= Solution
The <electric potential> and <magnetic vector potential> represent the fields through $\mathbf B=\nabla\times\mathbf A$ and $\mathbf E=-\nabla\phi-\partial_t\mathbf A$. They are not unique: a differentiable function $\chi$ gives
$$
\boxed{\mathbf A\mapsto\mathbf A+\nabla\chi,\qquad
\phi\mapsto\phi-\partial_t\chi,}
$$
without changing either field. <Gauge fixing> restricts this freedom to suit the sources, boundaries or symmetry; it does not alter observables.
The <Lorenz gauge> $\nabla\cdot\mathbf A+c^{-2}\partial_t\phi=0$ is <Lorentz covariant>. <Maxwell's equations> then reduce to sourced <wave equations>,
$$
(\nabla^2-c^{-2}\partial_t^2)\phi=-\rho/\epsilon_0,\qquad
(\nabla^2-c^{-2}\partial_t^2)\mathbf A=-\mu_0\mathbf J.
$$
Retarded solutions implement causal radiation conditions. This gauge is useful for radiation and relativistic calculations; residual transformations obey the homogeneous <wave equation>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[§18–6, Eqs. (18.23)–(18.25): gauge choice and potential wave equations]{ref}
The <Coulomb gauge> $\nabla\cdot\mathbf A=0$ instead gives $\nabla^2\phi=-\rho/\epsilon_0$. Its <electric potential> is an instantaneous Poisson solution, but the accompanying <magnetic vector potential> makes the full physical <electric field> causal. It is useful in separating longitudinal <electric fields> from transverse radiation. In a source-free <vacuum> region, suitable radiation <boundary conditions> permit <radiation gauge> $\phi=0$, $\nabla\cdot\mathbf A=0$, leaving two transverse plane-wave <polarizations>. These extra conditions are not a gauge choice applicable unchanged to arbitrary nonzero charge distributions.
\b[Different gauges redistribute information between potentials; the same boundary-value problem gives the same <electric fields> and <magnetic fields>.]
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