Combination of two identical capacitors, a resistor $R$ and a $DC$ voltage source of voltage $6\; V$ is used in an experiment on $C-R$ circuit. It is found that for a parallel combination of the capacitor the time in which the voltage of the fully charged combination reduces to half its original voltage is $10\; s$. For series combination the time needed for reducing the voltage of the fully charged series combination by half is
AIEEE 2011, Diffcult
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Time constant for parallel combination $=2 R C$
Time constant for series combination $=\frac{R C}{2}$
In first case :
$V=V_{0} e^{\frac{t_{1}}{2 R C}}=\frac{V_{0}}{2}$
In second case :
$V=V_{0} e^{\frac{t_{2}}{(R C / 2)}}=\frac{V_{0}}{2}$
$\frac{t_{1}}{2 R C}=\frac{t_{2}}{(R C / 2)}$
$\Rightarrow \quad t_{2}=\frac{t_{1}}{4}=\frac{10}{4}=2.5\; sec$
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