In AM wave, carrier power is given by :

Options

A. `P_(c)=(2E)_(C)^(2)/(R)`
B. `P_(c)=(E)_(C)^(2)/(R)`
C. `P_(c)=(E)_(C)^(2)/(2R)`
D. `P_(c)=(E)_(C)^(2)/sqrt (2(R))`

Answer

To find the carrier power (PC) in an Amplitude Modulated (AM) wave, we can follow these steps:

### Step 1: Understand the formula for carrier power
The carrier power (PC) in an AM wave is given by the formula:
\[ P_C = \frac{E_{RMS}^2}{R} \]
where:
- \( E_{RMS} \) is the root mean square (RMS) value of the carrier signal.
- \( R \) is the resistance across which the power is measured.

### Step 2: Relate RMS value to peak value
The RMS value of the carrier signal can be related to its peak value (EC) using the formula:
\[ E_{RMS} = \frac{E_C}{\sqrt{2}} \]
where:
- \( E_C \) is the peak value of the carrier signal.

### Step 3: Substitute the RMS value into the power formula
Substituting the expression for \( E_{RMS} \) into the carrier power formula, we get:
\[ P_C = \frac{\left(\frac{E_C}{\sqrt{2}}\right)^2}{R} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ P_C = \frac{E_C^2}{2R} \]

### Step 5: Final expression for carrier power
Thus, the final expression for the carrier power in an AM wave is:
\[ P_C = \frac{E_C^2}{2R} \]

### Step 6: Identify the correct option
Now, we can check which option matches this expression. The correct answer is:
\[ P_C = \frac{E_C^2}{2R} \]
This corresponds to option C.

### Summary
The carrier power in an AM wave is given by:
\[ P_C = \frac{E_C^2}{2R} \]

---