= Solution
In <electrostatic equilibrium> a conductor has $\mathbf E=0$ throughout its bulk and constant <electric potential> on each connected component. Otherwise free charge would continue moving. <Gauss's law> then excludes bulk excess charge; it resides on surfaces. Just outside a conductor the tangential field vanishes and the normal jump is
$$
\boxed{\mathbf E_{\rm out}=\frac{\sigma}{\epsilon_0}\widehat{\mathbf n}}
$$
for a vacuum interface with outward normal $\widehat{\mathbf n}$. Surface charge rearranges in an external field to enforce these <boundary conditions>. Greater curvature often concentrates charge, but the actual distribution is a global boundary-value problem.
An empty closed cavity inside a conductor has constant potential and zero field, even in an external electrostatic field. If a charge $q$ is placed in the cavity, a <Gaussian surface> in the metal shows that the cavity wall carries total induced charge $-q$; its detailed distribution depends on geometry. An initially neutral isolated conductor then has $+q$ on its exterior surface. A grounded conductor can exchange charge with <Earth>, so this isolation condition no longer applies.
A useful example is a charge $q$ at height $d$ above an infinite grounded plane. In the physical half-space, the <method of images> replaces the induced charges by an image $-q$ at height $-d$, giving the attractive <force> $q^2/(16\pi\epsilon_0d^2)$. The image is a mathematical construction, not an actual charge behind the conductor. \b[Equipotential surfaces and <charge conservation> determine conductor fields together.]
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