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  4. At time t = 0, a battery of 10 V is connected across points A and B in the given circuit. If the capacitors have no charge initially, at what time (in second) does the voltage across them become 4 V? [Take: In 5 = 1.6, In 3 = 1.1]

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Q. At time t = 0, a battery of 10 V is connected across points A
and B in the given circuit. If the capacitors have no charge
initially, at what time (in second) does the voltage across
them become 4 V? [Take : In 5 = 1.6, In 3 = 1.1]Physics Question Image

IIT JEEIIT JEE 2010Electrostatic Potential and Capacitance

Solution:

Voltage across the capacitors will increase from 0 to 10 V
exponentially. The voltage at time t will be given by
$ V=10(1-e^{-t/ \tau} C) $
Here $\tau_c=C_{net}R_{net} $
$ =(1 \times10^6)(4 \times 10^{-6})=4s $
$\therefore V=10(1-e^{-t/4}) $
Substituting V = 4 volt, we have
$4=10(1-e^{-t/4}) \, or \, e^{-t/4}=0.6= \frac {3}{5} $
Taking log both sides we have,
$\, \, \, \, \, \, \, \, \, \, - \frac {t}{4}=In \, 3-In \, 5 $
or t = 4 (In 5 - In 3)= 2 s
Hence, the answer is 2 .