6.5 RC Circuits
LEARNING OBJECTIVES
- Describe the charging process of a capacitor
- Describe the discharging process of a capacitor
- List some applications of circuits
When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging? This question and several other phenomena that involve charging and discharging capacitors are discussed in this module.
Circuits with Resistance and Capacitance
An RC circuit is a circuit containing resistance and capacitance. As presented in Capacitance, the capacitor is an electrical component that stores electric charge, storing energy in an electric field.
Figure 6.5.1(a) shows a simple circuit that employs a dc (direct current) voltage source , a resistor , a capacitor , and a two-position switch. The circuit allows the capacitor to be charged or discharged, depending on the position of the switch. When the switch is moved to position , the capacitor charges, resulting in the circuit in part (b). When the switch is moved to position , the capacitor discharges through the resistor.
(Figure 6.5.1)
Charging a Capacitor
We can use Kirchhoff’s loop rule to understand the charging of the capacitor. This results in the equation . This equation can be used to model the charge as a function of time as the capacitor charges. Capacitance is defined as , so the voltage across the capacitor is V_C=q/C. Using Ohm’s law, the potential drop across the resistor is , and the current is defined as .
This differential equation can be integrated to find an equation for the charge on the capacitor as a function of time.
Let , then . The result is
Simplifying results in an equation for the charge on the charging capacitor as a function of time:
A graph of the charge on the capacitor versus time is shown in Figure 6.5.2(a). First note that as time approaches infinity, the exponential goes to zero, so the charge approaches the maximum charge and has units of coulombs. The units of are seconds, units of time. This quantity is known as the time constant:
At time the charge is equal to of the maximum charge . Notice that the time rate change of the charge is the slope at a point of the charge versus time plot. The slope of the graph is large at time and approaches zero as time increases.
As the charge on the capacitor increases, the current through the resistor decreases, as shown in Figure 6.5.2(b). The current through the resistor can be found by taking the time derivative of the charge.
At time , the current through the resistor is . As time approaches infinity, the current approaches zero. At time , the current through the resistor is .
(Figure 6.5.2)
Figure 6.5.2(c) and Figure 6.5.2(d) show the voltage differences across the capacitor and the resistor, respectively. As the charge on the capacitor increases, the current decreases, as does the voltage difference across the resistor