For which set of values of `X_(L), X_(C) and R`, a series LCR circuit will be in resonance ?

Options

A. `X_(L)=10 Omega, X_(C)=5 Omega, R = 10 Omega`
B. `X_(L)=5 Omega, X_(C)=10Omega, R = 10 Omega`
C. `X_(L) = 10 Omega, X_(C)= 10 Omega, R = 5 Omega`
D. `X_(L) = 5 Omega, X_(C) = 3 Omega, R = 4 Omega`

Answer

To determine the set of values of \(X_L\), \(X_C\), and \(R\) for which a series LCR circuit will be in resonance, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Resonance in LCR Circuit**:
- In a series LCR circuit, resonance occurs when the inductive reactance \(X_L\) is equal to the capacitive reactance \(X_C\). This is the condition for maximum current in the circuit.

2. **Condition for Resonance**:
- The mathematical condition for resonance can be expressed as:
\[ X_L = X_C \]

3. **Impedance at Resonance**:
- At resonance, the total impedance \(Z\) of the circuit is minimized and is equal to the resistance \(R\):
\[ Z = R \]
- The formula for impedance in a series LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
- For resonance, since \(X_L = X_C\), the equation simplifies to:
\[ Z = R \]

4. **Conclusion**:
- Therefore, for a series LCR circuit to be in resonance, the values must satisfy:
\[ X_L = X_C \quad \text{and} \quad Z = R \]
- This means that the reactances must be equal, and the impedance is purely resistive at resonance.