question_answer

A capacitor of \[20\,\mu \,F\] capacity charged up to 500 V is connected in parallel with another capacitor of \[10\,\mu \,F\] which is charged up to 200 V. Their common potential is:                                                    [BHU PMT-2004]

A)  500 V                                   

B)  400 V

C)  300 V                                   

D)  200 V

Correct Answer: B

Solution :

                 Let the charges on capacitors be \[{{q}_{1}},\,\,{{q}_{2}}\] then \[{{q}_{1}}={{C}_{1}}\,{{V}_{1}},\,{{q}_{2}}={{C}_{2}}\,{{V}_{2}}\] Total charge \[q={{q}_{1}}+{{q}_{2}}\]                                                 \[={{C}_{1}}\,{{V}_{1}}+{{C}_{2}}\,{{V}_{2}}\] Let the equivalent potential be \[V\] and since capacitors are connected in parallel their equivalent capacitance is                                 \[C={{C}_{1}}+{{C}_{2}}\] \[\therefore \]  \[q=VC={{C}_{1}}\,{{V}_{1}}+{{C}_{2}}\,{{V}_{2}}\]                                 \[=V\left( {{C}_{1}}+{{C}_{2}} \right)\] \[\Rightarrow \]               \[V=\frac{{{C}_{1}}\,{{V}_{1}}+{{C}_{2}}\,{{V}_{2}}}{{{C}_{1}}+{{C}_{2}}}\] Given, \[{{C}_{1}}=20\,\mu F\], \[V=500\,v,\] \[{{C}_{2}}=10\,\mu F\],                                 \[{{V}_{2}}=200\,volt\] \[\therefore \]  \[V=\frac{20\times 500\times {{10}^{-6}}+10\times 200\times {{10}^{-6}}}{\left( 20+10 \right)\times {{10}^{-6}}}\]                                 \[=\frac{12000\times {{10}^{-6}}}{30\times {{10}^{-6}}}=400\,\text{volt}\] Note : Capacitors are combined in parallel when we require a large capacitance to a small potential.