Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 4 / f / Solution

The velocity representation givesThereforeIn the launching region, $r\simeq r_0\ll r_a$ and $A\ll 1$, so part (d) givesConsequently

The modified Bernoulli kinetic energy isThus a decrease in ${\Phi }_{cg}$ is shared between poloidal acceleration and relative toroidal motion. If $R$ is approximately constant, the same potential drop produces a poloidal kinetic increase smaller by the factor $1/(1+R^2)$ than the estimate that neglects the toroidal term. If $R$ varies strongly, that term can absorb or return energy, so monotonic decrease of ${\Phi }_{cg}$ alone no longer proves an equally direct increase of $u_p^2/2$.