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.

Lesson Summary

• 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. 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 the capacitor = 12QV

Energy stored in the spring = 12Fx

Energy stored in the spring = 12kx²

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 the connections using contributions from members of the class.

For capacitor decay:

Q = Q₀e^- t/_CR_

Current, Δ QQ = 1CR

Time constant, CR: time for charge to fall by 1e

T _½ CR = ln(2)

For radioactive decay:

N = N₀e^{-  λ_N_}

Activity, Δ NN = λN

Half life, T _½ : time for no. of atoms to fall by 12.

T _½ λ = ln(2)

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.)

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)