A capacitor, a resistance of 2.5 is in series with a 12 V battery. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. [Given (2) = 0.693] [2005]

Question

A $4 μ F$ capacitor, a resistance of 2.5 $M Ω$ is in series with a 12 V battery. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. [Given $l n$ (2) = 0.693] [2005]

Smart Achievers · Accepted Answer

(1)

At any instant of time $t$ during the charging process, the transient current in the circuit is

$M Ω$ $∴$ Potential difference across resistance $R$

$V_{R} = [\frac{V_{0}}{R} e^{- t / R C}] \times R$

$= V_{0} e^{- t / R C}$ $\dots (i)$

$∴$ Potential difference across C

$V_{C} = V_{0} - V_{0} e^{- t / R C} = V_{0} (1 - e^{- t / R C})$ $\dots (i i)$

$∵$ $V_{C} = 3 V_{R}$

$∴$ $V_{0} (1 - e^{- t / R C}) = 3 V_{0} . e^{- t / R C}$

$\Rightarrow 1 - e^{- t / R C} = 3 e^{- t / R C}$ $\Rightarrow 1 = 4 e^{- t / R C}$

Taking logarithm on both sides

$\log_{e} 1 = 2 \log_{e} 2 + (- \frac{t}{R C})$

$\Rightarrow 0 = 2 \times 2.303 \log_{10} 2 - \frac{t}{R C}$

$\Rightarrow t = (2 \times 2.303 \log_{10} 2) \times 2.5 \times 10^{6} \times 4 \times 10^{- 6}$

$∴$ $t = 13.86 s$

Solution

Q.
A $4 μ F$ capacitor, a resistance of 2.5 $M Ω$ is in series with a 12 V battery. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. [Given $l n$ (2) = 0.693] [2005]

1 13.86 s

2 6.93 s

3 7 s

4 14 s

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