Two identical resistors of magnitude R and two identical capacitors of magnitude C are used to form an RC circuit. In which case the time constant of the RC circuit is the highest?

Options

A. Two resistors and two capacitors connected in series with a power supply.
B. Two resistors and two capacitors connected in parallel with a power supply.
C. Two resistors connected in parallel, two capacitors connected in parallel, and those two combinations are in series with a power supply
D. Two resistors connected in series, two capacitors connected in parallel and those two combinations are connected in series with a power supply.

Answer

To determine in which configuration the time constant of the RC circuit is the highest, we need to analyze the combinations of resistors and capacitors.

### Step-by-Step Solution:

1. **Understanding Time Constant**:
The time constant (τ) of an RC circuit is given by the formula:
\[ \tau = R_{eq} \times C_{eq} \]
where \( R_{eq} \) is the equivalent resistance and \( C_{eq} \) is the equivalent capacitance.

2. **Resistor Combinations**:
- **Series Combination**: When two identical resistors \( R \) are connected in series, the equivalent resistance is:
\[ R_{eq} = R + R = 2R \]
- **Parallel Combination**: When two identical resistors \( R \) are connected in parallel, the equivalent resistance is:
\[ \frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{R} \implies R_{eq} = \frac{R}{2} \]

3. **Capacitor Combinations**:
- **Series Combination**: When two identical capacitors \( C \) are connected in series, the equivalent capacitance is:
\[ \frac{1}{C_{eq}} = \frac{1}{C} + \frac{1}{C} \implies C_{eq} = \frac{C}{2} \]
- **Parallel Combination**: When two identical capacitors \( C \) are connected in parallel, the equivalent capacitance is:
\[ C_{eq} = C + C = 2C \]

4. **Finding Maximum Time Constant**:
- **Case 1**: Resistors in series and capacitors in series:
\[ R_{eq} = 2R, \quad C_{eq} = \frac{C}{2} \implies \tau = (2R) \times \left(\frac{C}{2}\right) = RC \]
- **Case 2**: Resistors in series and capacitors in parallel:
\[ R_{eq} = 2R, \quad C_{eq} = 2C \implies \tau = (2R) \times (2C) = 4RC \]
- **Case 3**: Resistors in parallel and capacitors in series:
\[ R_{eq} = \frac{R}{2}, \quad C_{eq} = \frac{C}{2} \implies \tau = \left(\frac{R}{2}\right) \times \left(\frac{C}{2}\right) = \frac{RC}{4} \]
- **Case 4**: Resistors in parallel and capacitors in parallel:
\[ R_{eq} = \frac{R}{2}, \quad C_{eq} = 2C \implies \tau = \left(\frac{R}{2}\right) \times (2C) = RC \]

5. **Conclusion**:
The highest time constant occurs in **Case 2**, where the resistors are in series and the capacitors are in parallel:
\[ \tau = 4RC \]

### Final Answer:
The time constant of the RC circuit is the highest when the resistors are connected in series and the capacitors are connected in parallel.

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