The Time Constant

Time constant (τ) — the product of resistance and capacitance:

τ = R C

measured in seconds. Ohms multiplied by farads genuinely gives seconds, which is worth checking once as a reassurance that the formula is dimensionally sound.

What it means — after one time constant, the charge has fallen to 1/e of its initial value, about 37%. Equivalently, it has lost 63% of what it started with.

After successive time constants — 37%, then 13.5%, then 5.0%, then 1.8%. A capacitor is usually treated as fully discharged after about 5τ, by which point under 1% remains.

Half-life — the time for the charge to halve is

t₁⁄₂ = 0.69 τ

which comes from ln 2 = 0.693. The same relationship appears in radioactive decay, because it is the same mathematics.

Example: τ = RC tells you directly how to design a timing circuit. A 100 kΩ resistor with a 100 μF capacitor gives τ = 10⁵ × 10⁻⁴ = 10 s. To make a circuit slower, increase either component; to make it faster, decrease either. This is the basis of timing in everything from windscreen-wiper intervals to camera shutters.

Why a larger R or C means a slower discharge — a larger resistance restricts the current, so the charge leaves more slowly. A larger capacitance means more charge is stored for the same voltage, so there is more to remove at a given rate. Both lengthen the process.

Finding τ from a graph — read off the time at which the quantity has fallen to 37% of its initial value, or take the gradient of a ln-plot, where the gradient is −1/τ. The gradient method uses all the data rather than a single point and is the more reliable of the two.