Amplitude Modulation

The given amplitude modulated (AM) wave equation is: $ , we can rewrite the second term: \) and the upper and lower sideband frequencies and .
These are related to the angular frequencies by .
The carrier angular frequency is rad/s, so the carrier frequency is .
The modulating angular frequency is rad/s, so the modulating frequency is .
The upper sideband angular frequency is rad/s.
The upper sideband frequency is .
The lower sideband angular frequency is rad/s.
The lower sideband frequency is .
The ratio of the upper to the lower sideband frequencies is: \) $ The ratio of the upper to the lower sideband frequencies is .

Step 1: Modulation Index Formula The modulation index for an amplitude-modulated wave is given by: where: - V (maximum amplitude), - V (minimum amplitude). Step 2: Calculating Conclusion Thus, the correct answer is:

- Modulation index ( ) is defined as:
- Substitute the values:
- This quantifies the extent of modulation in amplitude modulated waves.

The D-layer of the ionosphere reflects low frequency electromagnetic waves during the daytime.

Frequency of the signal .
For efficient transmission and reception, the length of the antenna (L) is typically related to the wavelength of the signal.
A common minimum length for an antenna (like a quarter-wave monopole or a half-wave dipole) is or .
The question asks for "minimum length", often referring to .
First, calculate the wavelength .
The speed of electromagnetic waves (radio waves) is the speed of light .
The relation is .
For a quarter-wave monopole antenna, the minimum length is .
If it were a half-wave dipole, m.
Since 75 m is an option and often considered a practical "minimum" for efficient radiation (e.
g.
, for a Marconi antenna or a simple whip antenna grounded), this is the likely intended answer.
This matches option (3).

Frequency of the signal .
For efficient transmission and reception, the length of the antenna (L) is typically related to the wavelength of the signal.
A common minimum length for an antenna (like a quarter-wave monopole or a half-wave dipole) is or .
The question asks for "minimum length", often referring to .
First, calculate the wavelength .
The speed of electromagnetic waves (radio waves) is the speed of light .
The relation is .
For a quarter-wave monopole antenna, the minimum length is .
If it were a half-wave dipole, m.
Since 75 m is an option and often considered a practical "minimum" for efficient radiation (e.
g.
, for a Marconi antenna or a simple whip antenna grounded), this is the likely intended answer.
This matches option (3).

Step 1: Understand line of sight communication
The range depends on heights of transmitting and receiving antennas: Step 2: Given total height
Step 3: Maximize range
To maximize under constraint , use calculus or AM-GM inequality, maximum when Step 4: Conclusion
Height of receiving antenna should be for maximum range.

Step 1: Understand the relationship of wavelength with nuclear charge
For hydrogen-like atoms, the wavelength of emitted radiation is inversely proportional to the square of the atomic number ( ): Step 2: Given values
For He : ,
For H: , we need to find Step 3: Use ratio based on inverse square law
Step 4: Final Answer

Step 1: Modulation Index Formula The modulation index for an amplitude-modulated wave is given by: where: - V (maximum amplitude), - V (minimum amplitude). Step 2: Calculating Conclusion Thus, the correct answer is: