The power of a AM transmitter is 100 W. If the modulation index is 0.5 and the transmission is having signal side band, the percentage of useful power is

Options

A. `1.1` W
B. 11 W
C. `2.2` W
D. 22W

Answer

To solve the problem, we need to calculate the percentage of useful power in an AM transmitter given the total power, modulation index, and the relationship between total power and carrier power.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Total Power (P_total) = 100 W
- Modulation Index (m) = 0.5

2. **Use the Formula for Total Power in AM Transmission:**
The formula for total power in an AM transmission is given by:
\[ P_{total} = P_C \left(1 + \frac{m^2}{2}\right) \]
where \( P_C \) is the carrier power.

3. **Rearranging the Formula:**
We can rearrange the formula to find the carrier power \( P_C \):
\[ P_C = \frac{P_{total}}{1 + \frac{m^2}{2}} \]

4. **Calculate \( m^2 \):**
Since \( m = 0.5 \):
\[ m^2 = (0.5)^2 = 0.25 \]

5. **Substituting Values into the Formula:**
Now substitute \( m^2 \) into the formula:
\[ P_C = \frac{100}{1 + \frac{0.25}{2}} = \frac{100}{1 + 0.125} = \frac{100}{1.125} \]

6. **Calculate \( P_C \):**
\[ P_C = \frac{100}{1.125} \approx 88.89 \text{ W} \]

7. **Calculate Useful Power:**
The useful power in AM transmission is given by:
\[ P_{useful} = P_{total} - P_C \]
Thus,
\[ P_{useful} = 100 - 88.89 \approx 11.11 \text{ W} \]

8. **Calculate Percentage of Useful Power:**
The percentage of useful power is calculated as:
\[ \text{Percentage of Useful Power} = \left(\frac{P_{useful}}{P_{total}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage of Useful Power} = \left(\frac{11.11}{100}\right) \times 100 \approx 11.11\% \]

### Final Answer:
The percentage of useful power is approximately **11.11%**.