(e) exponential decay graph; constant-ratio property of such a graph.

The idea

An exponential decay graph shows a quantity decreasing at a rate proportional to its current value, producing a curve that asymptotically approaches zero. For a discharging capacitor, the charge Q, potential difference V or current I follows Q = Q₀e^(−t/CR). A key property is the constant-ratio property: for any equal time intervals, the ratio of the quantity at the end of the interval to that at the start is constant. For example, if the time interval equals the time constant τ = CR, the ratio is e^(−1) ≈ 0.37. If the interval is 2τ, the ratio is e^(−2) ≈ 0.14. This property allows identification of exponential decay from experimental data: successive values at equal time intervals should have a constant ratio. It also means that a graph of ln Q against t is a straight line with gradient −1/CR.

What earns the marks

• Recognise the shape of an exponential decay graph: a curve that decreases asymptotically towards zero.
• State that the rate of decay is proportional to the current value, leading to the equation x = x₀e^(−t/CR).
• Explain the constant-ratio property: for equal time intervals, the ratio of successive values is constant.
• Calculate the ratio for a given time interval, for example e^(−1) ≈ 0.37 for an interval of one time constant.
• Use the constant-ratio property to identify exponential decay from a table of data or to determine the time constant.

Common mistakes and how to put them right

• Thinking that exponential decay reaches zero in a finite time.

  Put it right: The graph approaches zero asymptotically; it never actually reaches zero in theory.

• Assuming that the ratio of successive values is constant for any time intervals, not just equal intervals.

  Put it right: The constant-ratio property applies only when the time intervals are equal.

• Confusing the constant ratio with a constant difference.

  Put it right: In exponential decay, the difference between successive values decreases, but the ratio remains constant for equal time intervals.

• Believing that the time constant is the time for the quantity to halve.

  Put it right: The half-life is τ ln 2 ≈ 0.693τ, not τ itself.

Exam tips

• To test for exponential decay, check whether the ratio of successive values at equal time intervals is constant.
• Remember that after one time constant the value is about 37% of the initial value, after two time constants about 14%, and after three about 5%.
• When plotting a graph to determine the time constant, use ln x against t; a straight line confirms exponential decay and its gradient gives −1/CR.