In an RLC alternating-current circuit, which expressions correctly define the inductive and capacitive reactances?

In an RLC alternating-current circuit, which expressions correctly define the inductive and capacitive reactances?

• A. X_L = ωL; X_C = ωC
• B. X_L = ωL; X_C = 1/(ωC)
• C. X_L = 1/(ωL); X_C = ωC
• D. X_L = L/ω; X_C = C/ω

Answer: (B)

The key idea is how reactance depends on frequency for inductors and capacitors. In an AC circuit, an inductor opposes changes in current with a reactance that grows as frequency increases, while a capacitor opposes voltage changes with a reactance that shrinks as frequency rises. These are quantified as X_L = ωL for the inductor and X_C = 1/(ωC) for the capacitor, where ω is the angular frequency and L and C are the inductance and capacitance. This pair matches the familiar impedance forms Z_L = jωL and Z_C = 1/(jωC), whose magnitudes are ωL and 1/(ωC) respectively. At a circuit level, the total reactance is X = X_L − X_C, so you can tune the frequency until X_L equals X_C (resonance), with ω0 = 1/√(LC). The other options propose wrong frequency dependencies (like X_C ∝ ω or X_L ∝ 1/ω, or inverses), which do not reflect the actual behavior of inductors and capacitors in AC circuits. Thus the correct expressions are X_L = ωL and X_C = 1/(ωC).

The key idea is how reactance depends on frequency for inductors and capacitors. In an AC circuit, an inductor opposes changes in current with a reactance that grows as frequency increases, while a capacitor opposes voltage changes with a reactance that shrinks as frequency rises. These are quantified as X_L = ωL for the inductor and X_C = 1/(ωC) for the capacitor, where ω is the angular frequency and L and C are the inductance and capacitance. This pair matches the familiar impedance forms Z_L = jωL and Z_C = 1/(jωC), whose magnitudes are ωL and 1/(ωC) respectively.

At a circuit level, the total reactance is X = X_L − X_C, so you can tune the frequency until X_L equals X_C (resonance), with ω0 = 1/√(LC). The other options propose wrong frequency dependencies (like X_C ∝ ω or X_L ∝ 1/ω, or inverses), which do not reflect the actual behavior of inductors and capacitors in AC circuits.

Thus the correct expressions are X_L = ωL and X_C = 1/(ωC).