Q.A network of four $10\ \mu\text{F}$ capacitors is connected to a $500\ \text{V}$ supply, as shown in Fig. 2.29.

The key idea is Capacitor Network Analysis — simplifying series and parallel combinations to find equivalent capacitance, then working backwards to find individual charges.

Step 1: Identify the network structure.

Capacitors $C_1$, $C_2$, and $C_3$ (each $10\ \mu\text{F}$) form a series path between nodes A and D. Capacitor $C_4$ (also $10\ \mu\text{F}$) is connected directly across A and D, in parallel with that series combination.

Step 2: Find equivalent capacitance of the series branch.

For three equal capacitors in series:

Step 3: Combine with the parallel capacitor.

$C_4$ is in parallel with $C_{\text{series}}$:

Step 4: Determine charges.

The supply voltage $V = 500\ \text{V}$ appears across both $C_4$ and the series branch. …

Reading the network (Fig. 2.29). $C_1$ (between A and B), $C_2$ (between B and C) and $C_3$ (between C and D) form a series chain running A to B to C to D. $C_4$ is connected directly across A and D, so it is in parallel with that series chain. The 500 V supply is applied across A-D.

1. Equivalent capacitance. Series combination of the three $10\ \mu\text{F}$ capacitors:

   $\frac{1}{C'} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} \implies C' = \frac{10}{3}\ \mu\text{F}$

   $C'$ in parallel with $C_4$:

   $C_{\text{eq}} = C' + C_4 = \frac{10}{3} + 10 = \frac{40}{3}\ \mu\text{F} \approx 13.3\ \mu\text{F}$

2. Charge on each capacitor. Series branch - $C_1, C_2, C_3$ share the same charge, equal to the charge on their equivalent $C'$ held across 500 V: …

Method: Reducing a Capacitor Network to Find Equivalent Capacitance and Individual Charges

This method applies to any circuit diagram showing several capacitors wired together across a supply, where you must find the equivalent capacitance and then the charge on each individual capacitor.

Steps

Step 1: Trace the circuit to identify series and parallel groups

Follow the wiring node by node. Capacitors sharing a single unbroken path — one after another, with no branch point in between — are in series. Capacitors connected across the exact same pair of nodes are in parallel. Most networks reduce in stages: a sub-group of series capacitors first, then combined in parallel with another branch, and so on.

Step 2: Reduce each sub-group using the standard combination formulas

Work from the innermost sub-group outward until a single equivalent capacitance for the whole network remains.

Step 3: Find the voltage across each branch …