builds a second transmitting antenna that broadcasts an identical electromagnetic wave in phase with the original one. The new antenna is \(8.12 \mathrm{~km}\) from your house. Does constructive or destructive interference occur at the receiving antenna of your radio? Show your calculations.

Step-by-Step Solution

Answer

Constructive interference occurs at your house.

Step 1: Identify the Known Values

We have two antennas: the first is located 7.00 km from your house, and the second is 8.12 km away. The frequency of the electromagnetic waves from both antennas is 536 kHz. Our task is to find out if the waves interfere constructively or destructively at your house.

Step 2: Calculate the Wavelength

The speed of light, which is the speed of electromagnetic waves, is approximately \(c = 3 \times 10^8 \, \text{m/s}\). The wavelength can be calculated using the formula:\[ \lambda = \frac{c}{f} \]Substitute the given frequency \(f = 536 \times 10^3 \, \text{Hz}\).\[ \lambda = \frac{3 \times 10^8}{536 \times 10^3} \approx 559 \text{ m} \]

Step 3: Calculate the Path Difference

The path difference \( \Delta d \) is the difference in distance from your house to each antenna:\[ \Delta d = d_2 - d_1 = 8.12 \, \text{km} - 7.00 \, \text{km} = 1.12 \, \text{km} = 1120 \, \text{m} \]

Step 4: Determine Interference Type

Interference is constructive if the path difference \( \Delta d \) is an integer multiple of the wavelength, i.e., \( n\lambda\) where \( n \) is an integer (0, 1, 2,...). It's destructive if it is a half-integer multiple, i.e., \( (n + 0.5)\lambda\).Let's check:\[ \frac{\Delta d}{\lambda} = \frac{1120}{559} \approx 2.0036 \]Since this is approximately 2, which is an integer, constructive interference occurs.

Key Concepts

Electromagnetic Waves

Electromagnetic waves are waves that can travel through the vacuum of outer space. They consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation. These fields are responsible for carrying energy and information over large distances.

For instance, when a radio station broadcasts a signal, it emits electromagnetic waves that travel through the air until they reach your radio. These waves include a broad range of frequencies, but in the context of radio, they usually fall within certain low-frequency ranges. Knowing how to work with these waves involves understanding their speed, which is the speed of light, approximately 3 imes 10^8 ext{ m/s}. This constant speed is crucial for calculating wavelengths and predicting how these waves will interact with each other.

This property allows electromagnetic waves to carry information swiftly, as in the exercise where the frequency determines how the wavelengths are calculated and contributes to understanding the interference patterns at play.

Constructive Interference

Constructive interference occurs when two or more waves overlap in such a way that their amplitudes add together, resulting in a wave of greater amplitude. This phenomenon happens under specific conditions: the waves must be in phase, meaning that their crests and troughs align.

In the context of radio waves from the two antennas, constructive interference happens when the path difference between the waves from these antennas is an integer multiple of their wavelength. For example, if the path difference is an exact number of wavelengths, the waves will reinforce each other, leading to a stronger reception signal.

Understanding constructive interference is key when setting up multiple antennas to enhance the strength and quality of the received signal. When planned carefully, antennas can maximize the beneficial overlap of electromagnetic waves, boosting clarity and reducing signal loss during transmission.

Destructive Interference

Destructive interference happens when waves combine to reduce or cancel each other's amplitudes. This type of interference occurs when the waves are out of phase; one wave's crest aligns with the other's trough.

For radio signals, when two waves meet and their path difference is a half-integer multiple of the wavelength (like (n + 0.5) ), the waves tend to nullify each other. This can result in poor signal quality or even complete signal cancellation.

Understanding and avoiding destructive interference is critical when setting up broadcasting equipment. It ensures that signals remain strong and clear by preventing conditions that would lead to signal degradation at the receiving end. Awareness of interference conditions can help in designing systems that minimize overlap, thus preserving signal integrity.

Wavelength Calculation

Calculating wavelength is essential in understanding wave behavior, particularly in predicting interference outcomes. The wavelength (\lambda) of a wave is determined using the formula:\[ \lambda = \frac{c}{f} \] where \(c\) represents the speed of light and \(f\) is the frequency.

In the given exercise, the frequency of 536 kHz leads to a specific wavelength. With the speed of light being 3 imes 10^8 ext{ m/s}, and substituting in the frequency, we find \(\lambda \approx 559 \text{ m}\). This value is crucial because it influences how the waves will interfere after traveling different distances.

Precision in calculating the wavelength helps in determining not just interference, but also the range and reach of the electromagnetic waves. Whether designing broadcasting systems or analyzing wave interactions, wavelength calculation remains a foundational tool.