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JEE Advanced 2023 Paper 1 · Physics · Electromagnetic Induction and Alternating Current · hard

A series circuit has a resistor R=30 Ω, an ideal inductor L=0.20 H and a capacitor C=20 muF, connected to a source of fixed rms voltage 60 V and…

One or more options correct

A series circuit has a resistor R=30 ΩR=30\ \Omega, an ideal inductor L=0.20 HL=0.20\ \mathrm{H} and a capacitor C=20 μFC=20\ \mu\mathrm{F}, connected to a source of fixed rms voltage 60 V60\ \mathrm{V} and variable angular frequency ω\omega. The resonant angular frequency is ω0\omega_0. Which of the following statements is/are correct?
  1. A. At ω=250 rad s−1\omega=250\ \mathrm{rad\,s^{-1}} the current lags behind the source voltage.
  2. B. The average power at ω=250 rad s−1\omega=250\ \mathrm{rad\,s^{-1}} is about half of the average power at resonance.
  3. C. The average power dissipated at ω=1000 rad s−1\omega=1000\ \mathrm{rad\,s^{-1}} is greater than that at ω=250 rad s−1\omega=250\ \mathrm{rad\,s^{-1}}.
  4. D. At resonance the rms voltage across the inductor is 200 V200\ \mathrm{V}.
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Answer: D

Resonance occurs at ω0=1/LC=500 rad s−1\omega_0=1/\sqrt{LC}=500\ \mathrm{rad\,s^{-1}}, where Z=RZ=R, so I0=60/30=2 AI_0=60/30=2\ \mathrm{A} and the power is I02R=120 WI_0^2R=120\ \mathrm{W}. At ω0\omega_0 the inductor voltage is I0ω0L=2×500×0.2=200 VI_0\omega_0L=2\times500\times0.2=200\ \mathrm{V}, so (D) is correct. At ω=250 rad s−1\omega=250\ \mathrm{rad\,s^{-1}}: XL=50 ΩX_L=50\ \Omega, XC=1/(250×20×10−6)=200 ΩX_C=1/(250\times20\times10^{-6})=200\ \Omega, net reactance −150 Ω-150\ \Omega, so the circuit is capacitive and the current leads the voltage; (A) is wrong. Then Z=302+1502=153 ΩZ=\sqrt{30^2+150^2}=153\ \Omega, I=0.392 AI=0.392\ \mathrm{A} and P=I2R=4.6 WP=I^2R=4.6\ \mathrm{W}, only about 4%4\% of the resonant power, so (B) is wrong. At ω=1000=ω02/250\omega=1000=\omega_0^2/250 the reactances are 200 Ω200\ \Omega and 50 Ω50\ \Omega, net +150 Ω+150\ \Omega, so ZZ and PP are exactly the same as at 250250; (C) is wrong. Correct option: D only.
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