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

Choices:  
A: 55 seconds  
B: 110 seconds  
C: 440 seconds  
D: 220 seconds    

Correct Answer is  D: 220 seconds  

To find the RC time constant with components in parallel:

- Capacitors in parallel add:  
  $$C_{\text{eq}} = C_1 + C_2 = 220\,\mu F + 220\,\mu F = 440\,\mu F$$
- Equal resistors in parallel halve:  
  $$R_{\text{eq}} = \frac{R}{2} = \frac{1\,\text{M}\Omega}{2} = 0.5\,\text{M}\Omega$$
- Time constant:  
  $$\tau = R_{\text{eq}} C_{\text{eq}} = (0.5\,\text{M}\Omega)(440\,\mu F) = 220\,\text{s}$$

Why the correct answer (D: 220 seconds) is correct:
- Using the correct parallel combinations and τ = RC gives 220 s.

Why the other choices are incorrect:
- 440 seconds: Uses 1 MΩ (forgot to halve the parallel resistors) with 440 µF: $$1\,\text{M}\Omega \times 440\,\mu F = 440\,\text{s}$$
- 110 seconds: Uses 0.5 MΩ with only one 220 µF capacitor (forgot to add capacitors): $$0.5\,\text{M}\Omega \times 220\,\mu F = 110\,\text{s}$$
- 55 seconds: Treats capacitors as if in series (incorrectly halved to 110 µF) and resistors correctly halved: $$0.5\,\text{M}\Omega \times 110\,\mu F = 55\,\text{s}$$

Useful formulas and tips:
- RC time constant: $$\tau = RC$$
- Capacitors: parallel adds; series uses reciprocals (for equal values, series halves).
- Resistors: series adds; parallel uses reciprocals (for equal values, parallel halves).
- Handy mental math: $$1\,\text{M}\Omega \times 1\,\mu F = 1\,\text{s}$$ so you can multiply the numeric parts directly: $$0.5 \times 440 = 220\,\text{s}.$$