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 regarding the modulation of a carrier wave, we will follow these steps:

### Step 1: Identify the given parameters
- Frequency of the message signal (fm) = 10 kHz = \(10 \times 10^3\) Hz
- Amplitude of the message signal (Am) = 20 volts
- Frequency of the carrier signal (fc) = 1000 kHz = \(1000 \times 10^3\) Hz = \(10^6\) Hz
- Amplitude of the carrier signal (Ac) = 40 volts

### Step 2: Calculate the Modulating Index (μ)
The modulating index (μ) is defined as the ratio of the amplitude of the message signal to the amplitude of the carrier signal:
\[ \mu = \frac{A_m}{A_c} = \frac{20 \text{ V}}{40 \text{ V}} = \frac{1}{2} = 0.5 \]

### Step 3: Write the equations for the message and carrier signals
The equation for the message signal can be expressed as:
\[ m(t) = A_m \sin(2\pi f_m t) = 20 \sin(2\pi (10 \times 10^3) t) \]
The equation for the carrier signal is:
\[ c(t) = A_c \sin(2\pi f_c t) = 40 \sin(2\pi (10^6) t) \]

### Step 4: Write the equation for the modulated signal
The equation for the amplitude-modulated wave (cmt) is given by:
\[ c_m(t) = A_c \left(1 + \mu \sin(2\pi f_m t)\right) \sin(2\pi f_c t) \]
Substituting the values we have:
\[ c_m(t) = 40 \left(1 + 0.5 \sin(2\pi (10 \times 10^3) t)\right) \sin(2\pi (10^6) t) \]

### Step 5: Simplify the modulated signal equation
This can be expanded to:
\[ c_m(t) = 40 \sin(2\pi (10^6) t) + 20 \sin(2\pi (10 \times 10^3) t) \sin(2\pi (10^6) t) \]

### Step 6: Identify the characteristics of the modulated wave
The modulated wave will have:
- A carrier frequency of \(1000 kHz\) (or \(10^6 Hz\))
- A modulation frequency of \(10 kHz\) (or \(10^4 Hz\))
- A peak amplitude that varies between \(20 V\) and \(60 V\) (from the modulation).

### Conclusion
The correct option regarding the carrier modulated wave is that it has a modulating index of \(0.5\) and the frequency components of the modulated signal will include both the carrier frequency and the modulation frequency.