= Solution
In a source-free <vacuum>, the <Maxwell equations> in <SI> are
$$
\nabla\cdot\mathbf E=0,\qquad \nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\qquad
\nabla\times\mathbf B=\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t},
$$
where $\epsilon_0$ and $\mu_0$ are the <vacuum permittivity> and <vacuum permeability>. Take the <curl> of the last equation, substitute <Faraday's law of induction>, and use the <curl of the curl identity>:
$$
\nabla\times(\nabla\times\mathbf B)
=\nabla(\nabla\cdot\mathbf B)-\nabla^2\mathbf B
=-\mu_0\epsilon_0\frac{\partial^2\mathbf B}{\partial t^2}.
$$
The zero <divergence> therefore gives the <wave equation>
$$
\boxed{\nabla^2\mathbf B=\frac1{c^2}\frac{\partial^2\mathbf B}{\partial t^2},\qquad c=\frac1{\sqrt{\mu_0\epsilon_0}}.}
$$
For a propagating <plane wave>, write $\mathbf B=\operatorname{Re}[\mathbf B_0e^{i(\mathbf k\cdot\mathbf r-\omega t)}]$. The <wave equation> gives $\omega^2=c^2|\mathbf k|^2$, and <Gauss's law for magnetism> gives $\mathbf k\cdot\mathbf B_0=0$. Thus the oscillating <magnetic field> is a <transverse wave>: it is perpendicular to the propagation <wavevector>. A static background <magnetic field>, if present, is separate from this propagating component.
Back to article page
