In AM wave total power of side bands is given by :

Options

A. `P_(S ) =(E_C ^(2 ))/(4R )`
B. `P_(s )=(4E _(C )^(2 ))/(R )`
C. `P_(s ) =(m^2 E _(c )^(2))/(4R )`
D. `P_(s )=(E _(c )^(2) )/(4m ^(2 R)`

Answer

To find the total power of the sidebands in an Amplitude Modulated (AM) wave, we can follow these steps:

### Step 1: Understand the Definitions
In AM waves, the total power is distributed among the carrier and the sidebands. The power of the sidebands (P_s) is a function of the modulation index (M) and the root mean square (RMS) value of the carrier signal (E_rms).

### Step 2: Use the Formula for Power of Sidebands
The formula for the power of the sidebands is given by:
\[ P_s = M_e \cdot \frac{E_{rms}^2}{R} \]
where:
- \( M_e \) is the modulation index,
- \( E_{rms} \) is the RMS value of the carrier signal,
- \( R \) is the resistance.

### Step 3: Substitute the Values
We know that:
- The RMS value of the carrier signal can be expressed as:
\[ E_{rms} = \frac{E_c}{\sqrt{2}} \]
where \( E_c \) is the peak value of the carrier signal.
- The modulation index \( M_e \) can be expressed as:
\[ M_e = \frac{M}{\sqrt{2}} \]
where \( M \) is the modulation index.

### Step 4: Substitute into the Power Formula
Now substitute these expressions into the formula for \( P_s \):
\[ P_s = \left(\frac{M}{\sqrt{2}}\right) \cdot \frac{\left(\frac{E_c}{\sqrt{2}}\right)^2}{R} \]

### Step 5: Simplify the Expression
Now simplify the expression:
\[ P_s = \frac{M}{\sqrt{2}} \cdot \frac{E_c^2}{2R} \]
\[ P_s = \frac{M \cdot E_c^2}{4R} \]

### Step 6: Final Result
Thus, the total power of the sidebands in an AM wave is given by:
\[ P_s = \frac{M^2 \cdot E_c^2}{4R} \]

### Conclusion
The correct answer is:
\[ P_s = \frac{M^2 \cdot E_c^2}{4R} \]