question_answer

A 4\[\mu F\] capacitor, a resistance of \[2.5M\Omega \] is in series with 12 V battery. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. \[\left[ Given\,\ell n(2)=0.693 \right]\]

A) \[13.86\,s\]                    

B) \[6.93\,s\]

C) \[7\,s\]                          

D) \[14\,s\]

Correct Answer: A

Solution :

\[4{{V}_{1}}=12\]or \[{{V}_{1}}=3V.\]

Time constant,\[\tau =CR\]\[=4\times {{10}^{-16}}\times 2.5\times {{10}^{6}}=10s\]

We have,\[V={{V}_{0}}\left( 1-{{e}^{-t/\tau }} \right)\]

Or \[9=12\left( 1-{{e}^{-t/10}} \right)\]

\[Or\,{{e}^{t/10}}=4\,or\,\frac{t}{10}=\ell n4\]

\[\therefore t=10\times 2\ell n2\]

=\[20\times 0.693=13.86s\]