The carrier freqeuncy of a station is 40 MHz. A resistor of 10k `Omega` and capacitor of 12 pF are available in the detector circuit. The possible value of C will be

Options

A. 12
B. 8.2
C. 5.6
D. All of these

Explanation

To solve the problem, we need to determine the possible values of capacitance (C) in the detector circuit based on the given parameters: a carrier frequency (Vc) of 40 MHz and a resistor (R) of 10 kΩ.

### Step-by-Step Solution:

1. **Convert the given values to standard units:**
- The carrier frequency (Vc) is given as 40 MHz. We convert this to Hertz (Hz):
\[ Vc = 40 \text{ MHz} = 40 \times 10^6 \text{ Hz} = 4 \times 10^7 \text{ Hz} \]
- The resistance (R) is given as 10 kΩ, which is:
\[ R = 10 \text{ kΩ} = 10 \times 10^3 \text{ Ω} = 10^4 \text{ Ω} \]
- The capacitance (C) is given as 12 pF, which is:
\[ C = 12 \text{ pF} = 12 \times 10^{-12} \text{ F} \]

2. **Calculate the time period (Tc) of the carrier frequency:**
- The time period (Tc) is the reciprocal of the frequency (Vc):
\[ Tc = \frac{1}{Vc} = \frac{1}{4 \times 10^7} \text{ seconds} \]
- Performing the calculation:
\[ Tc = 2.5 \times 10^{-8} \text{ seconds} \]

3. **Calculate the time constant (τ) of the RC circuit:**
- The time constant (τ) is given by the product of resistance (R) and capacitance (C):
\[ τ = R \times C \]
- Substituting the values:
\[ τ = 10^4 \text{ Ω} \times 12 \times 10^{-12} \text{ F} = 1.2 \times 10^{-7} \text{ seconds} \]

4. **Compare the time period (Tc) and time constant (τ):**
- For the circuit to be suitable for detection, the time period (Tc) must be less than the time constant (τ):
\[ Tc < τ \]
- Substituting the calculated values:
\[ 2.5 \times 10^{-8} < 1.2 \times 10^{-7} \]
- This inequality holds true, indicating that the circuit is capable of detecting the signal.

5. **Determine the possible values of capacitance (C):**
- Since the calculated τ is greater than Tc, any value of capacitance that keeps τ greater than Tc is acceptable. Thus, the possible values of C can include the given 12 pF and potentially other values as well.

6. **Conclusion:**
- The possible values of capacitance (C) in the detector circuit can be any value that satisfies the condition \( τ > Tc \). Therefore, all the options provided (if any) can be valid.