In an amplitude modulator circuit, the carrier wave is given by,
`C(t) =4 sin (20000 pi t)` while modulating signal is given by, `m(t) =2 sin (2000 pi t)`. The values of modulation index and lower side band frequency are :

Options

A. `0.5` and `10 kHz`
B. `0.4` and `10 kHz`
C. `0.3` and `9 kHz`
D. `0.5` and `9 kHz`

Answer

To solve the problem, we need to find the modulation index and the lower sideband frequency for the given amplitude modulator circuit.

### Step-by-step Solution:

1. **Identify the Carrier and Modulating Signals:**
- The carrier wave is given by:
\[ C(t) = 4 \sin(20000 \pi t) \]
- The modulating signal is given by:
\[ m(t) = 2 \sin(2000 \pi t) \]

2. **Extract Parameters from the Carrier Wave:**
- The general form of the carrier wave can be expressed as:
\[ C(t) = A_c \sin(\omega_c t) \]
- From the given carrier wave, we can identify:
- Amplitude of the carrier wave, \( A_c = 4 \)
- Angular frequency of the carrier wave, \( \omega_c = 20000 \pi \)

3. **Calculate the Carrier Frequency:**
- The relationship between angular frequency and frequency is given by:
\[ \omega = 2 \pi f \]
- Therefore, we can find the carrier frequency \( f_c \) as follows:
\[ \omega_c = 20000 \pi \implies f_c = \frac{\omega_c}{2 \pi} = \frac{20000 \pi}{2 \pi} = 10000 \text{ Hz} = 10 \text{ kHz} \]

4. **Extract Parameters from the Modulating Signal:**
- The general form of the modulating signal can be expressed as:
\[ m(t) = A_m \sin(\omega_m t) \]
- From the given modulating signal, we can identify:
- Amplitude of the modulating signal, \( A_m = 2 \)
- Angular frequency of the modulating signal, \( \omega_m = 2000 \pi \)

5. **Calculate the Modulating Frequency:**
- Using the same relationship as before, we find the modulating frequency \( f_m \):
\[ \omega_m = 2000 \pi \implies f_m = \frac{\omega_m}{2 \pi} = \frac{2000 \pi}{2 \pi} = 1000 \text{ Hz} = 1 \text{ kHz} \]

6. **Calculate the Modulation Index:**
- The modulation index \( \mu \) is defined as the ratio of the amplitude of the modulating signal to the amplitude of the carrier signal:
\[ \mu = \frac{A_m}{A_c} = \frac{2}{4} = 0.5 \]

7. **Calculate the Lower Sideband Frequency:**
- The lower sideband frequency (LSB) is calculated using the formula:
\[ \text{LSB} = f_c - f_m \]
- Substituting the values we found:
\[ \text{LSB} = 10 \text{ kHz} - 1 \text{ kHz} = 9 \text{ kHz} \]

### Final Results:
- Modulation Index \( \mu = 0.5 \)
- Lower Sideband Frequency \( \text{LSB} = 9 \text{ kHz} \)