= Solution
In source-free <vacuum>, <Maxwell's equations> read
$$
\nabla\cdot\mathbf E=0,\quad\nabla\cdot\mathbf B=0,\quad
\nabla\times\mathbf E=-\partial_t\mathbf B,\quad
\nabla\times\mathbf B=\mu_0\epsilon_0\partial_t\mathbf E.
$$
Take the <curl> of the third equation and use the <curl of the curl identity> $\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$. Then
$$
\boxed{\nabla^2\mathbf E=\mu_0\epsilon_0\partial_t^2\mathbf E,\qquad
c=\frac1{\sqrt{\mu_0\epsilon_0}}\simeq2.998\times10^8\,\mathrm{m\,s^{-1}}.}
$$
The prescribed <plane wave> travels in the $+z$ direction. Its <polarization> is along $x$, and <Faraday's law> requires $B_y=(k/\omega)E_x$ with $\omega=ck$. The wave's <magnetic flux density> is therefore
$$
\boxed{\mathbf B(z,t)=\frac{E_0}{c}\cos(kz-\omega t)\hat{\mathbf y},\qquad
B_0=3.34\times10^{-7}\,\mathrm T.}
$$
The direction is also fixed by $\mathbf E\times\mathbf B$ pointing along the propagation direction. A superposed static uniform <magnetic field> is not fixed by the oscillatory <electric field>; the expression above is the <magnetic field> belonging to the wave itself.
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