11.8: Resistor-Capacitor (RC) Circuits

Describe the equivalent capacitance of multiple capacitors.

11.8.A.1 A collection of capacitors in a circuit may be analyzed as though it were a single capacitor, with an equivalent capacitance eq C .

• 11.8.A.1.i The inverse of the equivalent capacitance of a set of capacitors connected in series is equal to the sum of the inverse of the individual capacitances.

• 11.8.A.1.ii The equivalent capacitance of two or more capacitors in series is less than the capacitance of the smallest capacitor.

11.8.A.2 The equivalent capacitance of two or more capacitors in parallel is the sum of the individual capacitances.

11.8.A.3 As a result of conservation of charge, each of the capacitors in series must have the same magnitude of charge on each plate.

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

11.8.B.1 The time constant is a significant feature of an RC circuit.

• 11.8.B.1.i The time constant of an RC circuit is a measure of how quickly the capacitor will charge or discharge and is described with the equation

• 11.8.B.1.ii For a charging capacitor, the time constant represents the time required for the capacitor to reach ~63% of the final asymptotic value.

• 11.8.B.1.iii For a discharging capacitor, the time constant represents the time required for the capacitor to reach ~37% of its value before discharging.

11.8.B.2 The potential difference across a capacitor and the current in the branch of the circuit containing the capacitor change over time as the capacitor charges and discharges, but both will reach a steady state after a long time interval.

• 11.8.B.2.i Immediately after a potential difference is applied to an uncharged capacitor, charges can easily flow to or from the plates of the capacitor.

• 11.8.B.2.ii As a capacitor charges, changes to the potential difference across the capacitor affect the charge on the plates of the capacitor, the current passing through the circuit branch in which the capacitor is located, and the electric potential energy stored in the capacitor.

• 11.8.B.2.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 have asymptotes that are determined by analyzing the circuit at steady-state.

• 11.8.B.2.iv After a long time, a charging capacitor approaches a state of being fully charged, eventually reaching a maximum potential difference with zero current in the circuit branch in which the capacitor is located.

• 11.8.B.2.v Immediately after the switch in a discharging capacitor circuit is closed, the amount of charge on the capacitor plates and the energy stored in the capacitor begin to decrease.

• 11.8.B.2.iv As a capacitor discharges, the amount of charge on the capacitor, the potential difference across the capacitor, and the current in the circuit branch of the capacitor each continue to decrease until steady-state is reached.

11.8.B.3 After either charging or discharging for times much greater than the time constant, the capacitor and circuit branch in which the capacitor is located approach steady-state conditions.

• 11.8.B.3.i After a long time, a charging capacitor approaches a state of being fully charged. At this point, the potential difference across the capacitor is at a maximum value with zero current in the circuit branch in which the capacitor is located.

• 11.8.B.3.ii After a long time, a discharging capacitor approaches a state of having zero charge on the capacitor. At this point, the potential difference across the capacitor is zero with zero current in the circuit branch in which the capacitor is located.