(d) graphical methods and spreadsheet modelling of the equation ΔQ/Δt =…

The idea

The differential equation ΔQ/Δt = −Q/CR describes the rate of change of charge Q on a discharging capacitor, where C is capacitance, R is resistance and CR is the time constant τ. The negative sign shows that charge is decreasing. Graphical methods include plotting Q against t to obtain an exponential decay curve, or plotting ln Q against t to obtain a straight line of gradient −1/CR. The gradient of the Q–t graph at any instant equals ΔQ/Δt, so the initial gradient is −Q₀/CR. Spreadsheet modelling involves using a small time step Δt to update Q iteratively: Q_new = Q_old + (ΔQ/Δt) × Δt, where ΔQ/Δt is calculated from the current Q. This numerical method approximates the exponential solution and allows the effect of changing C or R to be explored. The accuracy improves as Δt is made smaller.

What earns the marks

• Interpret the differential equation ΔQ/Δt = −Q/CR as the rate of change of charge on a discharging capacitor.
• Explain that the negative sign indicates a decreasing charge and that CR is the time constant τ.
• Use a graph of Q against t to find the gradient at a point, which equals ΔQ/Δt, and relate the initial gradient to −Q₀/CR.
• Use a graph of ln Q against t to obtain a straight line of gradient −1/CR and hence determine τ.
• Describe how a spreadsheet can model the equation by iterative steps: Q_new = Q_old + (ΔQ/Δt) × Δt, with ΔQ/Δt calculated from the current Q.
• Explain that reducing the time step Δt improves the accuracy of the numerical model.

Common mistakes and how to put them right

• Ignoring the negative sign in ΔQ/Δt = −Q/CR.

  Put it right: The negative sign shows that charge is decreasing; when modelling, ensure the change in charge is subtracted from the current value.

• Confusing the gradient of a Q–t graph with the time constant.

  Put it right: The gradient at a point is ΔQ/Δt, not τ; the time constant is found from the gradient of ln Q against t, which is −1/CR.

• Using a large time step in a spreadsheet model and expecting accurate results.

  Put it right: The numerical approximation improves as Δt becomes smaller; a large Δt leads to significant error.

• Assuming the differential equation applies to charging as well as discharging.

  Put it right: This form with a negative sign applies to discharging; for charging, the rate of change is positive and the equation is ΔQ/Δt = (Q₀ − Q)/CR.

Exam tips

• When using a spreadsheet model, always calculate the rate of change using the current value of Q, then update Q by adding (ΔQ/Δt) × Δt.
• To find τ from a graph, plot ln Q against t for a discharging capacitor; the gradient is −1/CR, so τ = −1/gradient.
• Remember that the initial gradient of a Q–t graph for a discharging capacitor is −Q₀/CR, which can be used to estimate τ if Q₀ is known.