The LC product of a tuned amplifier circuit require to generate a carrier wave of 1 MHz for amplitude modulation is

Options

A. `1.5xx10^(-14)s`
B. `1.2xx10^(-12)s`
C. `3.2xx10^(-12)s`
D. `2.5xx10^(-14) s`

Explanation

To solve the problem of finding the LC product of a tuned amplifier circuit required to generate a carrier wave of 1 MHz for amplitude modulation, we can follow these steps:

### Step 1: Understand the formula
The frequency \( f \) of a tuned amplifier circuit is given by the formula:
\[ f = \frac{1}{2\pi \sqrt{LC}} \]
where \( L \) is the inductance and \( C \) is the capacitance.

### Step 2: Rearrange the formula
We need to find the product \( LC \). We can rearrange the formula to express \( LC \):
\[ LC = \frac{1}{(2\pi f)^2} \]

### Step 3: Substitute the frequency
Given that the frequency \( f \) is 1 MHz, we convert this to Hz:
\[ f = 1 \text{ MHz} = 10^6 \text{ Hz} \]

### Step 4: Calculate \( LC \)
Now we substitute \( f \) into the rearranged formula:
\[ LC = \frac{1}{(2\pi \times 10^6)^2} \]

### Step 5: Calculate the numerical value
Calculating \( 2\pi \):
\[ 2\pi \approx 6.2832 \]
Now, calculate \( (2\pi \times 10^6)^2 \):
\[ (6.2832 \times 10^6)^2 \approx 39.4784 \times 10^{12} \approx 3.94784 \times 10^{13} \]
Now, take the reciprocal:
\[ LC \approx \frac{1}{3.94784 \times 10^{13}} \approx 2.53 \times 10^{-14} \]

### Step 6: Final answer
Thus, the LC product required to generate a carrier wave of 1 MHz for amplitude modulation is approximately:
\[ LC \approx 2.5 \times 10^{-14} \text{ seconds} \]

### Conclusion
The correct option is \( 2.5 \times 10^{-14} \) seconds.

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