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  4. In the following circuits, it is given that R1=1 Ω, R2=2 Ω, C1=2 μ F and C2=4 μ F. <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/c650a17eeef58e69b6b1c26469548b6a-.png /> The time constants (in μ s) for the circuits I, II, III are, respectively,

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Q. In the following circuits, it is given that $R_{1}=1\, \Omega, R_{2}=2\, \Omega$, $C_{1}=2\, \mu F$ and $C_{2}=4 \,\mu F$.
image
The time constants (in $\mu$ s) for the circuits $I, II, III$ are, respectively,

JEE AdvancedJEE Advanced 2006

Solution:

Circuit (I): We have
$\tau_{1}=\frac{C_{1} C_{2}}{C_{1}+C_{2}} \times \frac{R_{1} \times R_{2}}{R_{1}+R_{2}}$
$=\frac{8}{6} \times \frac{2}{3}=\frac{8}{9} \mu S$
image
Circuit (II): We have
$\tau_{2}=\left(C_{1}+C_{2}\right)\left(R_{1}+R_{2}\right)=18 \,\mu S$
Circuit (III): We have
$\tau_{3}=\left(C_{1}+C_{1}\right)\left(\frac{R_{1} R_{2}}{R_{1}+R_{2}}\right)=4\, \mu S$