# Episode 130: R-C circuits and other systems

There are many examples of exponential changes, both in physics and elsewhere. Your specification may require that you make a detailed comparison of the energy stored by a capacitor and a spring and of exponential decay in radioactivity and capacitors.

- Discussion: Energy stored (20 minutes)
- Discussion: Exponential decrease (20 minutes)
- Student questions: Exponential decrease (30 minutes)

**Discussion: Energy stored**

Comparing the energy stored by capacitors and springs: The key point in the discussion is that the graphs of ‘charge against pd’ for a capacitor and ‘force against extension’ for a spring are both straight lines through the origin. For capacitors, the energy stored is the area under the charge/pd graph (episode 128). A similar argument can be used to show that the energy stored in a spring is the area under the force/extension graph.

It follows that there are similar equations:

Energy stored in a capacitor = **1/2 QV = 1/2 CV ^{2}**

= **1/2 Fx = 1/2 kx ^{2}**

Although it is not specifically mentioned in the specifications, the energy can be released steadily but there are many occasions where oscillations occur. Students are likely to have seen this for a spring but may not have seen any electrical circuits involving oscillations. The section could be concluded with a demonstration of this.

Episode 130-1: Electrical oscillations (Word, 108 KB)

**Discussion: Exponential decrease**

Comparing exponential decay for radioactivity and capacitors: You could build up the table shown below using contributions from members of the class.

Any such comparison needs to highlight the similarities in the patterns for two very different physical processes by comparing the graphs of the decays. (This is a good point to remind pupils that testing for exponentials, either by a ‘constant ratio property’ or from a log graph, is an important skill.)

Capacitors | Radioactivity | |

Basic equation | Q = Q_{o} e^{-t/RC} | N = N_{o} e^{-l}t |

Rate of decay | current I = dQ/dt = –Q/RC | activity A = dN/dt = –lN |

Characteristic 'time' | Time constant = RC = time for charge to fall by 1/e T | Half life = T lT |

**Student questions: Exponential decrease**

This worksheet has a good survey of a number of processes involving exponential decay: radioactivity, capacitor discharge and more.

Episode 130-2: Exponential changes (Word, 75 KB)

**Download this episode**Episode 130: R-C circuits and other systems (Word, 186 KB)