Question

In the circuit shown, $L=1\mu H,C=1\mu F$ and $R=1k\Omega$. They are connected in series with an a.c. source $V=V_0\sin \omega t$ as shown. Which of the following options is/are correct?

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

Answer 1

Answer: (B), (D)

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

At resonant frequency, current and voltage will be in same phase.
Resonant frequency, ${\omega }_0=\frac{1}{\sqrt{LC}}={10}^{6}rad$
When $\omega >>{10}^{6}rad.s^{-1}$, reactance for inductor will be much higher as compared to that of capacitor. Hence the circuit behaves like inductor not capacitor. Thus option A is incorrect.

When $\omega \sim 0$, reactance of the capacitor will be infinite i.e $X_C\approx \infty$ and thus the impedence of the circuit$\left(Z\right)$ will be infinite. Hence the current flowing through the circuit, $I=\frac{V}{Z}=0$

Hence option B is correct

At resonating frequency, current flowing through the circuit will be in phase with voltage. Since resonating frequency is ${10}^6$, option C is incorrect.

Resonant frequency, ${\omega }_0=\frac{1}{\sqrt{LC}}$

which is independent of R. Hence option D is correct.