Question

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

Answer 1

Answer: (B), (C)

$L =1\, \mu H \,\,\,C =\mu F \,\,\,R =1 k \Omega$
Inductive reactance $X _{ L }=\omega L$
Capacitive reactance $X _{ C }=\frac{1}{\omega c }$
(A) at $\omega \rightarrow 0$
$ X _{ L } \rightarrow 0 $
$X _{ C } \rightarrow \infty$
$i \rightarrow 0$
(B) at $\omega>>10^{6}$
$ X _{ L }> X _{ C }$
Circuit behaves as inductor.
(C) Current $\&$ voltage are in same phase if $X _{ L }= X _{ C }$
$f =\frac{1}{2 \pi \sqrt{L C}}$
which is independent of $R$.
(D) Resonance frequency $\omega=\frac{1}{\sqrt{L C}}=10^{6} rad / s$.