Charging and Discharging

Discharging through a resistor — the capacitor drives a current through R. As charge leaves, the p.d. falls, so the current falls too, so the charge leaves more slowly still. The rate of change is proportional to what remains, which is the definition of exponential decay:

Q = Q₀ e^(−t / RC)

with the same form for current and voltage, since both are proportional to Q:

I = I₀ e^(−t / RC) and V = V₀ e^(−t / RC)

Charging — the charge and voltage rise towards their final values while the current decays exponentially:

Q = Q₀ (1 − e^(−t / RC))

Note the difference: during charging, the current is largest at the very start, when the capacitor is empty and offers no opposing voltage, and falls to zero when fully charged.

Example: the initial charging current is set entirely by the resistor, since the capacitor initially has no p.d. across it: I₀ = V_supply / R. This is why a large capacitor connected directly to a supply can draw a dangerous surge of current at the instant of connection.

Graph shapes — all four curves approach their final value asymptotically, never quite arriving. The curve is steepest at the start and flattens continuously.

A defining property of exponential decay — the quantity falls by the same fraction in every equal time interval. If the charge halves in 5 s, it halves again in the next 5 s, and again in the next. There is no fixed amount lost per second; there is a fixed proportion.

Identifying exponential decay experimentally — plot the natural logarithm. Taking logs of Q = Q₀e^(−t/RC) gives

ln Q = ln Q₀ − t / RC

so a graph of ln Q against t is a straight line with gradient −1/RC and intercept ln Q₀. A straight log plot is the evidence that the decay is genuinely exponential, and its gradient gives the time constant.