Editing Gauss's law (section)

===Deriving Coulomb's law from Gauss's law===

Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the [[Curl (mathematics)|curl]] of {{math|'''E'''}} (see [[Helmholtz decomposition]] and [[Faraday's law of induction|Faraday's law]]). However, Coulomb's law ''can'' be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).

{{math proof|title=Outline of proof
|proof=
Taking {{mvar|S}} in the integral form of Gauss's law to be a spherical surface of radius {{mvar|r}}, centered at the point charge {{mvar|Q}}, we have

<math display="block">\oint_S\mathbf{E}\cdot d\mathbf{A} = \frac{Q}{\varepsilon_0} </math>

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
<math display="block">4\pi r^2\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r}) = \frac{Q}{\varepsilon_0}</math>
where {{math|'''r̂'''}} is a [[unit vector]] pointing radially away from the charge. Again by spherical symmetry, {{math|'''E'''}} points in the radial direction, and so we get
<math display="block">\mathbf{E}(\mathbf{r}) = \frac{Q}{4\pi \varepsilon_0} \frac{\hat{\mathbf{r}}}{r^2}</math>
which is essentially equivalent to Coulomb's law. Thus the [[inverse-square law]] dependence of the electric field in Coulomb's law follows from Gauss's law.
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