In an RLC series circuit at resonance, the voltages across the resistor and the inductor are 30 V and 40 V…

Analyze the question

We have a series RLC circuit at resonance. We know V_R = 30 V and V_L = 40 V, and we need V_C.

Concept

At resonance in a series RLC circuit:

• The inductive reactance equals the capacitive reactance: X_L = X_C
• The same current I flows through all components in series.
• So V_L = I·X_L and V_C = I·X_C must be equal in magnitude.
• V_L and V_C are 180° out of phase, so they cancel. The supply voltage equals V_R, so the circuit behaves as purely resistive.

Solution

1. At resonance: X_L = X_C
2. Since the current is the same: V_L = I·X_L = I·X_C = V_C
3. Therefore V_C = V_L = 40 V

Check: The net reactive voltage is V_L − V_C = 40 − 40 = 0. The supply voltage is V = √(V_R² + (V_L − V_C)²) = √(30² + 0²) = 30 V, which is consistent with resonance.

Why the other options are wrong

• A (50 V) is √(30² + 40²), which is what you would get if the circuit were NOT at resonance and you combined V_R and V_L.
• B (30 V) is just the resistor voltage.
• C (70 V) is the simple sum 30 + 40, which wrongly ignores phase.

Answer: D (40 V).

Exam tip: "At resonance" means X_L = X_C, so V_L = V_C. Whenever you see resonance in a series RLC circuit, the inductor and capacitor voltages are equal and cancel, and the supply voltage equals the resistor voltage.