Correct Option: (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Step 1 (Governing Law & Definitions): [0.25 Mark]
In a series LCR circuit, the total impedance $Z$ is given by:
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$Where $X_L = \omega L$ (inductive reactance) and $X_C = \frac{1}{\omega C}$ (capacitive reactance).
Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]
- At resonance, the condition is defined as $X_L = X_C$.
- Substituting the expressions: $\omega L = \frac{1}{\omega C}$.
- Rearranging for $\omega$: $\omega^2 = \frac{1}{LC} \implies \omega = \frac{1}{\sqrt{LC}}$.
- Substituting $X_L = X_C$ into the impedance formula: $$Z = \sqrt{R^2 + (0)^2} = R$$
Step 3 (Final Concluded Answer): [0.25 Mark]
Since the impedance $Z$ becomes equal to the resistance $R$ ($Z = R$), the phase angle $\phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right) = 0$. Thus, the circuit behaves as a pure resistive circuit. Both statements are correct and logically linked.
💡 EL10 Concept Insight: Imagine the inductor and capacitor as two kids playing tug-of-war; at resonance, they pull with equal strength in opposite directions, effectively canceling each other out so that only the resistor is left to "feel" the current.