DATA You set up the circuit shown in Fig. 26.20 , where C=5.00 ×10^-6 F. At time t=0, you close

step 1 So for this problem, Kirchhoff's rules apply to this circuit. And so part A wants us to graph Q

DATA You set up the circuit shown in Fig. 26.20 , where...

DATA You set up the circuit shown in Fig. 26.20 , where $C=5.00 \times 10^{-6} \mathrm{~F}$. At time $t=0,$ you close the switch and then measure the charge $q$ on the capacitor as a function of the current $i$ in the resistor. Your results are given in the table: (a) Graph $q$ as a function of $i$. Explain why the data points, when plotted this way, fall close to a straight line. Find the slope and $y$ -intercept of the straight line that gives the best fit to the data. (b) Use your results from part (a) to calculate the resistance $R$ of the resistor and the emf $\mathcal{E}$ of the battery. (c) At what time $t$ after the switch is closed is the voltage across the capacitor equal to $10.0 \mathrm{~V} ?$ (d) When the voltage across the capacitor is $4.00 \mathrm{~V},$ what is the voltage across the resistor?

DATA You set up the circuit shown in Fig. 26.20 , where $C=5.00 \times 10^{-6} \mathrm{~F}$. At time $t=0,$ you close the switch and then measure the charge $q$ on the capacitor as a function of the current $i$ in the resistor. Your results are given in the table:
(a) Graph $q$ as a function of $i$. Explain why the data points, when plotted this way, fall close to a straight line. Find the slope and $y$ -intercept of the straight line that gives the best fit to the data. (b) Use your results from part (a) to calculate the resistance $R$ of the resistor and the emf $\mathcal{E}$ of the battery. (c) At what time $t$ after the switch is closed is the voltage across the capacitor equal to $10.0 \mathrm{~V} ?$ (d) When the voltage across the capacitor is $4.00 \mathrm{~V},$ what is the voltage across the resistor?

Key Concepts

Time Constant in RC Circuits

The time constant, represented by the product RC, is a key concept in RC circuits as it determines how quickly the capacitor charges or discharges. It is the time required for the capacitor to charge to about 63.2% of its maximum charge when charging, or to drop to 36.8% when discharging. The time constant encapsulates the dynamic response of the system and is essential for predicting the transient behavior of the circuit following a change such as closing a switch.

Linearization of Exponential Functions in Circuit Analysis

In many experiments and analyses, the exponential behavior of RC circuits is transformed or manipulated to yield linear relationships that are easier to analyze. For example, plotting one variable against another (like charge versus current) may produce a linear graph, enabling simple extraction of circuit parameters such as resistance and the battery’s electromotive force from the slope and intercept of the graph. This method is a common strategy in experimental physics and engineering to simplify data interpretation.

RC Circuit Transients

RC circuit transients refer to the time-dependent behavior of circuits that contain resistors and capacitors. When a circuit is switched, the charge on the capacitor does not instantly reach its final value; instead, it gradually changes over time. This behavior is governed by a differential equation, and the rate of change is determined by the product of the resistance and capacitance (the time constant). Understanding these transients is crucial when studying how circuits respond to sudden changes, such as turning a battery on or off.

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law (KVL) states that the sum of all the potential differences (voltage drops and rises) around any closed loop in a circuit must equal zero. In the context of an RC circuit, this means that the battery voltage is distributed between the resistor (where the voltage drop follows Ohm’s law) and the capacitor (where the voltage is related to the charge on the capacitor). This principle is fundamental for deriving the equations that describe the transient charging and discharging processes.

Exponential Charging of a Capacitor

When a capacitor charges through a resistor, its voltage and charge follow an exponential behavior rather than increasing linearly. The charge on the capacitor as a function of time can be expressed by an exponential formula where the final value of the charge is reached asymptotically. This behavior results from the nature of the differential equation that describes how current flows into the capacitor, and it is characterized by a time constant, which defines the speed of this charging process.

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