Compute `LC` product of a tuned amplifer circuit required to generate a carrier `wave` of `1 MHz` for amplitude modulation

Options

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

Explanation

To compute 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 Resonance Frequency Formula
The resonance frequency \( f \) of an LC circuit is given by the formula:
\[ f = \frac{1}{2\pi\sqrt{LC}} \]
where:
- \( f \) is the frequency in hertz (Hz),
- \( L \) is the inductance in henries (H),
- \( C \) is the capacitance in farads (F).

### Step 2: Convert Frequency to Standard Units
The frequency given is \( 1 \text{ MHz} \). We need to convert this to hertz:
\[ 1 \text{ MHz} = 1 \times 10^6 \text{ Hz} \]

### Step 3: Rearrange the Formula to Find LC Product
We need to find the product \( LC \). Rearranging the resonance frequency formula gives:
\[ LC = \frac{1}{(2\pi f)^2} \]

### Step 4: Substitute the Frequency Value
Now, substitute \( f = 1 \times 10^6 \text{ Hz} \) into the formula:
\[ LC = \frac{1}{(2\pi \times 10^6)^2} \]

### Step 5: Calculate the Value
Calculating \( (2\pi \times 10^6)^2 \):
\[ 2\pi \approx 6.2832 \quad \text{so} \quad 2\pi \times 10^6 \approx 6.2832 \times 10^6 \]
Now squaring this value:
\[ (6.2832 \times 10^6)^2 \approx 39.4784 \times 10^{12} \]
Thus,
\[ LC = \frac{1}{39.4784 \times 10^{12}} \approx 2.53 \times 10^{-14} \text{ seconds} \]

### Step 6: Final Result
The product \( LC \) is approximately:
\[ LC \approx 2.5 \times 10^{-14} \text{ seconds} \]

### Conclusion
The `LC` product of the tuned amplifier circuit required to generate a carrier wave of `1 MHz` for amplitude modulation is:
\[ LC \approx 2.5 \times 10^{-14} \text{ seconds} \]

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