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 the configuration of two identical resistors (R) and two identical capacitors (C) that yields the highest time constant in an RC circuit, we can follow these steps:

### Step 1: Understand the Time Constant Formula
The time constant (τ) for an RC circuit is given by the formula:
\[ \tau = R_{\text{equivalent}} \times C_{\text{equivalent}} \]
where \( R_{\text{equivalent}} \) is the equivalent resistance and \( C_{\text{equivalent}} \) is the equivalent capacitance.

### Step 2: Analyze the Resistor Configuration
To maximize the equivalent resistance:
- When resistors are connected in series, the equivalent resistance is:
\[ R_{\text{equivalent}} = R + R = 2R \]
This configuration will yield the maximum resistance.

### Step 3: Analyze the Capacitor Configuration
To maximize the equivalent capacitance:
- When capacitors are connected in parallel, the equivalent capacitance is:
\[ C_{\text{equivalent}} = C + C = 2C \]
This configuration will yield the maximum capacitance.

### Step 4: Combine the Configurations
Now, we combine the two configurations:
- Connect the two resistors in series to get \( R_{\text{equivalent}} = 2R \).
- Connect the two capacitors in parallel to get \( C_{\text{equivalent}} = 2C \).

### Step 5: Calculate the Maximum Time Constant
Substituting the equivalent values into the time constant formula:
\[ \tau = R_{\text{equivalent}} \times C_{\text{equivalent}} = (2R) \times (2C) = 4RC \]
This shows that the time constant is maximized at \( 4RC \).

### Conclusion
To achieve the highest time constant in the RC circuit:
- The two resistors should be connected in series.
- The two capacitors should be connected in parallel.
- This combination should then be connected across a power supply.

Thus, the correct configuration for the highest time constant is:
**Two resistors in series and two capacitors in parallel.**

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