question_answer

Two capacitors having capacity C and \[\frac{C}{2}\] are connected to a V volt battery, as shown in figure. Then work done charging both the capacitors fully is  

A)  \[\frac{3}{4}C{{V}^{2}}\]                 

B)  \[\frac{1}{4}C{{V}^{2}}\]

C)  \[2\,C{{V}^{2}}\]                  

D)  \[\frac{1}{2}\,C{{V}^{2}}\]

Correct Answer: A

Solution :

Here capacitors are connected in parallel so potential difference across both capacitors will remain same   \[\therefore \,\,\,\,{{Q}_{1}}=CV\,\,and\,\,{{Q}_{2}}\,\,=\,\,\frac{C}{2}\ V\] also      \[\operatorname{Q}={{Q}_{1}}+{{Q}_{2}}=CV+\frac{C}{2}V=\frac{3}{2}CV\] Work done in charging capacitor is \[=\,\,\frac{1}{2}\,QV\] \[=\,\,\,\frac{1}{2}\times \left( \frac{3}{2}\,\,CV \right)\times V=\frac{3}{4}C{{V}^{2}}\]