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Exponential processes

Exponential processes. Learning outcomes. Exponential processes describe radioactive decay as an exponential process describe capacitor charge and discharge as exponential processes, including the time constant, RC

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Exponential processes

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  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

  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

  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. 1Make a table and evaluate e-x for integers in the range 2Draw a graph of e-x against x and find gradients at x = 1, 2, 3, 4 If then ln and e are inverse operations.

  9. Exponential function e is the number (approximately 2.718281828) 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 (AdvPhys OHT) Question sheet: Radioactive decay with exponentials (AdvPhys 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. 2Re-plot as log yagainstx (or lny 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 moodIOP (booklet freely available from the National STEM Centre eLibrary)

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