A wave of frequency of 10 kHz and amplitude 20 volts is being used to modulate amplitude of carrier wave of frequency 1000 kHz and peak voltage 40 volts. Choose correct option regarding carrier modulated wave.

Options

A. Equation of carrier modulated wave is `40{1+0.25sin20pi xx10^(3)t}sin(2pi xx10^(6)t)` and modulation index is 0.2
B. Equation of carrier modulated wave is `40(1+0.5sin20pixx10^(5)t}sin(2pixx10^(4)t)` and modulation index is 0.5
C. Equation of carried modulated wave is `40{1+0.5sin20pixx10^(3)t}sin(2pi xx10^(6)t)` and modulation index is 2
D. Equation of carrier modulated wave is `40{1+0.5sin20pixx10^(3)t)sin(2pixx10^(6)t)` and depth of modulation is 0.5

Explanation

To solve the problem, we need to find the modulation index and the equation of the carrier modulated wave. Let's break it down step by step.

### Step 1: Identify the given values
- Frequency of the modulating wave (\(f_m\)): 10 kHz = \(10 \times 10^3\) Hz
- Amplitude of the modulating wave (\(A_m\)): 20 volts
- Frequency of the carrier wave (\(f_c\)): 1000 kHz = \(1000 \times 10^3\) Hz = \(10^6\) Hz
- Amplitude of the carrier wave (\(A_c\)): 40 volts

### Step 2: Calculate the modulation index (\( \mu \))
The modulation index is given by the formula:
\[ \mu = \frac{A_m}{A_c} \]
Substituting the values:
\[ \mu = \frac{20 \text{ volts}}{40 \text{ volts}} = 0.5 \]

### Step 3: Write the equation of the carrier modulated wave
The equation for the amplitude modulated wave can be expressed as:
\[ s(t) = A_c \left(1 + \mu \sin(\omega_m t)\right) \sin(\omega_c t) \]
Where:
- \( \omega_m = 2\pi f_m \)
- \( \omega_c = 2\pi f_c \)

Calculating \( \omega_m \) and \( \omega_c \):
\[ \omega_m = 2\pi \times 10 \times 10^3 = 20\pi \times 10^3 \text{ rad/s} \]
\[ \omega_c = 2\pi \times 10^6 = 2\pi \times 10^6 \text{ rad/s} \]

### Step 4: Substitute the values into the equation
Substituting \(A_c\), \(\mu\), \(\omega_m\), and \(\omega_c\) into the equation:
\[ s(t) = 40 \left(1 + 0.5 \sin(20\pi \times 10^3 t)\right) \sin(2\pi \times 10^6 t) \]

### Step 5: Simplify the equation
This can be rewritten as:
\[ s(t) = 40 \left(1 + 0.5 \sin(20\pi \times 10^3 t)\right) \sin(2\pi \times 10^6 t) \]
This shows the carrier wave modulated by the signal wave.

### Conclusion
The modulation index is \(0.5\) and the equation of the modulated wave is:
\[ s(t) = 40 \left(1 + 0.5 \sin(20\pi \times 10^3 t)\right) \sin(2\pi \times 10^6 t) \]