Fermi mechanism and cosmic ray energetics: One mechanism discussed relative to the production of very energetic cosmic rays, is that charged particles can interact with time-dependent magnetic fields acting over long distances, eventually gaining substantial amounts of energy. Assume that particles of fundamental charge e experience magnetic fields of strength B due to shock waves which are moving at near relativistic speeds (v ∼ c) and interact with them over a physical distance scale d. The dimensional analysis problem is then how that energy, E, could depend on e, c, B, and d. As always, write E ∝ c α e β Bγ d δ , and solve for α, β, γ, δ by matching dimensions.