Solution
= Solution
The instantaneous <beam lifetime> is
$$
\boxed{\tau=-\frac{N}{dN/dt},}
$$
where losses make $dN/dt<0$. If the fractional loss rate is constant, $N(t)=N_0e^{-t/\tau}$ 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.