Q.In a circuit a cell of emf $E$ is connected through a key $K_1$ to a capacitor $C_1$; a second key $K_2$ then connects $C_1$ to another capacitor $C_2$. The top plate of $C_1$ is joined to the cell's positive terminal through $K_1$ and, through $K_2$, to the top plate of $C_2$, while the bottom plates of both capacitors share a common return wire to the cell. Initially $K_1$ is closed and $K_2$ is open, so $C_1$ charges up to the cell's emf while $C_2$ stays uncharged. Then $K_1$ is opened and $K_2$ is closed (the order is important). Taking $Q_1'$ and $Q_2'$ as the final charges on $C_1$ and $C_2$, $V_1$ and $V_2$ as their final voltages, and $Q$ as the charge originally stored on $C_1$, then

Capacitor Network Analysis

Imagine you have a bucket of water and a pipe. The bigger the bucket, the more water it can hold for a given water pressure. A capacitor does the same thing with electric charge — it stores charge when a voltage is applied. The "size" of the bucket is called capacitance ($C$), measured in farads (F).

Now, what happens when you connect several buckets together with pipes? That's a capacitor network. The analysis is about finding one equivalent bucket (a single capacitor) that behaves exactly like the whole network.

The Core Idea: Charge and Voltage Must Match

When you connect capacitors, two things are always true:

• Charge conservation: Total charge in a closed part of the circuit stays the same unless a battery pushes more in.
• Voltage is shared: The voltage across each capacitor depends on how they're wired.

There are only two basic ways to connect them. Everything else is a combination of these two.

Series Connection: One Path, Shared Charge

Connect capacitors end-to-end, like train cars. The same current flows through each, so each capacitor stores the same amount of charge $Q$.

But the total voltage across the combination is the sum of the individual voltages:

Since $V = Q/C$ for each capacitor, we get:

Cancel $Q$ (it's the same everywhere):

Intuition: Adding capacitors in series makes the equivalent capacitance smaller than the smallest individual one. Why? Because you're effectively making the "bucket" longer and narrower — harder to fill.

Parallel Connection: Multiple Paths, Same Voltage

Connect capacitors side by side, like buckets with their bottoms connected by a wide pipe. Each capacitor sees the same voltage $V$ across it.

But the total charge stored is the sum of charges on each:

Since $Q = CV$ for each:

Cancel $V$:

Intuition: Adding capacitors in parallel makes the equivalent capacitance larger — you're just adding more bucket area. Easy to fill.

How to Analyze Any Network

1. Spot the pattern: Look for capacitors that are clearly in series (only two terminals, no branching between them) or clearly in parallel (both ends connected together).
2. Replace step by step: Replace each simple series or parallel group with its equivalent capacitor. Redraw the circuit after each step.
3. Repeat until you have one capacitor.

A Worked Example

Suppose you have three capacitors: $C_1 = 2\ \mu\text{F}$, $C_2 = 3\ \mu\text{F}$, $C_3 = 6\ \mu\text{F}$. $C_2$ and $C_3$ are in parallel, and that combination is in series with $C_1$. …