A resistance (R) = 12 Omega, inductance (L) = 2 henry and capacitive reactance C = 5 mF are connected in series to an ac generator

Options

A. at resonance, the circuit impedance is zero.
B. at resonance, the circuit impedance is 12Omega
C. the resonance frequency of the circuit is 1//2pi.
D. at resonance, the inductive reactance is less than the capacitive reactance

Answer

To solve the problem step by step, we will analyze the given parameters and check the options provided.

### Given:
- Resistance (R) = 12 Ω
- Inductance (L) = 2 H
- Capacitance (C) = 5 mF = 5 × 10^-3 F

### Step 1: Calculate the inductive reactance (XL)
Inductive reactance (XL) is given by the formula:
\[ X_L = 2 \pi f L \]
However, we need to find the resonance frequency first, so we will come back to this after finding the resonance frequency.

### Step 2: Calculate the capacitive reactance (XC)
Capacitive reactance (XC) is given by the formula:
\[ X_C = \frac{1}{2 \pi f C} \]
Again, we will need the frequency to calculate this.

### Step 3: Find the resonance frequency (fr)
At resonance, the inductive reactance (XL) equals the capacitive reactance (XC):
\[ X_L = X_C \]
The resonance frequency is given by:
\[ f_r = \frac{1}{2 \pi \sqrt{LC}} \]
Substituting the values of L and C:
\[ f_r = \frac{1}{2 \pi \sqrt{2 \times 5 \times 10^{-3}} = \frac{1}{2 \pi \sqrt{0.01}} = \frac{1}{2 \pi \times 0.1} = \frac{1}{0.2 \pi} \approx \frac{5}{\pi} \]

### Step 4: Calculate the impedance (Z) at resonance
At resonance, the impedance (Z) is given by:
\[ Z = R \]
Thus,
\[ Z = 12 \, \Omega \]

### Step 5: Check the options
1. **Option 1:** At resonance, the circuit impedance is zero.
- This is incorrect because at resonance, the impedance is equal to the resistance (R).

2. **Option 2:** At resonance, the circuit impedance is 12 Ω.
- This is correct as we calculated Z = R = 12 Ω.

3. **Option 3:** At resonance, the frequency of the circuit is \( \frac{1}{2 \pi} \).
- This is incorrect as we calculated the resonance frequency as \( \frac{5}{\pi} \).

### Final Answers:
- Option 1: Incorrect
- Option 2: Correct
- Option 3: Incorrect