You have five capacitors To store maximum amount of energy when using a source with terminal voltage V.

Options

A. You need to connect them all in series
B. You need to connect them all in parallel
C. You need to connect three capacitors in parallel and the rest in series with that combination
D. You need to connect two capacitors in parallel and the rest in series with that combination

Answer

To solve the problem of maximizing energy storage using five capacitors with a fixed terminal voltage \( V \), we can follow these steps:

### Step 1: Understand the energy stored in a capacitor
The energy \( U \) stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance and \( V \) is the voltage across the capacitor.

### Step 2: Analyze the effect of capacitance on energy storage
Since the terminal voltage \( V \) is fixed, the energy stored in the capacitors depends only on the capacitance \( C \). To maximize the energy stored, we need to maximize the equivalent capacitance \( C_{eq} \).

### Step 3: Determine the equivalent capacitance for different configurations
1. **Series Configuration**: For capacitors connected in series, the equivalent capacitance is given by:
\[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots \]
For five identical capacitors (each with capacitance \( C \)):
\[ \frac{1}{C_{eq}} = \frac{5}{C} \implies C_{eq} = \frac{C}{5} \]

2. **Parallel Configuration**: For capacitors connected in parallel, the equivalent capacitance is:
\[ C_{eq} = C_1 + C_2 + C_3 + \cdots \]
For five identical capacitors:
\[ C_{eq} = 5C \]

3. **Mixed Configurations**:
- **Three in Parallel and Two in Series**: The equivalent capacitance for three capacitors in parallel is:
\[ C_{eq1} = C + C + C = 3C \]
For the two capacitors in series:
\[ \frac{1}{C_{eq2}} = \frac{1}{C} + \frac{1}{C} \implies C_{eq2} = \frac{C}{2} \]
The total equivalent capacitance for this configuration is:
\[ C_{total} = C_{eq1} + C_{eq2} = 3C + \frac{C}{2} = 3C + 0.5C = 3.5C \]

- **Two in Parallel and Three in Series**: The equivalent capacitance for two capacitors in parallel is:
\[ C_{eq1} = C + C = 2C \]
For the three capacitors in series:
\[ \frac{1}{C_{eq2}} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C} \implies C_{eq2} = \frac{C}{3} \]
The total equivalent capacitance for this configuration is:
\[ C_{total} = C_{eq1} + C_{eq2} = 2C + \frac{C}{3} = 2C + 0.333C = 2.333C \]

### Step 4: Compare the equivalent capacitances
- All in series: \( C_{eq} = \frac{C}{5} \)
- All in parallel: \( C_{eq} = 5C \)
- Three in parallel and two in series: \( C_{eq} = 3.5C \)
- Two in parallel and three in series: \( C_{eq} = 2.333C \)

### Step 5: Conclusion
The maximum equivalent capacitance occurs when all five capacitors are connected in parallel, yielding \( C_{eq} = 5C \). Therefore, the configuration that stores the maximum amount of energy is when all capacitors are connected in parallel.

### Final Answer
The correct option is: **Connect all capacitors in parallel.**

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