levo

Creating Models

Iterative numerical models of exponential change

  • 3 things to know
  • Components 01, 02 and 03

Overview

Recall questions on step-by-step solutions of dQ/dt = −Q/(RC) and radioactive decay, calculating single steps, why such models predict too fast a fall, improving accuracy with smaller steps and the drawback of doing so, choosing a step size, checking against N = N₀exp(−λt) and in a spreadsheet, the graphical tangent method, and negative outputs.

What you need to know

Answer each one before you reveal it.

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Key term

Explain the idea behind solving dQ/dt = −Q/(RC) by a step-by-step numerical method.

Key term

Why does a step-by-step model of decay always predict too fast a fall?

Key term

State the recipe for one step of a numerical model of capacitor discharge.

Questions from this subtopic

Each one with the model answer underneath.

  1. 1

    Explain the idea behind solving dQ/dt = −Q/(RC) by a step-by-step numerical method.

    Model answer

    The rate of change is calculated from the present charge, assumed constant over a small time step, used to find the change in charge, and the whole process is repeated from the new value.

  2. 2

    State the recipe for one step of a numerical model of capacitor discharge.

    Model answer

    Compute ΔQ = −(Q/RC)Δt, add it to Q to get the new charge, advance the time by Δt and repeat.

  3. 3

    A capacitor holds 1.0 × 10⁻³ C with R = 1.0 kΩ and C = 100 μF. How much charge is lost in a first step of 0.010 s?

    Model answer

    ΔQ = −(Q/RC)Δt = −(1.0 × 10⁻³ / 0.10) × 0.010 = −1.0 × 10⁻⁴ C.

  4. 4

    A model step removes 1.0 × 10⁻⁴ C from a stored charge of 1.0 × 10⁻³ C. What is the charge at the end of that step?

    Model answer

    The charge is 1.0 × 10⁻³ − 1.0 × 10⁻⁴ = 9.0 × 10⁻⁴ C.

  5. 5

    Why does a step-by-step model of decay always predict too fast a fall?

    Model answer

    The rate is held at its value from the start of the step, but the true rate falls during the step, so too much is subtracted each time.

  6. 6

    How is the accuracy of an iterative model improved?

    Model answer

    Use a smaller time step, so that the rate changes very little during each step.

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The two charges .
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AThe neutron has lepton number 0.
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There are 2π rad in a full circle.
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