Problem 16 - Applied Physics

Question

An AM radio station broadcasts a signal with a wavelength of \(237 \mathrm{~m}\). Find its frequency in \(\mathrm{kHz}\).

Step-by-Step Solution

Answer

The frequency is approximately 1266 kHz.

Step 1: Understand the Relationship

The formula to relate the speed of light (c), frequency (f), and wavelength (λ) is given by \( c = f \cdot \lambda \), where \( c = 3 \times 10^8 \) meters per second. We will use this to find the frequency.

Step 2: Rearrange the Formula

From the equation \( c = f \cdot \lambda \), solve for frequency \( f \) by rearranging it to \( f = \frac{c}{\lambda} \).

Step 3: Substitute Known Values

Substitute \( c = 3 \times 10^8 \) meters per second and \( \lambda = 237 \) meters into the rearranged formula: \( f = \frac{3 \times 10^8 \, \text{m/s}}{237 \, \text{m}} \).

Step 4: Calculate the Frequency

Perform the division: \( f = \frac{3 \times 10^8}{237} \approx 1.266 \times 10^6 \) Hz.

Step 5: Convert Hertz to Kilohertz

Since 1 kilohertz (kHz) is 1000 hertz (Hz), convert the frequency to kilohertz: \( f = \frac{1.266 \times 10^6}{1000} \).

Step 6: Finalize the Frequency in Kilohertz

Complete the conversion to find \( f \approx 1266 \) kHz.

Key Concepts

Wavelength and Frequency Relationship

Understanding the connection between wavelength and frequency is crucial in radio broadcasting. Wavelength refers to the distance between two consecutive points of a wave, like crest to crest or trough to trough. Frequency, on the other hand, is the number of wave cycles that pass a given point in one second, measured in hertz (Hz).
These two properties are inversely related. When the wavelength is longer, the frequency is lower, and vice versa. This is because the speed of a wave is constant, so if the distance (wavelength) between waves increases, fewer waves (frequency) can pass a point in the same timeframe.

• The formula that connects them is: \( c = f \cdot \lambda \)
• Where \( c \) is the speed of light, \( f \) is the frequency, and \( \lambda \) is the wavelength.

The speed of light \( c \), in a vacuum, is approximately \( 3 \times 10^8 \) meters per second, a fundamental constant used in these calculations.

Speed of Light

The speed of light is a universal constant, denoted by the symbol \( c \). It represents the speed at which electromagnetic waves, including radio waves, travel in a vacuum.
This speed is roughly \( 3 \times 10^8 \) meters per second or about 300,000 kilometers per second. It is an essential part of understanding both frequency and wavelength, especially in radio wave applications.

• Because the speed is constant, changes in wavelength directly affect frequency.
• The speed of light is used in the standard equation for calculating frequency: \( f = \frac{c}{\lambda} \).

In mediums other than a vacuum, the speed of light can be slower, but for simplicity and most calculations related to radio waves, the constant speed is used.

Frequency Conversion

Converting frequencies from hertz (Hz) to kilohertz (kHz) is often necessary in radio communication to simplify numbers and make them more understandable.
Since 1 kHz is equivalent to 1,000 Hz, this conversion frequently appears in scenarios like radio broadcasting, where precise tuning to specific frequencies is required.To convert from hertz to kilohertz:

• Divide the frequency in hertz by 1,000.
• For example, a frequency calculated as \( 1.266 \times 10^6 \) Hz would become \( 1266 \) kHz.

This straightforward conversion aids both broadcasters and listeners in easily identifying specific stations and frequencies.

AM Radio Broadcasting

AM (Amplitude Modulation) radio broadcasting is a method of transmitting audio signals by modulating the amplitude of radio waves.
This is one of the oldest radio broadcast technologies, still widely used today for various stations, particularly for talk shows and news due to its long-range capabilities. Key characteristics of AM radio include:

• Broadcasting frequencies typically range from 530 to 1710 kHz.
• Uses long wavelengths, allowing signals to travel over long distances, especially at night.
• Best suited for broadcasting human voice or simpler music due to its larger reach and reliability, even though it has a lower sound quality compared to FM radio.

Learning about these characteristics helps in understanding why AM radio stations operate on certain frequencies and how they reach their audience effectively.