A more complicated simple circuit is an RC circuit --- a combination of a resistor and a capacitor. These are used frequently to extend the length of time that charge flows out of a capacitor.

In a previous demo, you saw that capacitors hold charge.

If there is no resistance, then ALL of the charge (energy stored) will "explode" from the capacitor all at once. This is how a camera flash works. But if you can control the release of that energy by channeling the "flow of stored charge" (ie: current) through a capacitor, the capacitor will take a lot longer to become fully discharged. The light bulbs used in the capacitance demonstration serve as resistors in this way.

Another practical application of resistors and capacitors in series is to keep a circuit energized in case of a power failure. My clock radio will keep its time setting even if I unplug it from the wall for as long as 15 seconds. Some telephone answering machines will likewise keep your messages for a long time if the electricity from the wall is interrupted.

The mathematics is not complicated, and simply uses the relation that current, i = dq/DT

You perform the loop rule using V = iR for the resistors and V = Q/C for the capacitor.

Then, take the time derivative of the whole equation, where dq/DT becomes i.

Now you have a differential equation of i and DI/DT

Solving, you get an exponential relation:

Current at any time = (current at time zero) exp[-(t/RC)]

RC is known as the "Time Constant", governing how fast the capacitor discharges.

(See the textbook, section 28-8 if you are interested in the details.)