A series LCR circuit has L = 5.0 H, C = 80 μF, R = 40 Ω and a variable-frequency 230 V AC source. At which pair of angular frequencies is the circuit power half its resonance value?
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A series LCR circuit has L = 5.0 H, C = 80 μF, R = 40 Ω and a variable-frequency 230 V AC source. At which pair of angular frequencies is the circuit power half its resonance value?

G1 Physics · Accepted Answer

Step 1: Resonant angular frequency ω₀ = 1/√(LC) = 1/√(5×80×10⁻⁶) = 50 rad/s. Step 2: At half-power, impedance magnitude increases to √2 R; hence |ωL−1/(ωC)|=R. Step 3: Near resonance, half-power frequencies satisfy ω≈ω₀±R/(2L) = 50±40/(10). Step 4: Thus ω₁≈46 rad/s and ω₂≈54 rad/s. Final answer: (C) 46 and 54 rad/s.