In AM wave the amplitude of each side hand frequency is

Options

A. `E_(c)`
B. `m E_(c)`
C. `(m E_(c))/(2)`
D. ` 2m E_(c)`

Answer

To solve the question regarding the amplitude of each sideband frequency in an amplitude modulated (AM) wave, we can follow these steps:

### Step 1: Understand the Components of AM Wave
An amplitude modulated wave consists of a carrier wave and a modulating signal. The carrier wave has a constant amplitude and frequency, while the modulating signal varies in amplitude.

### Step 2: Write the Equations for the Waves
The equations for the modulating signal and the carrier wave can be expressed as:
- Modulating Signal: \( A_m \sin(\omega_m t) \)
- Carrier Wave: \( A_c \sin(\omega_c t) \)

### Step 3: Combine the Waves
The resultant AM wave can be expressed as:
\[ C_m(t) = A_c \sin(\omega_c t) + A_m \sin(\omega_m t) \]
However, to find the sidebands, we need to express this in a different form.

### Step 4: Use Trigonometric Identities
Using the product-to-sum identities, we can express the combined wave:
\[ C_m(t) = A_c \sin(\omega_c t) + \frac{A_m}{2} \left( \sin((\omega_c + \omega_m)t) - \sin((\omega_c - \omega_m)t) \right) \]

### Step 5: Identify the Sidebands
From the above expression, we can see that the resulting wave consists of:
- A carrier frequency at \( \omega_c \)
- An upper sideband frequency at \( \omega_c + \omega_m \)
- A lower sideband frequency at \( \omega_c - \omega_m \)

### Step 6: Determine the Amplitude of Sidebands
The amplitude of each sideband frequency is given by:
\[ \text{Amplitude of each sideband} = \frac{A_m}{2} \]

### Step 7: Relate to Modulation Index
The modulation index \( \mu \) (or \( m \)) is defined as:
\[ \mu = \frac{A_m}{A_c} \]
Thus, the amplitude of each sideband can also be expressed in terms of the modulation index:
\[ \text{Amplitude of each sideband} = \frac{\mu A_c}{2} \]

### Conclusion
The amplitude of each sideband frequency in an AM wave is:
\[ \frac{\mu A_c}{2} \]