question_answer

A capacitor is initially connected to a battery of EMF 3 V. At \[t=0,\] switch is thrown to B state. Now charge on capacitor at any instant is given by:

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

B) \[\text{q}\,\text{=}\,\text{C}\,\text{(V}\,-\,\text{2}\,\text{V}{{\text{e}}^{\text{-t/RC}}}\text{)}\]

C)  \[\text{q}\,\text{=}\,\text{C}\,\text{(V}\,+\,\text{2}\,\text{V}{{\text{e}}^{\text{-t/2RC}}}\text{)}\]

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

Correct Answer: C

Solution :

\[Q={{Q}_{0}}{{e}^{-t/RC}}+CV(1-{{e}^{-t/RC}})\] \[=\left( CV+2CV\,{{e}^{-t/2RC}} \right)\] \[i=\frac{d\theta }{dt}=2CV\times \frac{{{e}^{-t/2RC}}}{2RC}\]