Question

What is the minimum length of antenna required to transmit a radio signal of frequency $20MHz$?
$\begin{align}
  & \text{A}\text{. 5}\text{.0m} \\
 & \text{B}\text{. 7}\text{.5m} \\
 & \text{C}\text{. 2}\text{.0m} \\
 & \text{D}\text{. 3}\text{.75m} \\
\end{align}$

Answer 1

Hint: To transmit a signal properly the height of the antenna should be comparable to the wavelength of the signal which is to be transferred (at least one-fourth in length). So the antenna can properly sense the time variation of the signal.

Formula Used:
$\text{Wavelength=}\dfrac{\text{wave velocity}}{\text{frequency}}$
Velocity of an electromagnetic wave is equal to the speed of light. So wave velocity
$v=3\times {{10}^{8}}m{{s}^{-1}}$.
Minimum antenna height is , $h=\dfrac{\text{wavelength of wave}}{4}$

Complete step-by-step answer:
For proper transfer of signal (modulation) the height of the antenna should be comparable to the wavelength of the signal which is to be transferred (at least one-fourth in length).
For a signal with frequency $20MHz$the wavelength is
$\text{Wavelength}=\dfrac{\text{wave velocity}}{\text{frequency}}=\dfrac{3\times {{10}^{8}}m{{s}^{-1}}}{20MHz}=\dfrac{3\times {{10}^{8}}}{20\times {{10}^{6}}Hz}=0.15\times {{10}^{2}}=15m$
The minimum height of the tower
$h=\dfrac{\text{wavelength of wave}}{4}=\dfrac{15m}{4}=3.75m$
So the correct option is $\text{D}\text{. 3}\text{.75m}$

Additional Information:
Modulation: Modulation is a process by which low frequency audio signals can be transmitted to large distances by carrying it with high frequency carrier waves. So by modulation, some characteristics of high frequency carrier waves like frequency, amplitude or phase angle are varied according to instantaneous low frequency audio signal to transmit the audio signal to a large distance.
Need of modulation: Message signal also called baseband signal refers to the band of frequency of original signal produced at source. Audio signals have bandwidth of \[20KHz\]. So the low frequency signal wave cannot be transferred directly to long distance.
Antenna length: To transmit a signal properly the height of the antenna should be comparable to the wavelength of the signal which is to be transferred (at least one-fourth in length). So the antenna can properly sense the time variation of the signal. Wavelength of signal is
$\text{Wavelength=}\dfrac{\text{wave velocity}}{\text{frequency}}$
As signals travels with speed of light$\left( 3\times {{10}^{8}} \right)$ also the frequency of audio signal is \[20KHz\]
So the wavelength of the signal is \[=\dfrac{3\times {{10}^{8}}\dfrac{m}{s}}{20KHz}=\dfrac{3\times {{10}^{8}}\dfrac{m}{s}}{20\times {{10}^{3}}Hz}=15000m=15km\]
So in order to transmit the audio signal the antenna length should be at least$5km$. But antennas of such height cannot be constructed or operated. If the carrier wave of frequency $1MHz$$\left( 1MHz={{10}^{6}}Hz \right)$ is used the antenna height is reduced to $200m-300m$.

Note: To transmit low frequency audio signal to a very large distance it is superimposed with high frequency carrier wave. The carrier wave should have at-least twice the highest component of the audio signal. So the high frequency signal is the signal which is actually transmitted and carries the information signal or audio signal so it is called a carrier wave.