= Solution
Differentiate $\mathbf u^E=\frac12\nabla\times\mathbf E$ with respect to time. The vacuum <Maxwell equations> give $\partial_t\mathbf E=c^2\nabla\times\mathbf B$, so
$$
\partial_t\mathbf u^E=\frac{c^2}{2}\nabla\times(\nabla\times\mathbf B).
$$
The <curl of the curl identity> is $\nabla\times(\nabla\times\mathbf B)=\nabla(\nabla\cdot\mathbf B)-\nabla^2\mathbf B$. Since $\nabla\cdot\mathbf B=0$,
$$
\boxed{\frac{\partial\mathbf u^E}{\partial t}=-\frac{c^2}{2}\nabla^2\mathbf B}.
$$
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