1. Exponential processes

2. Learning outcomes Exponential processes
describe radioactive decay as an exponential process
describe capacitor charge and discharge as exponential processes, including the time constant, RC
calculate the decay constant and predict later activity of a radioactive sample, using and
solve problems using and

3. Teaching challenges Showing how logarithms work
Getting from the probability of radioactive decay to
and the idea of equal ratios in equal times
Conveying the meaning of a time constant for a capacitor (Why does the product RC give a time?)
Developing confidence in solving problems with exponential equations

4. Exponential processes
What do the following have in common?
popping popcorn
the head on a glass of beer
sand falling through a hourglass timer
air leaking from a tyre
and these?
cell growth in a foetus
rabbit populations (good food supply, no disease or predators)
time series data on Internet traffic
nuclear chain reaction
sand falling through hourglass time NOT exponential, as constriction passes only few grains at a time. Water draining from a burette probably IS exponential, provided flow is unimpeded at stopcock.

5. Growth and decay amount changes with time
rate of increase proportional to amount
rate of decrease proportional to amount
A story: Doubling rice grains
TAP question sheet: Exponential changes
‘Doubling rice grains
shorten Doubling rice grains’ story – to save time.

6. Law of exponents In general,
Here x is called the ‘base’ of a number system.

7. Logarithms Logarithms use base 10. log 100 = y where 10y = 100
What is log 203? log 1505? log 5? log1? log(.01)?
Using logarithms changes multiplication to addition.
What is 203 x 1505?
Log and exponent (power) are inverse operations.
What is log (102.5)? log (10y)?

8. Natural logarithms Base is the number e.
1 Make a table and evaluate e-x for integers in the range
2 Draw a graph of e-x against x and find gradients at x = 1, 2, 3, 4
If then
ln and e are inverse operations.
Each table allocates finding of gradients one per person, results pooled and discussed.

9. Exponential function
e is the number (approximately ) such that the function ex equals its gradient for all values of x.
ex is sometimes written exp(x).

10. Radioactive decay Radioactive decay is a chance process.
Activity, A (number of atoms in a sample disintegrating per second, dN/dt) is directly proportional to the number of atoms present, N.
 is a constant characteristic of the atom (decay constant).
If N0 atoms present,
Handout: Smoothed-out radioactive decay (Adv Phys OHT)
Question sheet: Radioactive decay with exponentials (Adv Phys 90S)

11. Half life

12. Discharging a capacitor
At time t, during discharge of capacitor C, the p.d. across the capacitor is V and the charge on it is Q, then Q = VC.
The discharge current I at time t equals the rate of loss of charge [- sign indicating Q decreases as t increases]
V=IR
so [RC is the decay constant]
or which has solution

13. Charging a capacitor
in series with a resistor R
which has solution

14. Charging and discharging C
Graphs for Q, I and V against t all have the same general shape. How are they related?
I graph is the gradient of the Q graph (since I = dQ/dt).
V = Q/C

15. An exponential relationship?
Two tests for an exponential graph:
1 Half-life of radioactive materials is one instance of a more general pattern. In general, measured values have
equal ratios in equal intervals of time.
2 Re-plot as log y against x (or ln y against x).
An exponential relationship gives a straight line.
e.g.

16. Charging and discharging C
TAP experiment Analysing the discharge of a capacitor
Question sheets:
Charging capacitors
Discharge and time constants

17. Exponentials – more examples
Carbon-dating an archaeological artefact, by comparing C-14 in a sample with atmospheric C-14
Estimating the age of the Earth, from ratio of U-235 (t1/2 = 7.04 x108 y) to U-238 (t1/2 = 4.47x109 y)
Intensity of -rays with thickness of absorber
Calculating atmospheric pressure at altitude h
Decay of amplitude for a vibration

18. Endpoints Further reading Helen Reynolds, Wherefore ?
in Simon Carson (ed) (1999) Physics in mathematical mood IOP (booklet freely available from the National STEM Centre eLibrary)