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Electric Field

Three uncharg...

Question

Three uncharged capacitor of capacities $C_{1}, C_{2}, C_{3}$ are connected as shown in the figure. Then, the potential at O will be :

A

\frac{V_{A} + V_{B} + V_{D}}{C_{1} + C_{2} + C_{3}}

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B

\frac{V_{A} C_{1} + V_{B} C_{2} + V_{D} C_{3}}{C_{1} + C_{2} + C_{3}}

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C

\frac{V_{A} V_{B} + V_{B} V_{D} + V_{D} V_{A}}{C_{1} + C_{2} + C_{3}}

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D

\frac{V_{A} + V_{B} + V_{D}}{C_{1} C_{2} + C_{2} C_{3} + C_{3} C_{1}}

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Solution

The correct option is A $\frac{V_{A} C_{1} + V_{B} C_{2} + V_{D} C_{3}}{C_{1} + C_{2} + C_{3}}$
The potential at O be the common potential of each capacitors.
thus, $Q_{1} = C_{1} V_{O}, Q_{2} = C_{2} V_{O}$ and $Q_{3} = C_{3} V_{O}$
If $V_{A}, V_{B}$ and $V_{D}$ are given to capacitors then charges are
$Q_{1}^{'} = C_{1} V_{A}, Q_{2}^{'} = C_{2} V_{B}$ and $Q_{3}^{'} = C_{3} V_{D}$
As total charge is conserved so $Q_{1} + Q_{2} + Q_{3} = Q_{1}^{'} + Q_{2}^{'} + Q_{3}^{'}$
$C_{1} V_{O} + C_{2} V_{O} + C_{3} V_{O} = C_{1} V_{A} + C_{2} V_{B} + C_{3} V_{D}$
or $V_{O} = \frac{C_{1} V_{A} + C_{2} V_{B} + C_{3} V_{D}}{C_{1} + C_{2} + C_{3}}$

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