= Gauss's law
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{title2=$\oint_{\partial V}\mathbf E\cdot\mathbf n\,dS=Q_{\mathrm{enclosed}}/\varepsilon_0$}
{wiki}
= Gauss law
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{synonym}
<Gauss's law> relates the <electric field> on a <closed surface> to the net <electric charge> inside it. In <SI> units, with outward <unit normal> $\mathbf n$,
$$
\boxed{\oint_{\partial V}\mathbf E\cdot\mathbf n\,dS=\frac{Q_{\mathrm{enclosed}}}{\varepsilon_0},\qquad Q_{\mathrm{enclosed}}=\int_V\rho\,dV.}
$$
Here $\rho$ describes a volume <charge density>, and $\varepsilon_0$ is the <vacuum permittivity>; for ideal point or surface charges, use the enclosed <electric charge> directly rather than an ordinary volume-density integral. The left side is <electric flux>: the <surface integral> of the outward normal component of the <electric field>. Field directed outward contributes positively; field directed inward contributes negatively. The equality uses net signed <electric charge>, so positive and negative enclosed charges can cancel. The <Gaussian surface> is an imagined accounting boundary, not necessarily a material object. See https://openstax.org/books/university-physics-volume-2/pages/6-2-explaining-gausss-law[OpenStax, explaining Gauss's law].
Where the <electric field> and <charge density> are smooth, the local statement is $\nabla\cdot\mathbf E=\rho/\varepsilon_0$. The <divergence theorem> connects this <divergence> to the <surface integral>. This is the satisfying connection between local sources and global accounting: a surface can register how much net <electric charge> it encloses without revealing where that charge is located within it. A <point charge> requires care with the singular field at its position; the integral formulation remains useful with a boundary that avoids the charge.
The law fixes total <electric flux>, rather than every value of the <electric field>. In an uncharged region, a uniform <electric field> $\mathbf E=(E_0,0,0)$ crosses a cube of side $\ell$. Its left and right faces contribute $-E_0\ell^2$ and $+E_0\ell^2$; the other four faces contribute zero. Thus \b[the net electric flux is zero while the electric field is nonzero everywhere]. Charges outside a <Gaussian surface> can change the field on it without changing its net <electric flux>. Conversely, fields cannot generally be reconstructed from enclosed <electric charge> alone. <Electrostatics> also requires the <electric field> to be the negative <gradient> of an <electric potential>, with appropriate boundary information. This distinction is emphasized in https://www.feynmanlectures.caltech.edu/II_05.html[Feynman lectures II.5, application of Gauss' law].
\Image[/gaussian-flux-and-field.png]
{title=Zero net flux can coexist with a nonzero uniform field, and equal enclosed charge can give different field magnitudes on different spheres}
<Spherical symmetry> supplies the extra information needed for a particularly simple calculation. An isolated <spherically symmetric> charge distribution in vacuum produces a radial <electric field>, constant in magnitude on each concentric <sphere>. Its <electric flux> is therefore $4\pi r^2E_r(r)$, giving
$$
E_r(r)=\frac{Q_{\mathrm{enclosed}}(r)}{4\pi\varepsilon_0r^2}.
$$
Outside the distribution, doubling $r$ quadruples the <surface area> and divides the radial <electric field> by four, while the <electric flux> stays $Q/\varepsilon_0$. The inverse-square dependence is an area-dilution result. The <electric field of a uniformly charged ball> and <electric field of a uniformly charged spherical shell> show that the same total charge can give different interior fields. These symmetry arguments and their limits are developed in https://openstax.org/books/university-physics-volume-2/pages/6-3-applying-gausss-law[OpenStax, applying Gauss's law].
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