= Electrostatics
{wiki}
<Electrostatics> studies time-independent <electric charge> distributions and their <electric fields>. In <vacuum>, its local equations in <SI> units are
$$
\nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0},\qquad \nabla\times\mathbf E=\mathbf0,\qquad \mathbf E=-\nabla\Phi,\qquad \nabla^2\Phi=-\frac{\rho}{\epsilon_0}.
$$
Here $\rho$ is <charge density>, $\epsilon_0$ is <vacuum permittivity>, and $\Phi$ is <electric potential>. <Gauss's law> supplies the <divergence> equation; the static form of <Faraday's law of induction> supplies the zero <curl>. The <Poisson equation> becomes <Laplace's equation> where there is no <electric charge>. Prescribed <electric charges> give <electric fields> through <Coulomb's law> and the <superposition principle>; mobile <electric charges> on an <electrical conductor> instead require solving for their distribution together with the <electric potential>. https://farside.ph.utexas.edu/teaching/em/lectures/node40.html[Richard Fitzpatrick, Electrostatics and magnetostatics: field equations and boundary data]{ref}.
The useful distinction is between prescribed sources and prescribed <boundary conditions>. On a grounded <electrical conductor>, an external connection fixes its <electric potential> relative to an <electrical reference potential> and allows <electric charge> to enter or leave. On an isolated <electrical conductor>, total <electric charge> is fixed and its constant <electric potential> is an unknown. A neutral isolated <electrical conductor> can therefore have a nonzero local <surface charge density> through <electrostatic induction>. Equal <electric potentials> also do not imply equal <electric charges>: <charge sharing between distant connected spheres> gives a simple geometric example. https://www.feynmanlectures.caltech.edu/II_06.html#Ch6-S6[Feynman, Leighton and Sands, II.6: charged conductors and connected-sphere estimates]{ref}.
<Electrostatics> is useful because a spatial <boundary value problem> replaces the need to follow every moving <charge carrier> during <charge relaxation>. It describes the settled state of a <capacitor> and explains <electrostatic shielding> of sensitive equipment. Its assumptions must match the apparatus: stationary boundaries and sources, negligible time variation of the <magnetic field>, and enough time for <charge relaxation>. A metal carrying a maintained <electric current> through finite <electrical resistance> generally needs a nonzero internal <electric field> by <Ohm's law>; it is not a <conductor in electrostatic equilibrium>. Static macroscopic <electric charge> also does not mean that every <electron> is motionless. https://www.feynmanlectures.caltech.edu/II_07.html#Ch7-S1[Feynman, Leighton and Sands, II.7.1: conductor potentials as boundary-value problems]{ref}.
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