20.13

RC Circuits

Many electric circuits contain both resistors and capacitors. Figure 20-39 illustrates an example of a resistor–capacitor or RC circuit. Part a of the drawing shows the circuit at a time t after the switch has been closed and the battery has begun to charge up the capacitor plates. The charge on the plates builds up gradually to its equilibrium value of , where V0 is the voltage of the battery. Assuming that the capacitor is uncharged at time t = 0 s when the switch is closed, it can be shown that the magnitude q of the charge on the plates at time t is

| (20.20)

where the exponential e has the value of 2.718. … Part b of the drawing shows a graph of this expression, which indicates that the charge is q = 0 C when t = 0 s and increases gradually toward the equilibrium value of q0 = CV0. The voltage V across the capacitor at any time can be obtained from Equation 20.20 by dividing the charges q and q0 by the capacitance C, since V = q/C and V0 = q0/C.

Figure 20-39 Charging a capacitor.

The term RC in the exponent in Equation 20.20 is called the time constant of the circuit:

| (20.21)

The time constant is measured in seconds; verification of the fact that an ohm times a farad is equivalent to a second is left as an exercise (see conceptual question 16). The time constant is the amount of time required for the capacitor to accumulate 63.2% of its equilibrium charge, as can be seen by substituting in Equation 20.20; . The charge approaches its equilibrium value rapidly when the time constant is small and slowly when the time constant is large.
Figure 20-40a shows a circuit at a time t after a switch is closed to allow a charged capacitor to begin discharging. There is no battery in this circuit, so the charge +q on the left plate of the capacitor can flow counterclockwise through the resistor and neutralize the charge −q on the right plate. Assuming that the capacitor has a charge q0 at time t = 0 s when the switch is closed, it can be shown that

| (20.22)

where q is the amount of charge remaining on either plate at time t. The graph of this expression in part b of the drawing shows that the charge begins at q0 when t = 0 s and decreases gradually toward zero. Smaller values of the time constant RC lead to a more rapid discharge. Equation 20.22 indicates that when , the magnitude of the charge remaining on each plate is . Therefore, the time constant is also the amount of time required for a charged capacitor to lose 63.2% of its charge.

Figure 20-40 Discharging a capacitor.

The charging/discharging of a capacitor has many applications. Heart pacemakers, for instance, incorporate RC circuits to control the timing of voltage pulses that are delivered to a malfunctioning heart to regulate its beating cycle. The pulses are delivered by electrodes attached externally to the chest or located internally near the heart when the pacemaker is implanted surgically (see Figure 20-41). A voltage pulse is delivered when the capacitor discharges to a preset level, following which the capacitor is recharged rapidly and the cycle repeats. The value of the time constant RC controls the pulsing rate, which is about once per second.

Figure 20-41 This X-ray photograph shows a heart pacemaker that has been implanted surgically. (RNHRD NHS Trust/Stone/Getty Images, Inc.)

The physics of heart pacemakers.

The charging/discharging of a capacitor is also used in automobiles that have windshield wipers equipped for intermittent operation during a light drizzle. In this mode of operation, the wipers remain off for a while and then turn on briefly. The timing of the on–off cycle is determined by the time constant of a resistor–capacitor combination.

The physics of windshield wipers.

Check Your Understanding 5

The drawings show two different resistor–capacitor arrangements. The time constant for arrangement A is 0.20 s. What is the time constant for arrangement B? (The answer is given at the end of the book.)

Background: Capacitors, like resistors, can be connected in series and in parallel. The time constant for a given connection depends on the resistance and the equivalent capacitance of the capacitors.
For similar questions (including conceptual counterparts), consult Self-Assessment Test 20.2, which is described next.

Test your understanding of the material in Sections 20.8, 20.9, 20.10, 20.11, 20.12 and 20.13:

· | Circuits Wired Partially in Series and Partially in Parallel
· | Internal Resistance
· | Kirchhoff’s Rules
· | The Measurement of Current and Voltage
· | Capacitors in Series and in Parallel
· | RC Circuits

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