A circuit connected to an a c source

Question:

A circuit connected to an $a c$ source of $e m f e=e_{0} \sin (100 t)$

with $t$ in seconds, gives a phase difference of $\frac{\pi}{4}$ between

the $e m f e$ and current $i$. Which of the following circuits will exhibit this?

  1. (1) $\mathrm{RL}$ circuit with $\mathrm{R}=1 \mathrm{k} \Omega$ and $\mathrm{L}=10 \mathrm{mH}$

  2. (2) $R L$ circuit with $R=1 \mathrm{k} \Omega$ and $L=1 \mathrm{mH}$

  3. (3) $R C$ circuit with $R=1 \mathrm{k} \Omega$ and $C=1 \mu F$

  4. (4) $R C$ circuit with $R=1 \mathrm{k} \Omega$ and $C=10 \mu \mathrm{F}$.

Correct Option: , 4

Solution:

(4) $\omega=100 \mathrm{rad} / \mathrm{s}$

We know that

$\tan \phi=\frac{X_{C}}{R}=\frac{1}{\omega C R}$

or $\tan 45^{\circ}=\frac{1}{\omega C R}$

or $\omega \mathrm{CR}=1$

LHS: $\omega C R=10 \times 10 \times 10^{-6} \times 10^{3}=1$

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