To determine which combination should be selected for better tuning of an LCR circuit used for communication, we will calculate the quality factor (Q) for each given combination. The formula for Q is:
\[
Q = \frac{1}{R} \sqrt{\frac{L}{C}}
\]
where:
- \( R \) is the resistance,
- \( L \) is the inductance,
- \( C \) is the capacitance.
We will analyze each option one by one.
### Step 1: Calculate Q for Option A
- Given: \( R = 20 \, \Omega \), \( L = 1.5 \, H \), \( C = 35 \, \mu F = 35 \times 10^{-6} \, F \)
\[
Q_A = \frac{1}{20} \sqrt{\frac{1.5}{35 \times 10^{-6}}}
\]
Calculating the square root:
\[
Q_A = \frac{1}{20} \sqrt{\frac{1.5}{35 \times 10^{-6}}} = \frac{1}{20} \sqrt{42857.14} \approx \frac{1}{20} \times 207.08 \approx 10.35
\]
### Step 2: Calculate Q for Option B
- Given: \( R = 25 \, \Omega \), \( L = 2.5 \, H \), \( C = 45 \, \mu F = 45 \times 10^{-6} \, F \)
\[
Q_B = \frac{1}{25} \sqrt{\frac{2.5}{45 \times 10^{-6}}}
\]
Calculating the square root:
\[
Q_B = \frac{1}{25} \sqrt{\frac{2.5}{45 \times 10^{-6}}} = \frac{1}{25} \sqrt{55555.56} \approx \frac{1}{25} \times 235.7 \approx 9.43
\]
### Step 3: Calculate Q for Option C
- Given: \( R = 15 \, \Omega \), \( L = 3.5 \, H \), \( C = 30 \, \mu F = 30 \times 10^{-6} \, F \)
\[
Q_C = \frac{1}{15} \sqrt{\frac{3.5}{30 \times 10^{-6}}}
\]
Calculating the square root:
\[
Q_C = \frac{1}{15} \sqrt{\frac{3.5}{30 \times 10^{-6}}} = \frac{1}{15} \sqrt{116666.67} \approx \frac{1}{15} \times 341.74 \approx 22.77
\]
### Step 4: Calculate Q for Option D
- Given: \( R = 25 \, \Omega \), \( L = 3.5 \, H \), \( C = 45 \, \mu F = 45 \times 10^{-6} \, F \)
\[
Q_D = \frac{1}{25} \sqrt{\frac{3.5}{45 \times 10^{-6}}}
\]
Calculating the square root:
\[
Q_D = \frac{1}{25} \sqrt{\frac{3.5}{45 \times 10^{-6}}} = \frac{1}{25} \sqrt{77777.78} \approx \frac{1}{25} \times 279.51 \approx 11.18
\]
### Conclusion
Now we compare the values of Q:
- \( Q_A \approx 10.35 \)
- \( Q_B \approx 9.43 \)
- \( Q_C \approx 22.77 \)
- \( Q_D \approx 11.18 \)
The maximum value of Q is for Option C, which is approximately 22.77. Therefore, the combination that should be selected for better tuning of the LCR circuit used for communication is **Option C**.
