A carrier frequency of 1 MHz and peak value of 10 V is amplitude modulated with a signal frequency of 10 KHz with peak value of 0.5 V. What are the values of the modulation index and the side band frequencies ?

Options

A. 0.05 and 1 `pm` 0.010 MHz
B. 0.5 and 1 `pm` 0.010 MHz
C. 0.05 and 1 `pm` 0.05 MHz
D. 0.5 and 1 `pm` 0.005 MHz

Explanation

To solve the problem, we need to determine the modulation index and the sideband frequencies based on the given parameters of the carrier and modulating signals.

### Step 1: Identify the given values
- Carrier frequency (\(f_c\)) = 1 MHz
- Amplitude of carrier wave (\(A_c\)) = 10 V
- Modulating signal frequency (\(f_m\)) = 10 kHz = 0.01 MHz
- Amplitude of modulating wave (\(A_m\)) = 0.5 V

### Step 2: Calculate the modulation index
The modulation index (\(M\)) is calculated using the formula:
\[ M = \frac{A_m}{A_c} \]
Substituting the values:
\[ M = \frac{0.5 \, \text{V}}{10 \, \text{V}} = 0.05 \]

### Step 3: Calculate the sideband frequencies
The sideband frequencies consist of the upper sideband (USB) and lower sideband (LSB). They can be calculated as follows:
- Lower Sideband (LSB) frequency:
\[ f_{LSB} = f_c - f_m \]
Substituting the values:
\[ f_{LSB} = 1 \, \text{MHz} - 0.01 \, \text{MHz} = 0.99 \, \text{MHz} \]

- Upper Sideband (USB) frequency:
\[ f_{USB} = f_c + f_m \]
Substituting the values:
\[ f_{USB} = 1 \, \text{MHz} + 0.01 \, \text{MHz} = 1.01 \, \text{MHz} \]

### Step 4: Present the results
- Modulation index (\(M\)) = 0.05
- Sideband frequencies: 0.99 MHz (LSB) and 1.01 MHz (USB)

### Final Answer
- Modulation Index: 0.05
- Sideband Frequencies: 0.99 MHz and 1.01 MHz

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