First slide

Capacitance

Question

The capacitors $C_{1}, C_{2} a n d C_{3}$ are uncharged initially and connected to A, B and C with potentials $V_{1}, V_{2} a n d V_{3}$ respectively. The potential of the junction D is :

Moderate

A

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

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

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

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

Solution

From conservation of charge
$\begin{array}{l} (V_{A} - V_{D}) C_{1} = (V_{D} - V_{B}) C_{2} + (V_{D} - V_{C}) C_{3} \\ V_{1} C_{1} + V_{2} C_{2} + V_{C} C_{3} = V_{D} (C_{2} + C_{2} + C_{3}) \\ V_{D} = \frac{C_{1} V_{1} + C_{2} V_{2} + C_{3} V_{3}}{C_{1} + C_{2} + C_{3}} \end{array}$

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