Physics

Physics

Problem

A charged capacitor of capacitance CC is connected across a resistor of resistance RR at time t=0t=0, allowing it to discharge. After a time t=2.0 mst=2.0\,\text{ms} the resistor is instantaneously replaced by a different resistor of resistance 3R3R, and the discharge continues. Assume all connections are ideal and ignore any inductive effects. (a) On clearly labeled axes, sketch the qualitative shape of the current-versus-time graph from t=0t=0 until a time long after the substitution. Indicate on the time axis the instant t=2.0 mst=2.0\,\text{ms} and mark at least one point associated with a characteristic time constant before and after the substitution. (b) Explain how the substitution affects the instantaneous current just after t=2.0 mst=2.0\,\text{ms}. Justify your answer without performing explicit calculations. (c) Compare the total amount of charge that leaves the capacitor during the first 2.0 ms2.0\,\text{ms} with the total amount of charge that leaves during the next 4.0 ms4.0\,\text{ms} after the substitution. Provide qualitative reasoning that refers to the time constants involved. (d) Suppose instead that, at t=2.0 mst=2.0\,\text{ms}, the original resistor RR were replaced by a resistor of resistance 13R\tfrac13 R. Discuss how each of your answers in parts (a) through (c) would change, if at all, and explain the physical basis for any differences.
Intuitive explanation

This problem tests your understanding of RC discharge and how changing circuit parameters mid-process affects the exponential decay. The key insight is that the charge (and voltage) on the capacitor at the moment of resistor swap becomes the new initial condition for the second phase—the capacitor doesn't "reset," it just continues decaying at a different rate determined by the new time constant τ=3RC\tau = 3RC. You'll need to find the charge remaining after the first 2.0 ms, then use that as your starting point for the second discharge phase.

Intuitive explanation

This problem tests your understanding of RC discharge and how changing circuit parameters mid-process affects the exponential decay. The key insight is that the charge (and voltage) on the capacitor at the moment of resistor swap becomes the new initial condition for the second phase—the capacitor doesn't "reset," it just continues decaying at a different rate determined by the new time constant τ=3RC\tau = 3RC. You'll need to find the charge remaining after the first 2.0 ms, then use that as your starting point for the second discharge phase.

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