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Parallel and Series Combination of Capacitors

Question

Deduce an expression for equivalent capacitance C when three capacitors $C_{1}$ , $C_{2}$ and $C_{3}$ are connected in parallel.

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Solution

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Let the three capacitance connected in parallel be $C_{1}$ , $C_{2}$ and $C_{3}$ and supply voltage be $V$ .
Let the equivalent capacitance of the combination be $C_{e q}$
Charge on a capacitor is given by $Q = C V$
Since all the capacitors are connected in parallel, so voltage across each capacitor is $V$ .
Thus charge on $C_{1}$ , $Q_{1} = C_{1} V$
Charge on $C_{2}$ , $Q_{2} = C_{2} V$
Charge on $C_{3}$ , $Q_{3} = C_{3} V$
Charge on equivalent capacitance $Q_{e q} = C_{e q} V$
Now $Q_{e q} = Q_{1} + Q_{2} + Q_{3}$
$∴$ $C_{e q} V = C_{1} V + C_{2} V + C_{3} V$
$⟹ C_{e q} = C_{1} + C_{2} + C_{3}$

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