*Question E5B04:
What is the time constant of a circuit having two 220-microfarad capacitors and two 1-megohm resistors, all in parallel?*  

*Correct Answer is  D: 220 seconds*  

### AI Response: (may contain errors)  

To find the time constant of a circuit with capacitors and resistors in parallel, we use the formula for the time constant $\tau$ of an RC circuit:

$$
\tau = R_{\text{eq}} \times C_{\text{eq}}
$$

where:
- $R_{\text{eq}}$ is the equivalent resistance
- $C_{\text{eq}}$ is the equivalent capacitance

**Step 1: Calculate $R_{\text{eq}}$**

For resistors in parallel, the equivalent resistance $R_{\text{eq}}$ is given by:

$$
\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2}
$$

Given two 1-megohm resistors in parallel:

$$
\frac{1}{R_{\text{eq}}} = \frac{1}{1\,\text{M}\Omega} + \frac{1}{1\,\text{M}\Omega} = \frac{2}{1\,\text{M}\Omega}
$$

Thus, $R_{\text{eq}} = \frac{1}{2} \times 1\,\text{M}\Omega = 0.5\,\text{M}\Omega$.

**Step 2: Calculate $C_{\text{eq}}$**

For capacitors in parallel, the equivalent capacitance $C_{\text{eq}}$ is the sum of the capacitances:

$$
C_{\text{eq}} = C_1 + C_2
$$

Given two 220-microfarad capacitors in parallel:

$$
C_{\text{eq}} = 220\,\mu\text{F} + 220\,\mu\text{F} = 440\,\mu\text{F}
$$

**Step 3: Calculate the time constant $\tau$**

Now, substitute $R_{\text{eq}}$ and $C_{\text{eq}}$ into the time constant formula:

$$
\tau = R_{\text{eq}} \times C_{\text{eq}} = 0.5\,\text{M}\Omega \times 440\,\mu\text{F}
$$

Convert units for calculation:

- $0.5\,\text{M}\Omega = 0.5 \times 10^6\,\Omega$
- $440\,\mu\text{F} = 440 \times 10^{-6}\,\text{F}$

Thus:

$$
\tau = (0.5 \times 10^6) \times (440 \times 10^{-6}) = 220\,\text{seconds}
$$

**Conclusion:**

- **Correct Answer: 220 seconds (D)**: This is the correct calculation of the time constant using the given values.
- **Incorrect Choices:**
  - **55 seconds (A)**: This would result from incorrect calculations of $R_{\text{eq}}$ or $C_{\text{eq}}$.
  - **110 seconds (B)**: This might result from using only one resistor or capacitor in the calculation.
  - **440 seconds (C)**: This could result from incorrectly summing resistances instead of using the parallel formula.

Understanding how to calculate equivalent resistance and capacitance in parallel is key to solving this problem correctly.