Electrostatics 6 Question 3
3. A $4 \mu \mathrm{F}$ capacitor and a resistance of $2.5 \mathrm{M} \Omega$ are in series with $12 \mathrm{~V}$ battery. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. [Given, $\ln (2)=0.693$ ]
(2005, 2M)
(a) $13.86 \mathrm{~s}$
(b) $6.93 \mathrm{~s}$
(c) $7 \mathrm{~s}$
(d) $14 \mathrm{~s}$
$(1983,6 \mathrm{M})$
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Answer:
Correct Answer: 3. (a)
Solution:
- Given : $V_{C}=3 V_{R}=3\left(V-V_{C}\right)$
Here, $V$ is the applied potential.
$$ \begin{array}{ll} \therefore & V_{C}=\frac{3}{4} V \quad \text { or } \quad V\left(1-e^{-t / \tau_{c}}\right)=\frac{3}{4} V \\ \therefore & e^{-t / \tau_{c}}=\frac{1}{4} \end{array} $$
Here,
$$ \tau_{c}=C R=10 \mathrm{~s} $$
Substituting this value of $\tau_{c}$ in Eq. (i) and solving for $t$, we get
$$ t=13.86 \mathrm{~s} $$
$\therefore$ Correct answer is (a).
