Match the following, if $X_{\text{L}}$ and $X_{\text{C}}$ are inductive and capacitive reactances respectively:
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Remember the "CIVIL" mnemonic:
In a Capacitor, I (current) leads V (voltage).
In an Lnductor (inductor), V leads I (current lags).
At resonance ($X_{\text{L}} = X_{\text{C}}$), they are in phase.
Step 1: Understanding the Question:
This question asks us to match the relationships between inductive reactance ($X_{\text{L}}$) and capacitive reactance ($X_{\text{C}}$) with the phase relationships between current and voltage in an AC LCR series circuit.
Step 2: Key Concept and Approach:
The phase angle $\phi$ between voltage and current in an LCR circuit is:
\[ \tan\phi = \frac{X_{\text{L}} - X_{\text{C}}}{R} \]
- If $X_{\text{L}} = X_{\text{C}}$, the circuit is in resonance and behaves as a purely resistive circuit.
- If $X_{\text{L}} \lt X_{\text{C}}$, the capacitive reactance dominates, making the circuit capacitive.
- If $X_{\text{L}} \gt X_{\text{C}}$, the inductive reactance dominates, making the circuit inductive.
Step 3: Detailed Explanation:
• A. $X_{\text{L}} = X_{\text{C}}$:
Here, $\tan\phi = 0 \implies \phi = 0$.
The voltage and current are in phase.
Thus, A matches with I.
• B. $X_{\text{L}} \lt X_{\text{C}}$:
Here, $\tan\phi \lt 0$, indicating a capacitive phase shift.
In a capacitive circuit, the current leads the voltage.
Thus, B matches with III.
• C. $X_{\text{L}} \gt X_{\text{C}}$:
Here, $\tan\phi \gt 0$, indicating an inductive phase shift.
In an inductive circuit, the current lags behind the voltage.
Thus, C matches with II.
• This gives the matching: A-I, B-III, C-II.
Step 4: Final Answer:
The correct matching is A-I, B-III, C-II, which corresponds to Option (D).
