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Q.

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 during discharging is 10 s. for series combination, the time needed for reducing the voltage of the fully charged series combination during discharge by half is

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a

20 s

b

10 s

c

2.5 s

d

5 s

answer is D.

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Detailed Solution

If $C_{e}$ is the effective capacitance, then
$V_{C} = \frac{1}{2} V_{0}, \frac{q}{C_{0}} = \frac{q_{0}}{2 C_{0}}$
As $q = q_{0} (1 - e^{\frac{- t}{R C_{e}}})$
$\Rightarrow q_{0} (1 - e^{- t / R C_{e}}) = \frac{q_{0}}{2} \Rightarrow t = R C_{e} \ln 2$
For parallel grouping

Question Image

$\begin{array}{l} C_{e} = 2 C \\ ∴ t_{2} = 2 R C \ln 2 \end{array}$
For series grouping.

Question Image

$\begin{array}{l} C_{e} = \frac{C}{2} \\ ∴ t_{1} = \frac{R C}{2} \ln 2 \\ ∴ \frac{t_{2}}{t_{1}} = \frac{1}{4} t_{2} = 2.5 s \end{array}$

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