User:Inconspicuum/Physics (A Level)/Half-lives

The half life of something that is decaying exponentially is the time taken for the value of a decaying variable to halve.

Half Life of a Radioisotope
The most common use of half-lives is in radioactive decay. The activity is given by the equation:

$$A_t = A_0e^{-\lambda t}$$

At t=t½, At = ½A0, so:

$$\frac{A_0}{2} = A_0e^{-\lambda t_{\frac{1}{2}}} = \frac{A_0}{e^{\lambda t_{\frac{1}{2}}}}$$

$$2 = e^{\lambda t_{\frac{1}{2}}}$$

$$\ln{2} = \lambda t_{\frac{1}{2}}$$

Therefore:

$$t_{\frac{1}{2}} = \frac{\ln{2}}{\lambda}$$

It is important to note that the half-life is completely unrelated to the variable which is decaying. At the end of the half-life, all decaying variables will have halved. This also means that you can start at any point in the decay, with any value of any decaying variable, and the time taken for the value of that variable to halve from that time will be the half-life.

Half-Life of a Capacitor
You can also use this formula for other forms of decay simply by replacing the decay constant &lambda; with the constant that was in front of the t in the exponential relationship. So, for the charge on a capacitor, given by the relationship:

$$Q_t = Q_0e^{\frac{-t}{RC}}$$

So, substitute:

$$\lambda = \frac{1}{RC}$$

Therefore, the half-life of a capacitor is given by:

$$t_{\frac{1}{2}} = RC\ln{2}$$

Time Constant of a Capacitor
However, when dealing with capacitors, it is more common to use the time constant, commonly denoted &tau;, where:

$$\tau = RC = \frac{t_{\frac{1}{2}}}{\ln{2}}$$

At t = &tau;:

$$Q_t = Q_0e^{\frac{-RC}{RC}} = \frac{Q_0}{e}$$

So, the time constant of a capacitor can be defined as the time taken for the charge, current or voltage from the capacitor to decay to the reciprocal of e (36.8%) of the original charge, current or voltage.