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JEE Advanced 2018 Paper 2 · Physics · Electromagnetic Induction and Alternating Current · medium

A series circuit has a resistor R=30 Ω, an ideal inductor of reactance 80 Ω and an ideal capacitor of reactance 40 Ω at the supply frequency…

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A series circuit has a resistor R=30 ΩR=30\ \Omega, an ideal inductor of reactance 80 Ω80\ \Omega and an ideal capacitor of reactance 40 Ω40\ \Omega at the supply frequency, connected to a source of rms voltage 200 V200\ \mathrm{V} and variable frequency (the supply is at the frequency where the reactances have the stated values unless told otherwise). Match each quantity in List-I with its value in List-II.

List-I
(P) The rms current in the circuit at the stated frequency
(Q) The average power dissipated at the stated frequency
(R) The rms voltage across the capacitor at the stated frequency
(S) The average power dissipated if the frequency is tuned to resonance (source voltage unchanged)

List-II
(1) 160 V160\ \mathrm{V}
(2) about 1.33 kW1.33\ \mathrm{kW}
(3) 4 A4\ \mathrm{A}
(4) 480 W480\ \mathrm{W}
(5) 320 V320\ \mathrm{V}
  1. A. P→3, Q→4, R→1, S→2
  2. B. P→3, Q→4, R→5, S→2
  3. C. P→1, Q→4, R→3, S→2
  4. D. P→3, Q→2, R→1, S→4
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Answer: (A) P→3, Q→4, R→1, S→2

At the stated frequency the net reactance is X=XL−XC=40 ΩX=X_L-X_C=40\ \mathrm{\Omega}, so the impedance is Z=302+402=50 ΩZ=\sqrt{30^2+40^2}=50\ \mathrm{\Omega}. The rms current is I=V/Z=200/50=4 AI=V/Z=200/50=4\ \mathrm{A}, so (P) matches 3. The average power is dissipated only in the resistor: P=I2R=16×30=480 WP=I^2R=16\times30=480\ \mathrm{W}, so (Q) matches 4 (equivalently VIcos⁡ϕ=200×4×0.6VI\cos\phi=200\times4\times0.6). The capacitor voltage is VC=IXC=4×40=160 VV_C=IX_C=4\times40=160\ \mathrm{V}, so (R) matches 1; note 320 V=IXL320\ \mathrm{V}=IX_L is the inductor voltage, a distractor. At resonance XL=XCX_L=X_C, Z=R=30 ΩZ=R=30\ \mathrm{\Omega}, the current becomes 200/30=6.67 A200/30=6.67\ \mathrm{A} and the power is V2/R=40000/30≈1.33 kWV^2/R=40000/30\approx1.33\ \mathrm{kW}, so (S) matches 2. Therefore P→\to3, Q→\to4, R→\to1, S→\to2.
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