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Deduce an expression for equivalent capa...

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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Three capacitors `C_(1), C_(2) and C_(3)` are connected in parallel between the points A and B across a source of potential difference V. The potential difference between the plates of each capacitor will be same. Let it be V.

A charge +Q given to the point A by means of an electric source is distributed on all the three capacitors according to their capacitances. If the charges on the capacitors `C_(1), C_(2) and C_(3)` be `Q_(1), Q_(2) and Q_(3)` respectively, then
`Q_(1)= C_(1)V, Q_(2)= C_(2)V and Q_(3)= C_(3)V`
Total charge on all the three capacitors is `Q= Q_(1) + Q_(2) + Q_(3)`
`=C_(1)V + C_(2)V + C_(3)V`
`=V (C_(1) + C_(2) + C_(3))` ...(i)
If in place of all the three capacitors, only one capacitor is placed between A and B such that on giving it a charge the potential difference between its plates is V, then
Q= CV ...(ii)
where, C is the equivalent capacitance of `C_(1), C_(2) and C_(3)` connected in parallel. On comparing equations (i) and (ii), we have
`CV= V(C_(1) + C_(2) + C_(3))`
or `C= C_(1) + C_(2) + C_(3)`
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