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 that 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 connected 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 case the time constant of the RC circuit is the highest, we need to analyze how resistors and capacitors are connected in the circuit. The time constant (τ) of an RC circuit is given by the formula:

\[ \tau = R_{eq} \cdot C_{eq} \]

where \(R_{eq}\) is the equivalent resistance and \(C_{eq}\) is the equivalent capacitance.

### Step 1: Identify the configurations
We have two identical resistors (R) and two identical capacitors (C). The configurations can be:
1. Both resistors in series and both capacitors in series.
2. Both resistors in parallel and both capacitors in parallel.
3. Both resistors in series and both capacitors in parallel.
4. Both resistors in parallel and both capacitors in series.

### Step 2: Calculate equivalent resistance and capacitance for each configuration

1. **Both resistors in series and both capacitors in series:**
- \(R_{eq} = R + R = 2R\)
- \(C_{eq} = \frac{C}{2}\)
- Time constant: \(\tau = (2R) \cdot \left(\frac{C}{2}\right) = RC\)

2. **Both resistors in parallel and both capacitors in parallel:**
- \(R_{eq} = \frac{R}{2}\)
- \(C_{eq} = C + C = 2C\)
- Time constant: \(\tau = \left(\frac{R}{2}\right) \cdot (2C) = RC\)

3. **Both resistors in series and both capacitors in parallel:**
- \(R_{eq} = 2R\)
- \(C_{eq} = C + C = 2C\)
- Time constant: \(\tau = (2R) \cdot (2C) = 4RC\)

4. **Both resistors in parallel and both capacitors in series:**
- \(R_{eq} = \frac{R}{2}\)
- \(C_{eq} = \frac{C}{2}\)
- Time constant: \(\tau = \left(\frac{R}{2}\right) \cdot \left(\frac{C}{2}\right) = \frac{RC}{4}\)

### Step 3: Compare the time constants
From the calculations:
- Configuration 1: \(\tau = RC\)
- Configuration 2: \(\tau = RC\)
- Configuration 3: \(\tau = 4RC\)
- Configuration 4: \(\tau = \frac{RC}{4}\)

The highest time constant is found in **Configuration 3**, where the resistors are in series and the capacitors are in parallel, yielding \(\tau = 4RC\).

### Conclusion
The configuration that gives the highest time constant is when **two resistors are connected in series and two capacitors are connected in parallel**.

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