From 1’32” in the film Alom suggests that students can check the discharge curve is exponential by seeing if there is a constant half-life (the time taken for the potential difference to fall by a half). Or, better, to measure the time constant and if it that is… er… constant (the time constant is the time taken for the measured potential difference to drop to 1/e of its original value – see the graph above).
Earlier in the film, Alom chooses values for C and R which multiply to give a time constant which is measurable: neither too fast nor too slow. In this case:
\(C = 4700~\mathrm{\mu F}\)
\(R = 10~\mathrm{k\Omega}\)
So time constant \(\tau = 4700\times10^{-6} \cdot 10 \cdot 10^3 = 47~\mathrm{s}\)
The exponential equation:
\(V = V_{0}e^{-t/RC}\)
is one which students will need to be able to handle. Alom shows how to manipulate this into the form of a straight-line graph. First, take natural logs of both sides:
\(ln(V) = ln(V_0) – \frac{t}{RC}\)
Compare with:
\(y = mx + c\)
If \(ln(V)\) is plotted on the y-axis against \(t\) on the x-axis, then \(ln(V_0)\) will be the y-intercept and \(-\frac{1}{RC}\) the gradient, hence \(C\) if \(R\) is known.