The Time Constant

How long will it take a capacitor in a circuit to discharge completely?	How long will it take all of the pressure in the punctured tyre to leak away?	How long will it take all of the unstable atoms in the radioactive substance to decay?

Not only is it very difficult to answer these questions (as these decays are exponential) but the answer may not be particularly useful.

A very useful quantity is the "half life". This is the time it takes for an quantity to fall to half its initial value. The half life of a radioactive substance is a useful quantity to know.

In the case of a capacitor discharging the "time constant" is just as useful. This is the time it takes for the charge on the capacitor to fall to a value equal to the fraction e^-1 (or about 37%) of its initial value. It is equal to RC, the product of the capacitance and the resistance of the circuit, and has the units seconds.

Example
A 400 µF capacitor is charge using a 10V d.c. supply then discharged through a 5 kΩ resistor. calculate the time constant RC of the discharge. Will the capacitor be fully discharged after a minute?

RC = 5 x 10³  x  400 x 10^-6 = 10s       so after 10s the voltage across the capacitor will fall from 10V to around 3.7V

A minute is 10 time constants so the voltage will be 10 x (0.37)¹⁰ =  0.0005V, virtually nothing.