21.6 Dc circuits containing resistors and capacitors

Learning objectives

By the end of this section, you will be able to:

• Explain the importance of the time constant τ, and calculate the time constant for a given resistance and capacitance.
• Explain why batteries in a flashlight gradually lose power and the light dims over time.
• Describe what happens to a graph of the voltage across a capacitor over time as it charges.
• Explain how a timing circuit works and list some applications.
• Calculate the necessary speed of a strobe flash needed to “stop” the movement of an object over a particular length.

The information presented in this section supports the following AP® learning objectives and science practices:

• 5.C.3.6 The student is able to determine missing values and direction of electric current in branches of a circuit with both resistors and capacitors from values and directions of current in other branches of the circuit through appropriate selection of nodes and application of the junction rule. (S.P. 1.4, 2.2)
• 5.C.3.7 The student is able to determine missing values, direction of electric current, charge of capacitors at steady state, and potential differences within a circuit with resistors and capacitors from values and directions of current in other branches of the circuit. (S.P. 1.4, 2.2)

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 a number of other phenomena that involve charging and discharging capacitors are discussed in this module.

RC Circuits

An $\mathbf{\text{RC}}$ circuit    is one containing a resistor     $R$ and a capacitor     $C$ . The capacitor is an electrical component that stores electric charge.

[link] shows a simple $\text{RC}$ circuit that employs a DC (direct current) voltage source. The capacitor is initially uncharged. As soon as the switch is closed, current flows to and from the initially uncharged capacitor. As charge increases on the capacitor plates, there is increasing opposition to the flow of charge by the repulsion of like charges on each plate.

In terms of voltage, this is because voltage across the capacitor is given by $V_{\text{c}}=Q/C$ , where $Q$ is the amount of charge stored on each plate and $C$ is the capacitance    . This voltage opposes the battery, growing from zero to the maximum emf when fully charged. The current thus decreases from its initial value of $I_0=\frac{\text{emf}}{R}$ to zero as the voltage on the capacitor reaches the same value as the emf. When there is no current, there is no $\text{IR}$ drop, and so the voltage on the capacitor must then equal the emf of the voltage source. This can also be explained with Kirchhoff’s second rule (the loop rule), discussed in Kirchhoff’s Rules , which says that the algebraic sum of changes in potential around any closed loop must be zero.

The initial current is $I_0=\frac{\text{emf}}{R}$ , because all of the $\text{IR}$ drop is in the resistance. Therefore, the smaller the resistance, the faster a given capacitor will be charged. Note that the internal resistance of the voltage source is included in $R$ , as are the resistances of the capacitor and the connecting wires. In the flash camera scenario above, when the batteries powering the camera begin to wear out, their internal resistance rises, reducing the current and lengthening the time it takes to get ready for the next flash.