In an amplitude modulation, a modulating signal having amplitude of XV is superimposed with a carrier signal of amplitude YV in first case. Then, in second case the same modulating signal is superimposed with different carrier signal of amplitude 2YV. The ratio of modulation index in the two cases respectively will be :

Options

A. `2 : 1`
B. `1 : 2`
C. `1 : 1`
D. `4 : 1`

Answer

To solve the problem of finding the ratio of modulation indices in two cases of amplitude modulation, we can follow these steps:

### Step 1: Understand the Modulation Index
The modulation index (μ) in amplitude modulation is defined as the ratio of the amplitude of the modulating signal (Am) to the amplitude of the carrier signal (Ac). Mathematically, it is given by:
\[ \mu = \frac{A_m}{A_c} \]

### Step 2: Define the Parameters for Both Cases
In the first case:
- Amplitude of the modulating signal, \( A_m = X \)
- Amplitude of the carrier signal, \( A_c = Y \)

Thus, the modulation index for the first case (\( \mu_1 \)) is:
\[ \mu_1 = \frac{X}{Y} \]

In the second case:
- Amplitude of the modulating signal remains the same, \( A_m = X \)
- Amplitude of the carrier signal is now \( A_c = 2Y \)

Thus, the modulation index for the second case (\( \mu_2 \)) is:
\[ \mu_2 = \frac{X}{2Y} \]

### Step 3: Calculate the Ratio of Modulation Indices
Now, we need to find the ratio of the modulation indices \( \mu_1 \) and \( \mu_2 \):
\[ \frac{\mu_1}{\mu_2} = \frac{\frac{X}{Y}}{\frac{X}{2Y}} \]

### Step 4: Simplify the Ratio
To simplify this expression, we can multiply by the reciprocal of the denominator:
\[ \frac{\mu_1}{\mu_2} = \frac{X}{Y} \cdot \frac{2Y}{X} \]

Now, the \( X \) terms cancel out and the \( Y \) terms also cancel out:
\[ \frac{\mu_1}{\mu_2} = \frac{2Y}{Y} = 2 \]

### Step 5: Final Result
Thus, the ratio of the modulation indices in the two cases is:
\[ \frac{\mu_1}{\mu_2} = 2:1 \]

### Conclusion
The final answer is that the ratio of modulation indices in the two cases is \( 2:1 \).

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