Question

To transmit a signal of 3kHz frequency, the minimum length of antenna is ____ $Km$.
(A) \[25\,\,Km\]
(B) \[20\,\,Km\]
(C) \[50\,\,Km\]
(D) \[75\,\,Km\]

Answer 1

Hint: Wavelength is the interval in between similar points (that is adjoining peak) in the adjoining cycles of a waveform signal transmitted in space or through a wire. Wavelength is to the contrary related to frequency, which denotes the number of wave cycles per second.

Formula used:
The formula for the wavelength of radio signals to be transmitted or received is given by;
$l = \dfrac{\lambda }{4}$
Where, $l$ denotes the length of the transmitter, $\lambda $ denotes the wavelength of radio signals that are to be transmitted or received.

Complete step by step solution
The data given in the problem is;
The velocity of the signal is given as, $v = 3\,\,KHz$
That is taken as, $v = 3 \times {10^3}\,\,Hz$
The Key Concept Marconi antenna is a straight conductor of length equal to a quarter of the wavelength of radio signals to be transmitted or received. It depends on the frequency of the carrier wave and the direction of the beam.

Now we can find the wavelength of radio signals to be transmitted or received using the formula of speed of light.
That is;
$c = v\lambda $
Where, $c$ denotes the speed of light, $v$ denotes the velocity of the transmitted signal, $\lambda $ denotes the wavelength of radio signals that are to be transmitted or received. We know that the value for the speed of light is, $c = 3 \times {10^8}\,\,m\,{s^{ - 1}}$.
Substitute the values of speed of light and the velocity of the transmitted signal in the above formula.
$c = v\lambda $
$
  3 \times {10^8} = 3 \times {10^3} \times \lambda \\
  \lambda = \dfrac{{3 \times {{10}^8}}}{{3 \times {{10}^3}}} \\
 $
Cancel the like terms and simplify;
$\lambda = \dfrac{{{{10}^8}}}{{{{10}^3}}}$
Bring the denominator value to the numerator;
$
  \lambda = {10^8} \times {10^{ - 3}} \\
  \lambda = {10^5} \\
 $
For the minimum length of the antenna the formula for the wavelength of radio signals to be transmitted or received is given by;
 $l = \dfrac{\lambda }{4}$
Substitute the value of the wavelength of radio signals that is to be transmitted or received in the above formula;
\[
  l = \dfrac{{{{10}^5}}}{4} \\
  l = \dfrac{{100000}}{4} \\
  l = 25000\,\,m \\
  l = 25\,\,Km \\
 \]
Therefore, the minimum length of antenna to transmit a signal of $3\,\,KHz$ frequency is \[l = 25\,\,Km\]

Hence the option (A) \[l = 25\,\,Km\] is the correct answer.

Note: The greater the frequency of the signal, the lesser size of the wavelength. Instruments such as optical spectrometers or optical spectrum analysers can be used to detect wavelengths in the electromagnetic spectrum.