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JEE Advanced 2016 Paper 2 · Physics · Electromagnetic Induction and Alternating Current · medium

A series circuit has R=10 Ω, L=0.10 H and C=10 muF, connected to a variable-frequency source of rms voltage 100 V.

One or more options correct

A series circuit has R=10 ΩR=10\ \Omega, L=0.10 HL=0.10\ \mathrm{H} and C=10 μFC=10\ \mu\mathrm{F}, connected to a variable-frequency source of rms voltage 100 V100\ \mathrm{V}. Choose the correct statement(s).
  1. A. The quality factor of the circuit is 1010.
  2. B. At resonance, the rms voltage across the capacitor is 100 V100\ \mathrm{V}.
  3. C. At resonance, the average power dissipated is 500 W500\ \mathrm{W}.
  4. D. At an angular frequency of 1100 rad s−11100\ \mathrm{rad\,s^{-1}} the current leads the source voltage.
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Answer: A

Resonance occurs at ω0=1/LC=1/0.10×10−5=1000 rad s−1\omega_0=1/\sqrt{LC}=1/\sqrt{0.10\times10^{-5}}=1000\ \mathrm{rad\,s^{-1}}. The quality factor is Q=ω0L/R=100/10=10Q=\omega_0L/R=100/10=10; A is true. At resonance the impedance equals RR, so I=100/10=10 AI=100/10=10\ \mathrm{A} and the capacitor voltage is I/(ω0C)=10×100=1000 VI/(\omega_0C)=10\times100=1000\ \mathrm{V}, ten times the source voltage (the factor QQ); B is false. The power is I2R=100×10=1000 WI^2R=100\times10=1000\ \mathrm{W}, not 500 W500\ \mathrm{W}; C is false. At 1100 rad s−11100\ \mathrm{rad\,s^{-1}}, XL=110 ΩX_L=110\ \Omega and XC=1/(1100×10−5)=90.9 ΩX_C=1/(1100\times10^{-5})=90.9\ \Omega, so the net reactance is inductive, the current lags the voltage, and D is false. Only statement A holds, so the answer is option A alone. The key physical point is that at resonance the reactive voltages are QQ times the source voltage but cancel each other in the total.
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