 Backus' necessary condition for dynamo action

For an isolated bounded conductor of uniform magnetic diffusivity $\eta >0$, contained in a sphere of radius $R$ and matched to a decaying potential field in an insulating exterior, dynamo action requires maximum stretching rate $S\ge \eta {\pi }^2/R^2$. Here $S$ bounds the largest eigenvalue of the rate-of-strain tensor throughout the flow and time. Use fluid boundary conditions eliminating the stretching boundary term, for example a no-slip boundary condition, and no imposed energy input. The proof combines the magnetic free-decay spectral bound with the magnetic energy equation. This is a necessary condition, not a sufficiency criterion; changing magnetic boundary conditions changes the spectral constant.