If resistance R =100 Omega, inductance L = 12mH and capacitance C=15muF are connected in series to an AC sources, then at resonance the impedance of circuit is

Options

A. zero
B. 100Omega
C. 1000Omega
D. 10kOmega

Answer

To solve the problem of finding the impedance of a series RLC circuit at resonance, we can follow these steps:

### Step 1: Understand the concept of resonance in an RLC circuit
At resonance in a series RLC circuit, the inductive reactance (XL) is equal to the capacitive reactance (XC). This means that the reactive components cancel each other out, and the total impedance (Z) of the circuit is purely resistive and equal to the resistance (R).

### Step 2: Calculate the inductive reactance (XL)
The inductive reactance (XL) is given by the formula:
\[ XL = 2\pi f L \]
where:
- \(L\) is the inductance in henries (H)
- \(f\) is the frequency in hertz (Hz)

However, we do not need to calculate XL directly because at resonance, we will set it equal to XC.

### Step 3: Calculate the capacitive reactance (XC)
The capacitive reactance (XC) is given by the formula:
\[ XC = \frac{1}{2\pi f C} \]
where:
- \(C\) is the capacitance in farads (F)

Again, we will not need to calculate XC directly since we will set it equal to XL at resonance.

### Step 4: Apply the resonance condition
At resonance, we have:
\[ XL = XC \]
This implies that the total impedance (Z) of the circuit is:
\[ Z = R \]

### Step 5: State the impedance at resonance
Since the impedance at resonance is purely resistive, we conclude that:
\[ Z = R = 100 \, \Omega \]

### Final Answer
The impedance of the circuit at resonance is \(100 \, \Omega\).

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