11.8.B—Describe the behavior of a circuit containing combinations of resistors and capacitors

Describe the behavior of a circuit containing combinations of resistors and capacitors.

• The charge on a capacitor or the current in a resistor in an RC circuit can be described by a fundamental differential equation derived from Kirchhoff’s loop rule. Derived equation:
• The time constant ( ) is a significant feature of an RC circuit.
  • i. The time constant of an RC circuit is a measure of how quickly the capacitor will charge or discharge and is defined as
  • ii. For a charging capacitor, the time constant represents the time required for the capacitor’s charge to increase from zero to approximately 63 percent of its final asymptotic value.
  • iii. For a discharging capacitor, the time constant represents the time required for the capacitor’s charge to decrease from fully charged to approximately 37 percent of its initial value.
• The potential difference across a capacitor and the current in the branch of the circuit containing the capacitor each change over time as the capacitor charges and discharges, but both will reach a steady state after a long time interval.
  • i. Immediately after being placed in a circuit, an uncharged capacitor acts like a wire, and charge can easily flow to or from the plates of the capacitor.
  • ii. As a capacitor charges, changes to the potential difference across the capacitor affect the charge on the plates of the capacitor, the current in the circuit branch in which the capacitor is located, and the electric potential energy stored in the capacitor.
  • iii. The potential difference across a capacitor, the current in the circuit branch in which the capacitor is located, and the electric potential energy stored in the capacitor all change with respect to time and asymptotically approach steady state conditions.
  • iv. After a long time, a charging capacitor approaches a state of being fully charged, reaching a maximum potential difference at which there is zero current in the circuit branch in which the capacitor is located.
  • v. Immediately after a charged capacitor begins discharging, the amount of charge on the capacitor and the energy stored in the capacitor begin to decrease.
  • vi. As a capacitor discharges, the amount of charge on the capacitor, the potential difference across the capacitor, and the current in the circuit branch in which the capacitor is located all decrease until a steady state is reached.
  • vii. After either charging or discharging for times much greater than the time constant, the capacitor and the relevant circuit branch may be modeled using steady-state conditions.