 Solution

Use $k_{\mathrm{B}}{T}_p=1000$ eV and $k_{\mathrm{B}}{T}_b=100$ eV. The nonrelativistic beam speed and cross-sectional area areHere $V/{\sigma }_p=\sqrt{20}$, ${\sigma }_b^2/{\sigma }_p^2=0.1$, and the exponential in the requested threshold is $e^{-10}$. With $C=\sqrt{e}$,Converting beam density to the magnitude of its electric current givesThis is an order-of-magnitude estimate from the source's asymptotic formula. The numerical hierarchy is only moderate: $V{\sigma }_b/{\sigma }_p^2=\sqrt{2}$. Retaining the background slope at $V-{\sigma }_b$ instead gives about $9.1\times 10^4$ A as a sufficient positive-slope current at that chosen velocity. The difference quantifies the sensitivity of the exponential-tail approximation; an exact kinetic instability threshold requires the dispersion relation. The requested estimate is therefore on the scale of tens of kiloamperes, rather than a precisely determined current.