An engineer has five identical resistors of resistance R0=2.0kΩ and five identical capacitors of capacitance C0=5.0μF. She constructs three prototype timing circuits, each powered by the same ideal battery. In every circuit the resistor set is followed in series by the capacitor set, so current must pass through every resistor listed before reaching every capacitor listed.
• Circuit I: Three resistors in series followed by two capacitors in parallel.
• Circuit II: Two resistors in parallel followed by three capacitors in series.
• Circuit III: A ladder network of three identical sections; each section contains one resistor in series with one capacitor, and the three RC sections are connected in parallel with one another.
Assume all capacitors are initially uncharged.
(a) For each circuit, derive algebraic expressions for the equivalent resistance Req and equivalent capacitance Ceq in terms of R0 and C0.
(b) Compute numerical values for Req, Ceq, and the resulting time constant τ for each circuit.
(c) Rank the three circuits according to their time constants from smallest to largest, and explain the ordering with reference to the effects of series and parallel combinations on resistance and capacitance.
(d) Using any or all remaining unused components, propose a fourth circuit whose time constant is smaller than those of all three prototypes. Justify your design qualitatively and calculate its time constant to confirm that it meets the stated goal.