Series RLC at resonance: which relationship holds between inductive reactance and capacitive reactance at the resonant frequency?

Introduction / Context:
Resonance in a series RLC circuit is a cornerstone concept. At resonance, the reactive effects cancel, minimizing impedance and maximizing current for a given applied voltage. Understanding the equality of reactances is essential for filter design, tuning, and impedance matching.

Given Data / Assumptions:

• Series R–L–C elements with sinusoidal steady-state excitation.
• Resonant frequency defined by ω0 = 1 / sqrt(L * C).
• Ideal components (to focus on the core concept).

Concept / Approach:
At resonance in series RLC, X_L = ω0 * L and X_C = 1 / (ω0 * C) are equal in magnitude and opposite in sign. Their sum is zero, so the net reactive part of impedance is zero, leaving Z ≈ R (minimum magnitude).

Step-by-Step Solution:
1) Write reactances: X_L = ωL; X_C = 1/(ωC).2) At resonance: ω0 = 1/sqrt(LC).3) Substitute: X_L(ω0) = ω0L and X_C(ω0) = 1/(ω0C) = ω0L.4) Conclude: X_L equals X_C in magnitude; net reactance is zero.

Verification / Alternative check:
Measure impedance versus frequency: the minimum occurs at resonance where reactive terms cancel, confirming equality of magnitudes.

Why Other Options Are Wrong:
Options A and B contradict resonance; option D (difference equals resistance) is not a standard resonance condition; resistance is independent of reactive cancellation.

Common Pitfalls:
Confusing series with parallel resonance; in series, impedance is minimum at resonance, not maximum.

Final Answer:
Inductive reactance is equal to the capacitive reactance.