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 \(\frac{3}{4} \mathrm{~m}\).