question_answer

A capacitor is initially connected to a battery of \[emf3\operatorname{V}.\text{ At}\,t\text{ }=\text{ }0\], switch is thrown to 5 state. Now charge on capacitor at any instant is given by

A) \[q=C\left( V+2V\,\,\frac{-t}{{{e}^{RC}}} \right)\]        

B) \[q=C\left( V-2V\,\frac{-t}{{{e}^{RC}}} \right)\]

C) \[q=C\left( V-2V\,\frac{-t}{{{e}^{2RC}}} \right)\]

D) \[q=C\left( V+2V\,\frac{-2t}{{{e}^{RC}}} \right)\]

Correct Answer: C

Solution :

[c]

Initial charge on the capacitor \[{{Q}_{0}}=C\times 3V\]\[=3CV\]

\[-\frac{q}{C}+i\times 2R+V=0\]

or \[2Ri=\left( \frac{q}{C}-V \right)\] or \[2R\frac{dq}{dt}=\left( \frac{q}{C}-V \right)\] or \[\int\limits_{{{Q}_{0}}}^{q}{\frac{dq}{\left( \frac{q}{C}-V \right)}=\int_{0}^{t}{\frac{dt}{2R}}}\]

\[\frac{\left| \ell n\left( \frac{q}{C}-V \right) \right|_{{{Q}_{0}}}^{q}}{\left( \frac{1}{C} \right)}=\frac{t}{2R}\]

\[\left| \ell n\left( \frac{q}{C}-V \right) \right|_{{{Q}_{0}}}^{q}=\frac{t}{2CR}\] or \[\ell n\left( \frac{q}{C}-V \right)-\ell n\left( \frac{{{Q}_{0}}}{C}-V \right)=\frac{t}{2CR}\] or \[\ell n\left( \frac{q}{C}-V \right)-\ell n2V=\frac{t}{2CR}\]

or \[\ell n2V-\ell n\left( \frac{q}{C}-V \right)=-\frac{t}{2CR}\]

\[\ell n\left[ \frac{2V}{\left( \frac{q}{C}-V \right)} \right]=\frac{t}{2CR}\] or \[\frac{2V}{\left( \frac{q}{C}-V \right)}={{e}^{\frac{t}{2CR}}}\] or \[2V=\left( \frac{q}{C}-V \right){{e}^{\frac{t}{2CR}}}\] or \[q=CV-2CV{{e}^{-\frac{t}{2CR}}}\]