Question

For efficient transmission of a \(100 \mathrm{MHz}\) frequency wave, the minimum length of an antenna should be (A) \(3 \mathrm{~m}\) (B) \((3 / 4) \mathrm{m}\) (C) \(10 \mathrm{~m}\) (D) \(100 \mathrm{~m}\)

Short answer

The minimum length of the antenna for efficient transmission of a 100 MHz frequency wave is approximately 0.75 meters. The correct option is (B) 0.75 meters or \(\frac{3}{4} \mathrm{~m}\).

Step-by-step solution

Write down the given information

The frequency of the radio wave is given as \(100MHz\). We need to find the minimum length of an antenna (L) for efficient transmission.

Convert frequency to wavelength

To find the wavelength (λ), we need to use the formula for the speed of light (c), which is \[c = λf\], where c is the speed of light in a vacuum, approximately 299,792,458 m/s, λ is the wavelength, and f is the frequency. Given, frequency, \(f = 100MHz = 100 \times 10^{6} Hz\). Let's find the wavelength by rearranging the formula to solve for λ. \[λ = \frac{c}{f}\] Now, insert the given values into the formula: \[λ = \frac{299,792,458}{100 \times 10^{6}}\]

Calculate the wavelength

Now, calculate the wavelength: \[λ = \frac{299,792,458}{100 \times 10^{6}} = 2.99792458 \mathrm{~m}\] The wavelength of the radio wave is approximately 3 meters.

Determine the minimum antenna length

The minimum length of an antenna for efficient transmission is usually one-fourth of the wavelength (λ). So the minimum antenna length will be: \[L = \frac{λ}{4}\] Now, insert the calculated wavelength into this formula: \[L = \frac{2.99792458}{4}\]

Calculate the minimum antenna length

Calculate the minimum antenna length: \[L = \frac{2.99792458}{4} = 0.749481145 \mathrm{~m} \approx 0.75 \mathrm{~m}\] The minimum length of the antenna for efficient transmission of a 100 MHz frequency wave is approximately 0.75 meters. Comparing with the given options, the correct option is (B) 0.75 meters or \(\frac{3}{4} \mathrm{~m}\).

Key concepts

Radio Waves

Radio waves are a type of electromagnetic radiation with wavelengths in the electromagnetic spectrum longer than infrared light. These waves facilitate many forms of communication, such as broadcasting sounds for radios, gaming systems, and televisions. They are emitted by both natural sources like stars and artificial sources like radio transmitters.

One significant characteristic of radio waves is that they can create and receive signals over long distances without needing a medium. The information these waves carry, such as audio or digital data, is encoded within their frequency. This capability is what makes radio waves incredibly useful for many everyday technologies.

Wavelength Calculation

To understand wavelength, envision the distance between consecutive peaks of a wave. Wavelength is closely related to both the frequency and speed of light, and can be calculated using the formula:

• \[ \lambda = \frac{c}{f} \]

where \( \lambda \) is the wavelength, \( c \) is the speed of light, and \( f \) is the frequency of the wave.

For instance, if you have a radio wave frequency of \(100\) MHz, you can rearrange the formula to calculate its wavelength.

• Substituting the speed of light \(c = 299,792,458 \) m/s and frequency \(f = 100 \times 10^{6} \) Hz, you get \( \lambda = 3 \) meters.

This simple formula succinctly links frequency and wavelength, helping you find the distance a wave travels during one cycle.

Speed of Light

The speed of light is essential in wave calculations, particularly in converting frequency to wavelength. It is a fundamental constant of physics, symbolized by \( c \), with a value of approximately \(299,792,458\) meters per second (m/s).

This universal speed limit affects how we perceive and measure waves of all kinds, including radio waves. Because light travels so quickly, it allows radio waves to transmit data nearly instantaneously across vast distances.

The speed of light not only underpins the precise calculations of wave properties but also enables the synchronization necessary for efficient communication technologies. Accurate knowledge of this constant simplifies the conversion between frequency and wavelength, which is vital for designing antennas and other transmission equipment.

Frequency

Frequency defines the number of wave cycles that pass a point per unit time. It is measured in Hertz (Hz). In the context of radio waves, frequency determines the radio station or channel you tune into. Higher frequencies mean more cycles per second and consequently shorter wavelengths.

For radio communication, different frequencies are reserved for specific types of communication, providing structure and minimizing interference. A frequency of \(100 \) MHz, for example, translates to \(100\) million cycles per second.

Understanding frequency is crucial for engineers and technicians, especially when designing antennas. The length of an antenna is often based on fractions of the wavelength derived from the frequency, optimizing the antenna's ability to efficiently transmit or receive specific frequencies. Knowing how frequency interacts with wavelength and speed can lead to more effective communication systems.