Now let us consider a circuit in which a capacitor has an initial charge. The circuit below shows that there is now a resitor, switch, capacitor, but no battery. This time, the charged capacitor will be the source of emf.

While the switch is open, the potential difference across the capacitor is Q/C. Potential difference across the resistor is zero because no current is flowing in the circuit. As the switch is closed at time t=0, the capacitor begins to discharge through the resistor. At any time during the discharge, the potential difference across the resistor is equal to the potential difference across the capacitor, IR=Q/C (this is from Kirkoff's Second Law). As the capacitor discharges, the electric current across the resistor will approach its maximum value. Also, the charge on the capacitor and the potential across the capacitor will decrease exponentially with time. The electric current, charge, and potential change according to the following equations:

I(t)=(E/Rmax)[1-e-t/RC]
Q(t)=ECe-t/RC
V(t)=Ee-t/RC

It is important to remember that the six equations highlighted in this section are not the only equations that deal with RC circuits. Using V=IR for the maximum current, when the capacitor has no affect on the current; and using Q=CV for the charge and potential difference on the capacitor are both important equations in understanding RC ciruits.

Finally, it may be important to note that the quantity RC that appears in the denominator of the above exponents (as in

e-t/RC

) is a constant that is called the time constant of the circuit. The time constant represents the time it takes the current to decrease to 1/e of its original value.

Information provided by: http://www.msms.doe.k12.ms.us