For an amplitude modulated wave, the maximum amplitude is found to be 2 V. The modulation index is

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(4)/(3)`
D. `(1)/(2)`

Answer

To find the modulation index for an amplitude modulated wave given the maximum amplitude, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The maximum amplitude \( A_{max} \) of an amplitude modulated wave is given by:
\[ A_{max} = A_c + A_m \]
- The minimum amplitude \( A_{min} \) is given by:
\[ A_{min} = A_c - A_m \]
- The modulation index \( \mu \) is defined as:
\[ \mu = \frac{A_m}{A_c} \]

2. **Given Information**:
- The maximum amplitude \( A_{max} = 2 \, V \).
- We need to express \( A_c \) and \( A_m \) in terms of \( A_{max} \) and \( A_{min} \).

3. **Set Up the Equations**:
- From the maximum amplitude equation:
\[ A_{max} = A_c + A_m \quad \text{(1)} \]
- From the minimum amplitude equation:
\[ A_{min} = A_c - A_m \quad \text{(2)} \]

4. **Solve the Equations**:
- Adding equations (1) and (2):
\[ A_{max} + A_{min} = (A_c + A_m) + (A_c - A_m) = 2A_c \]
\[ A_c = \frac{A_{max} + A_{min}}{2} \]

- Subtracting equation (2) from equation (1):
\[ A_{max} - A_{min} = (A_c + A_m) - (A_c - A_m) = 2A_m \]
\[ A_m = \frac{A_{max} - A_{min}}{2} \]

5. **Substituting Values**:
- Since we only have \( A_{max} = 2 \, V \) and we need \( A_{min} \), we can assume \( A_{min} = 0 \, V \) (for maximum modulation).
- Thus:
\[ A_c = \frac{2 + 0}{2} = 1 \, V \]
\[ A_m = \frac{2 - 0}{2} = 1 \, V \]

6. **Calculate the Modulation Index**:
- Now substituting \( A_m \) and \( A_c \) into the modulation index formula:
\[ \mu = \frac{A_m}{A_c} = \frac{1}{1} = 1 \]

### Final Answer:
The modulation index \( \mu \) is **1**.