Capacitor in Parallel Derivation

Here is a dc circuit with a battery using two electrolytic capacitors connected in parallel. The goal is to derive the formula for an equivalent capacitor. Each capacitor has holds a charge q, has a voltage V across it, and has a capacitance C, as shown in the diagram above.

The voltage V1 across capacitor C1 is q1 / C1.

The voltage V2 across capacitor C2 is q2 / C2.

The total voltage Vtot across the parallel combination is qtot / Ctot. This is the ratio of total charge and total capacitance.

The total charge qtot is the sum of q1 and q2. By substituting the previous expressions into this equation, we get the following expression.

This is the same as the previous equation but in terms of C and V, thus eliminating the charge q.

Capacitors in parallel have the same voltage; therefore, Vtot is the same as V1 and V2.

Hence, we divide throughout by V in the previous equation, thus cancelling out V1 and V2, giving the final expression below.

Capacitors in Parallel Formula

This is the formula for capacitors in parallel.

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