Consider the resistor R and capacitance C in the circuit loop in figure 2.1. Notice that there is no source.
Figure 2.1: RC circuit.
We start with a differential version of Kirchoff's voltage law.
When applied to our circuit
where is the voltage drop across the capacitor and is the voltage drop across the resistor.
The change in the voltage drop across the capacitor is given by our previous expression,
The change in the voltage drop across the resistor can be obtained from Ohm's law
Substituting these changes in voltage into Kirchoff's equation gives
where the current due to the flow of charge on or off the capacitor is the same as through the resistor.
Now we need some initial conditions. Notice that although the capacitor behaves as an open circuit to DC, current must flow to charge or discharge the capacitor. Lets take the case where the capacitor is initially charged and then the circuit is closed and the charge is allowed to drain off the capacitor (eg. closing a switch). The resulting current will flow through the resistor.
Solving for the current we obtain
where is the initial current given by Ohm's law
Using a time dependent version of Ohm's law we can solve for the voltage across the resistor
where is the initial voltage across the capacitor and is the commonly defined time constant of the decay. You should also be able to solve for the voltage across the capacitor and charge on the capacitor.
For the case of an applying voltage being suddenly placed into the circuit (inserting a battery) the capacitor is initially not charged and the voltage across the capacitor is
In the first case, current and voltage exponentially decay away with time constant when the switch is closed. The charge flows off the capacitor and through the resistor. The energy initially stored in the capacitor is dissipated in the resistor.
In the second case the capacitor charges to a voltage until no current flows and hence the voltage drop across the resistor is zero. Energy from the battery is stored in the capacitor.
In both cases the characteristic RC time constant occurs. In general this is true of all resistor-capacitor combinations and will be important throughout the course.