Question

Deduce the formula of equivalent capacity of three condensers connected in parallel.

Hint:

In parallel the voltage across each capacitor is the same; add the charges \( Q = C_1V + C_2V + C_3V \) and use \( C_p = Q/V \).

Answer 1

Step 1 (Set-up): Let three capacitors of capacitances \( C_1 \), \( C_2 \) and \( C_3 \) be joined in parallel and connected to a battery of potential difference \( V \). In a parallel combination, the left plates are joined to one common point and the right plates to another common point.

Step 2 (Key property): Because each capacitor is connected directly across the same two terminals, the potential difference across every capacitor is the same and equal to the applied voltage:
\[ V_1 = V_2 = V_3 = V \]
Step 3 (Charge on each capacitor): Using the defining relation \( Q = C V \) for a capacitor,
\[ Q_1 = C_1 V,\qquad Q_2 = C_2 V,\qquad Q_3 = C_3 V \]
Step 4 (Total charge): The battery supplies a total charge equal to the sum of the charges on the individual capacitors:
\[ Q = Q_1 + Q_2 + Q_3 = C_1 V + C_2 V + C_3 V = (C_1 + C_2 + C_3)\,V \]
Step 5 (Equivalent capacitance): If the combination is replaced by a single equivalent capacitor \( C_p \) carrying the same total charge \( Q \) at the same voltage \( V \), then by definition \( Q = C_p V \). Comparing with Step 4,
\[ C_p V = (C_1 + C_2 + C_3)\,V \] Cancelling \( V \) from both sides,
\[\boxed{C_p = C_1 + C_2 + C_3}\]
Thus in a parallel combination the equivalent capacitance is the simple sum of the individual capacitances, and it is always larger than the greatest of them.