= Solution
In microscopic SI form, <Maxwell's equations> couple charge <density> $\rho$ and current <density> $\mathbf J$ to electric and <magnetic fields>:
$$
\nabla\cdot\mathbf E=\frac\rho{\epsilon_0},\qquad
\nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\partial_t\mathbf B,\qquad
\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\partial_t\mathbf E.
$$
Gauss's electric law expresses electric flux from charge and reduces to the inverse-square Coulomb field for a point charge. Gauss's magnetic law gives zero net <magnetic flux> through every closed surface; <magnetic field> lines do not begin or end on magnetic monopoles in this theory.
Faraday's induction law says changing <magnetic flux> produces a circulating <electric field>. For a fixed loop, its circulation is minus the rate of change of enclosed <magnetic flux>, explaining the sign of induction and <Lenz's law>. For example, changing a primary coil's <electric current> changes the flux through a stationary secondary coil and induces a <voltage>. A moving wire also has the magnetic part of the <Lorentz force> on its charges; the fixed-loop field equation and the full motional-emf calculation must be distinguished.
The Ampère-Maxwell law relates magnetic circulation to conduction current and <displacement current>. The latter is essential around a charging capacitor: a surface cutting the lead and another passing through the gap must predict the same circulation. Taking the divergence gives
$$
\nabla\cdot\mathbf J+\partial_t\rho=0,
$$
using Gauss's electric law. <Charge conservation> therefore fixes the consistency of the time-dependent equations. In matter, the equivalent macroscopic equations use $\mathbf D$ and $\mathbf H$, free sources and constitutive relations. The <Lorentz force> <density> $\rho\mathbf E+\mathbf J\times\mathbf B$ describes exchange of <momentum> with charges.
In source-free vacuum, take the curl of Faraday's law and use $\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$. Both fields satisfy
$$
\boxed{\nabla^2\mathbf E=\mu_0\epsilon_0\partial_t^2\mathbf E,\qquad
\nabla^2\mathbf B=\mu_0\epsilon_0\partial_t^2\mathbf B,
\qquad c=\frac1{\sqrt{\mu_0\epsilon_0}}.}
$$
For a <plane wave>, $\omega=ck$, $\mathbf k\cdot\mathbf E=\mathbf k\cdot\mathbf B=0$ and $\mathbf B=\hat{\mathbf k}\times\mathbf E/c$. The fields are transverse, mutually perpendicular and in phase. Their <energy density> and <energy flux> are
$$
u=\frac{\epsilon_0E^2}{2}+\frac{B^2}{2\mu_0},\qquad
\mathbf S=\frac{\mathbf E\times\mathbf B}{\mu_0},\qquad
\partial_tu+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E.
$$
This is <Poynting's theorem>: field <energy> is transported and can do <work> on matter. \b[<Electromagnetic radiation> is a propagating, energy-carrying solution of the coupled electric and <magnetic field> equations.]
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