A series circuit consists of a… | JEE Advanced 2016 P1

A series circuit consists of a resistor of 30 Ω, an ideal inductor of 15 mH and an ideal capacitor of 200/3 muF. It is connected to a source of rms…

Numerical answer

A series circuit consists of a resistor of

30\ \Omega

, an ideal inductor of

15\ \mathrm{mH}

and an ideal capacitor of

\tfrac{200}{3}\ \mu\mathrm{F}

. It is connected to a source of rms voltage

100\ \mathrm{V}

whose angular frequency is set to exactly three times the resonant angular frequency of the circuit. The reading, in amperes, of an ideal rms ammeter in the circuit is

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Answer: 2

The resonant angular frequency is

\omega_0=\dfrac{1}{\sqrt{LC}}=\dfrac{1}{\sqrt{15\times10^{-3}\times\frac{200}{3}\times10^{-6}}}=\dfrac{1}{\sqrt{10^{-6}}}=1000\ \mathrm{rad\,s^{-1}}

, so the source runs at

\omega=3000\ \mathrm{rad\,s^{-1}}

. The reactances are

X_L=\omega L=3000\times15\times10^{-3}=45\ \Omega

and

X_C=\dfrac{1}{\omega C}=\dfrac{1}{3000\times\frac{200}{3}\times10^{-6}}=5\ \Omega

. The net reactance is

X=X_L-X_C=40\ \Omega

, so the impedance is

Z=\sqrt{R^2+X^2}=\sqrt{30^2+40^2}=50\ \Omega

. The rms current is

I=V/Z=100/50=2\ \mathrm{A}

, and since the ammeter reads rms values the reading is

2

. The circuit is inductive with

\cos\phi=0.6

, a useful check. As a further check, the resonant current would have been

100/30=3.33\ \mathrm{A}

, larger than the

2\ \mathrm{A}

obtained away from resonance, as expected.

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