Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S19 / 3 / b / i / Solution

The instantaneous beam lifetime iswhere losses make $dN/dt<0$. If the fractional loss rate is constant, $N(t)=N_0{e}^{-t/\tau }$ and $\tau$ is the time to fall to $1/e$ of the starting population. At fixed revolution frequency, the circulating current has the same exponential decay. Density-dependent loss mechanisms can make the instantaneous lifetime vary, so a single exponential need not fit every fill.