In the circuit shown, L=1\,{H},C=1\,{F} and R=1\,{k}.They are connected in series with an a.c. source V=V₀ω t as shown. Which of the following options is/are correct? [JEE Adv 2017 P1]

At ω=0 the current flowing through the circuit becomes nearly zero

At ω 10^6\,{rad}\,{s}^{-1}, the circuit behaves like a capacitor

The frequency at which the current will be in phase with the voltage is independent of R

The current will be in phase with the voltage if ω=10^4\,{rad}\,{s}^{-1}

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Alternating Current

Section: Physics

Question : 6 of 13

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In the circuit shown,

L=1\,\mu\mathrm{H},C=1\,\mu\mathrm{F}

R=1\,\mathrm{k}\Omega

V=V_0\sin\omega t

At $\omega=0$ the current flowing through the circuit becomes nearly zero
At $\omega\gg 10^6\,\mathrm{rad}\,\mathrm{s}^{-1}$ , the circuit behaves like a capacitor
The frequency at which the current will be in phase with the voltage is independent of R
The current will be in phase with the voltage if $\omega=10^4\,\mathrm{rad}\,\mathrm{s}^{-1}$

Solution:

\omega\sim 0, X_L\sim 0

and

X_C\rightarrow\infty

∴ Current will be nearly zero

\omega\gg 10^6, X_L\gg 1

and

X_C\rightarrow 0

Hence circuit does not behave like a capacitor.

At resonance frequency

(\omega_0)

i.e. when

X_C=X_L

current will be in phase with voltage and frequency is independent of R

Resonant frequency

\omega_0=\frac{1}{\sqrt{LC}}=10^6

rad/s

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