# A circuit connected to an ac source of emf $e=e_{0} \sin(100 t)$ with t in seconds, gives a phase difference of $\frac{\pi}{4}$ betweeen the emf e and current i. Which of the following circuit will exhibit this? Option 1) RL circuit with $R=1 \;k\Omega$ and  L=10 mH Option 2) RL circuit with $R=1 \;k\Omega$ and  L=1 mH Option 3) RC circuit with $R=1 \;k\Omega$ and  $C=1\mu F$ Option 4) RC circuit with $R=1 \;k\Omega$ and  $C=10\mu F$

$w=100^{-1}$

for phase difference of $\frac{\pi}{4}\Rightarrow \left | X_{L}-X_{C} \right |=R$

if  $\;X_{C}=0 \Rightarrow X_{L}=R \Rightarrow wL=R \Rightarrow L=\frac{R}{w}$

if $\;X_{L}=0 \Rightarrow X_{C}=R= \frac{1}{wC} \Rightarrow C=\frac{1}{wR}$

for option (4)

$R=1k\Omega$

$C=\frac{1}{wR}=\frac{1}{\frac{100}{5}\times 10^{3} \Omega }= 10^{-5}_{s/ \Omega}\Rightarrow 10 \times 0^{-6}_{s/ \Omega}\Rightarrow 10\mu F$

Option 1)

RL circuit with $R=1 \;k\Omega$ and  L=10 mH

Option 2)

RL circuit with $R=1 \;k\Omega$ and  L=1 mH

Option 3)

RC circuit with $R=1 \;k\Omega$ and  $C=1\mu F$

Option 4)

RC circuit with $R=1 \;k\Omega$ and  $C=10\mu F$

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