#### Question

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

#### Solution

An expression for effective capacitance in the parallel grouping of capacitors :

Consider three capacitors of capacitance C_{1} , C_{2} and C_{3} are connected in parallel.

Let Q_{1}, Q_{2} and Q_{3} be the charges deposited on the capacitors as shown in the figure.

Suppose a potential difference ‘V ’ is applied across the combination. Then, the potential difference between the plates of each capacitor is V but charges on each capacitor are different. Since different current flows through different branches, so the charges are given by

`Q_1 = C_1V.Q_2 = C_2V. Q_3 = C_3V` ....(i)

From the principle of conservation of charge

`Q = Q_1 + Q_2 + Q_3`

`Q = C_1V + C_2V + C_3V` [From equation (i)]

`∴ Q = V(C_1 + C_2 + C_3)`

If these capacitors are replaced by a single capacitor of capacity C_{P} such that `Q = C_PV` then using equation (ii) we have,

`C_PV = V(C_1+C_2+C_3)`

`C_P = C_1+C_2+C_3`