= Magnetic free-decay spectral bound
{title2=$\int_D|\nabla\times\mathbf B|^2\geq\frac{\pi^2}{R^2}\int_{\mathbb R^3}|\mathbf B|^2$}
For a <solenoidal vector field> carrying current only in a conductor contained in a sphere of radius $R$, match the exterior field to a decaying potential field and include exterior <magnetic energy>. The smallest free-decay <eigenvalue> of the enclosing sphere is $\pi^2/R^2$, from its dipolar poloidal mode. The corresponding variational estimate gives the displayed bound. Continuity and the usual insulating-interface magnetic conditions are part of the admissible field class. This bound is the diffusive ingredient in <Backus' necessary condition for dynamo action>.
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