= Solution
For the oscillatory magnetic field, <Faraday's law> gives $\mathbf B=(1/\omega)\mathbf k\times\mathbf E$. Indeed its time derivative equals $-\nabla\times\mathbf E$, and the dispersion relation makes it satisfy the remaining <Maxwell equations>. Thus
$$
\boxed{\mathbf B(z,t)=\frac{kE_0}{\sqrt2\,\omega}(-1,1,0)\cos(kz-\omega t).}
$$
For $\omega=ck$, its amplitude is $E_0/c$. The <magnetic field>, <electric field> and direction of propagation are mutually perpendicular. A constant background magnetic field could be added; the expression above is the wave-associated field.
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