DATA You set up the circuit shown in Fig. 26.22 a, where R=196 Ω. You close the switch at time t

step 1 So for this problem, when the switch S is closed, the current begins to flow as the capacitor pl

DATA You set up the circuit shown in Fig. 26.22 a, where...

DATA You set up the circuit shown in Fig. $26.22 \mathrm{a}$, where $R=196 \Omega$. You close the switch at time $t=0$ and measure the magnitude $i$ of the current in the resistor $R$ as a function of time $t$ since the switch was closed. Your results are shown in Fig. $\mathbf{P 2 6 . 7 8},$ where you have chosen to plot $\ln i$ as a function of $t$. (a) Explain why your data points lie close to a straight line. (b) Use the graph in Fig. P26.78 to calculate the capacitance $C$ and the initial charge $Q_{0}$ on the capacitor. (c) When $i=0.0500 \mathrm{~A},$ what is the charge on the capacitor? (d) When $q=0.500 \times 10^{-4} \mathrm{C},$ what is the current in the resistor?

DATA You set up the circuit shown in Fig. $26.22 \mathrm{a}$, where $R=196 \Omega$. You close the switch at time $t=0$ and measure the magnitude $i$ of the current in the resistor $R$ as a function of time $t$ since the switch was closed. Your results are shown in Fig. $\mathbf{P 2 6 . 7 8},$ where you have chosen to plot $\ln i$ as a function of $t$. (a) Explain why your data points lie close to a straight line. (b) Use the graph in Fig. P26.78 to calculate the capacitance $C$ and the initial charge $Q_{0}$ on the capacitor.
(c) When $i=0.0500 \mathrm{~A},$ what is the charge on the capacitor?
(d) When $q=0.500 \times 10^{-4} \mathrm{C},$ what is the current in the resistor?

Key Concepts

Relationship Between Current and Capacitor Charge

The instantaneous current in the circuit is related to the rate at which the capacitor loses its charge, expressed by the differential equation i = –dq/dt. This relationship is fundamental in connecting the exponential decay of the capacitor’s charge to that of the measured current, allowing for conversions between charge and current at any point in time.

Exponential Decay in RC Circuits

In an RC circuit, when the switch is closed the capacitor discharges through the resistor, and both the capacitor’s charge and the circuit’s current decay exponentially over time. This behavior is expressed by equations such as q(t) = Q? exp(–t/RC) for the charge and i(t) = (Q?/RC) exp(–t/RC) for the current, where Q? is the initial charge and RC is the time constant.

Logarithmic Linearization

By taking the natural logarithm of the exponentially decaying current, ln[i(t)] = ln(i?) – t/RC, the relationship becomes linear with respect to time. Plotting ln(i) against time results in a straight line, whose slope (–1/RC) and y-intercept (ln(i?)) can be used to extract the RC circuit parameters.

Time Constant

The time constant, denoted by ? = RC, represents the characteristic time over which the capacitor discharges. It quantifies the rate of exponential decay; a larger time constant means slower decay. The linear plot of ln(i) versus t allows for the experimental determination of ? from the slope of the line.

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