Summarizing DC Circuits Containing Resistors and Capacitors

Summary

• An \(\text{RC}\) circuit is one that has both a resistor and a capacitor.
• The time constant \(\tau \) for an \(\text{RC}\) circuit is \(\tau =\text{RC}\).
• When an initially uncharged (\({V}_{0}=0\) at \(t=0\)) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source; as a function of time,

  \(V=\text{emf}(1-{e}^{-t/\text{RC}})\text{(charging).}\)

• Within the span of each time constant \(\tau \), the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically.
• If a capacitor with an initial voltage \({V}_{0}\) is discharged through a resistor starting at \(t=0\), then its voltage decreases exponentially as given by

  \(V={V}_{0}{e}^{-t/\text{RC}}\text{(discharging).}\)

• In each time constant \(\tau \), the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically.

Glossary

RC circuit

a circuit that contains both a resistor and a capacitor

capacitor

an electrical component used to store energy by separating electric charge on two opposing plates

capacitance

the maximum amount of electric potential energy that can be stored (or separated) for a given electric potential