Question

If a message signal of frequency '\[{f_m}\]' is amplitude modulated with a carrier signal of frequency '\[{f_c}\]' and radiated through an antenna, the wavelength of the corresponding signal in air is:
[Given, C is the speed of electromagnetic waves in vacuum/air]
A. \[C/{f_c} + {f_m}\]
B. \[C/{f_c} - {f_m}\]
C. \[C/{f_m}\]
D. \[C/{f_c}\]

Answer 1

Hint: In this question, we need to find the wavelength of an amplitude modulated signal whose message frequency is \[{f_m}\] and the frequency of the carrier signal is \[{f_c}\]. For this, we have to use the equation of amplitude modulated wave which is given below.

Formula used:
The equation of amplitude modulated wave is given by \[y{\text{ }} = {\text{ }}({A_c}\; + {\text{ }}{A_m}\;sin{\text{ }}{\omega _m}t){\text{ }}sin{\text{ }}{\omega _c}t\]
Here, \[{A_c}\] is the amplitude of the carrier signal, \[{A_m}\] is the amplitude of the message signal, \[{\omega _m}\] is the angular frequency of the message signal, \[{\omega _c}\] is the angular frequency of the carrier signal and \[t\] is the time period.
Also, the wavelength of a signal is given by
\[\lambda = \dfrac{v}{f}\]
Here, \[\lambda \] is the wavelength, \[v\] is the speed and \[f\] is the frequency.

Complete step by step solution:
We know that the equation of the amplitude modulated wave is given by \[y{\text{ }} = {\text{ }}({A_c}\; + {\text{ }}{A_m}\;sin{\text{ }}{\omega _m}t){\text{ }}sin{\text{ }}{\omega _c}t\]. …. (1)
Now, we will find the wavelength of the respective signal.
But \[\lambda = \dfrac{v}{f}\]…. (2)
Here, \[v = C\]
From equation (1), we can say that
\[f = {f_c}\]
From equation (2), we get
\[\lambda = \dfrac{C}{{{f_c}}}\]
That means the wavelength of an amplitude-modulated signal is \[C/{f_c}\].

Therefore, the correct option is (D).

Additional Information: Amplitude modulation is a method of transmitting a wave signal by modulating its amplitude. It is also known as AM. Amplitude modulation is commonly used in telecommunications.

Note: Here, students generally make mistakes in writing the equation of amplitude-modulated wave. If so, then the wavelength of a signal will be wrong. Also, it is necessary to mention the correct formula for the wavelength of a signal.