A charged capacitor is discharged through a resistor. The...

A charged capacitor is discharged through a resistor. The current $I(t)$ through this resistor, determined by measuring the voltage $V_{R}(t)=I(t) R$ with an oscilloscope, is shown in the graph. The total energy dissipated in the resistor is $2.0 \times 10^{-4} \mathrm{J} .$ (a) Find the capacitance $C$, the resistance $R,$ and the initial charge on the capacitor. [Hint: You will need to solve three equations simultaneously for the three unknowns. You can find both the initial current and the time constant from the graph.] (b) At what time is the stored energy in the capacitor $5.0 \times 10^{-5} \mathrm{J} ?$ (FIGURE CAN'T COPY)

Key Concepts

RC Circuit

An RC circuit consists of a resistor and a capacitor connected together, where the discharge or charge of the capacitor through the resistor is governed by differential equations. The behavior of the circuit is characterized by the relationship between voltage, current, and the time-dependent decay processes inherent in the resistor-capacitor system.

Time Constant

The time constant, denoted as ? and defined as the product of resistance and capacitance (? = RC), is a critical parameter in RC circuits. It represents the timescale over which the circuit's voltage and current decay exponentially, with values typically falling to about 37% of their initial values after one time constant.

Exponential Decay

Exponential decay describes processes where quantities decrease at a rate proportional to their current value. In the context of an RC circuit, both the current and the stored charge on the capacitor decay exponentially over time, following the form e^(?t/RC), which simplifies the analysis of the circuit’s behavior over time.

Energy Stored in a Capacitor

The energy stored in a capacitor is given by the formula E = ½ C V², where C is the capacitance and V the voltage across the capacitor. This stored energy diminishes as the capacitor discharges through the resistor, being converted into heat, highlighting the interplay between electrical energy storage and energy dissipation.

Integration of Power for Energy Analysis

Determining the total energy dissipated in a resistor involves integrating the instantaneous power (P = I²R) over the time of discharge. This process leverages calculus to account for the continuously varying current in the circuit, and the integration often exploits the properties of exponential decay.

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