= Solution
A <linear force-free magnetic field> has a spatially constant proportionality factor, $\nabla\times\mathbf B=\lambda\mathbf B$. Taking its <curl> and using $\nabla\cdot\mathbf B=0$ gives
$$
-\nabla^2\mathbf B
=\nabla\times(\nabla\times\mathbf B)
=\lambda\nabla\times\mathbf B=\lambda^2\mathbf B.
$$
Hence
$$
\boxed{\nabla^2\mathbf B+\lambda^2\mathbf B=0.}
$$
The constant $\lambda$ has dimensions of inverse length in this convention. If it varied in space, the extra term $\nabla\lambda\times\mathbf B$ would remain, so the same <Helmholtz equation> would not follow merely from a <Force-free magnetic field>[nonlinear force-free condition].
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