Calculation of time constants including their determination from graphical…

The idea

The time constant τ of a capacitor circuit is the product of resistance and capacitance, τ = RC, and equals the time for charge, potential difference or current to fall to 1/e of its initial value during discharge, about 37%. Calculate it directly from component values, ensuring ohms and farads are used so τ is in seconds. From graphical data, plot ln V against time for discharge; the gradient is −1/RC, so τ is the negative reciprocal of the gradient. Alternatively read the time at which V falls to V₀/e. For charging, τ is the time to reach 1 − 1/e of the final value, about 63%. Always state units and interpret the physical meaning of the value obtained.

What earns the marks

• States τ = RC and substitutes resistance in ohms and capacitance in farads to obtain a time in seconds.
• Explains that during discharge the quantity falls to 1/e of its initial value in one time constant, approximately 37%.
• For graphical determination, plots ln V against time and uses the gradient, since gradient = −1/RC, so τ = −1/gradient.
• Reads the time at which V = V₀/e from an exponential decay graph as an alternative to the logarithmic method.
• For charging, identifies τ as the time to reach 1 − 1/e of the final value, approximately 63%.
• Checks that the calculated or graphical value is consistent with the component values and includes the correct unit, the second.

Common mistakes and how to put them right

• Using resistance in kilohms or capacitance in microfarads without converting to ohms and farads, giving a time constant wrong by powers of ten.

  Put it right by converting all values to SI base units before multiplying.

• Taking the gradient of a ln V against time graph as τ rather than as −1/RC.

  Put it right by recognising that τ is the negative reciprocal of the gradient.

• Assuming the capacitor fully discharges in one time constant.

  Put it right by stating that one time constant leaves about 37% of the initial value, and full discharge is approached only after several time constants.

Exam tips

• Show the rearrangement from gradient = −1/RC to τ = −1/gradient explicitly so the method is clear.
• When reading a value from a graph, mark the construction lines on the graph and quote the coordinates used.
• Give the final answer to an appropriate number of significant figures with the unit s, and comment briefly on whether it is physically sensible.