First slide

Capacitance

Question

Three uncharged capacitors of capacities C₁, C₂and C₃ are connected as shown in the figure to one another and the points A, B and C are at potentials V₁, V₂ and V₃ respectively. Then the potential at O will be

Moderate

A

$\frac{V_{1} C_{1} + V_{2} C_{2} + V_{3} C_{3}}{C_{1} + C_{2} + C_{3}}$
B

$\frac{V_{1} + V_{2} + V_{3}}{C_{1} + C_{2} + C_{3}}$
C

$\frac{V_{1} (V_{2} + V_{3})}{C_{1} (C_{2} + C_{3})}$
D

$\frac{V_{1} V_{2} V_{3}}{C_{1} C_{2} C_{3}}$

Solution

If potential at O is V then
$\begin{array}{rcl} (V_{1} - V) C_{1} & = & (V - V_{2}) C_{2} + (V - V_{3}) C_{3} \\ (C_{1} + C_{2} + C_{3}) V & = & V_{1} C_{1} + V_{2} C_{2} + V_{3} C_{3} \\ V & = & \frac{V_{1} C_{1} + V_{2} C_{2} + V_{3} C_{3}}{(C_{1} + C_{2} + C_{3})} \end{array}$

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