https://file.aiquickdraw.com/mathbot/file/1722140340030yqqx13ns.png

To analyze the circuit and determine the equivalent capacitance and electrical energy stored, let's break down the configuration step by step.

### 1. Identify the Configuration
- There are two sections in the circuit:
  1. The left section contains three capacitors of \(4 \text{ F}\) each in parallel.
  2. The right section contains two capacitors of \(4 \text{ F}\) and \(2 \text{ F}\) in series.
- These two sections are connected in series with each other and the 12V voltage source.

### 2. Calculate Equivalent Capacitance

#### Left Section (Parallel Capacitors)
Capacitors in parallel add directly:
\[
C_{\text{left}} = C_1 + C_2 + C_3 = 4 \text{ F} + 4 \text{ F} + 4 \text{ F} = 12 \text{ F}
\]

#### Right Section (Series Capacitors)
For series capacitors:
\[
\frac{1}{C_{\text{right}}} = \frac{1}{C_4} + \frac{1}{C_5} = \frac{1}{4 \text{ F}} + \frac{1}{2 \text{ F}} = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}
\]
\[
C_{\text{right}} = \frac{4}{3} \text{ F} \approx 1.33 \text{ F}
\]

### 3. Total Equivalent Capacitance
The left and right sections are in series, so we use the series formula again:
\[
\frac{1}{C_{\text{eq}}} = \frac{1}{C_{\text{left}}} + \frac{1}{C_{\text{right}}} = \frac{1}{12 \text{ F}} + \frac{1}{\frac{4}{3} \text{ F}} = \frac{1}{12} + \frac{3}{4 \times 3} = \frac{1}{12} + \frac{3}{12} = \frac{4}{12} = \frac{1}{3}
\]
\[
C_{\text{eq}} = 3 \text{ F}
\]

### 4. Calculate Energy Stored
The energy stored in a capacitor is given by:
\[
E = \frac{1}{2} C V^2
\]
Where \(V = 12 \text{ V}\) and \(C_{\text{eq}} = 3 \text{ F}\):
\[
E = \frac{1}{2} \times 3 \times 12^2 = \frac{1}{2} \times 3 \times 144 = 216 \text{ joules}
\]

Therefore, the electrical energy stored in the circuit is 216 joules.

Do you want any further details or have any other questions?

Here are some related questions you might be interested in:
1. How to determine the voltage across each capacitor in a series-parallel circuit?
2. What is the impact of adding an additional capacitor in parallel with an existing capacitor?
3. How to calculate the charge stored on each capacitor in a series-parallel circuit?
4. How does changing the voltage source affect the energy stored in capacitors?
5. What are the differences in energy storage between capacitors in series versus parallel?

**Tip:** When dealing with complex circuits, always simplify step by step, combining capacitors in series and parallel until you reach a single equivalent capacitance.

Learn how to analyze capacitance in series-parallel circuits with a detailed step-by-step solution. This problem involves calculating equivalent capacitance and electrical energy stored in a complex circuit configuration. Suitable for high school students studying physics or electrical engineering.

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