Antenna current of an AM broadcast transmitter modulated by `50%` is `11 A`. The carrier current is

Options

A. 9.25 A
B. 22 A
C. 10.3 A
D. 5.5 A

Answer

To find the carrier current \( I_C \) of an AM broadcast transmitter given the antenna current \( I_T \) and the modulation percentage, we can use the formula for the antenna current in terms of the carrier current and modulation index:

\[ I_T = I_C \sqrt{1 + \frac{M^2}{2}} \]

Where:
- \( I_T \) is the antenna current,
- \( I_C \) is the carrier current,
- \( M \) is the modulation index.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Antenna current \( I_T = 11 \, A \)
- Modulation percentage = 50%, which means \( M = \frac{50}{100} = 0.5 \)

2. **Substitute the Modulation Index:**
- Calculate \( M^2 \):
\[ M^2 = (0.5)^2 = 0.25 \]

3. **Substitute Values into the Formula:**
- Plug in the values into the formula:
\[ 11 = I_C \sqrt{1 + \frac{0.25}{2}} \]

4. **Calculate the Expression Inside the Square Root:**
- Calculate \( \frac{0.25}{2} = 0.125 \)
- Therefore:
\[ 1 + 0.125 = 1.125 \]

5. **Rewrite the Equation:**
- Now the equation becomes:
\[ 11 = I_C \sqrt{1.125} \]

6. **Solve for \( I_C \):**
- Rearranging gives:
\[ I_C = \frac{11}{\sqrt{1.125}} \]

7. **Calculate \( \sqrt{1.125} \):**
- \( \sqrt{1.125} \approx 1.06066 \)

8. **Final Calculation of \( I_C \):**
- Now calculate \( I_C \):
\[ I_C \approx \frac{11}{1.06066} \approx 10.38 \, A \]

9. **Round the Answer:**
- Rounding to one decimal place gives:
\[ I_C \approx 10.3 \, A \]

### Final Answer:
The carrier current \( I_C \) is approximately \( 10.3 \, A \).

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