State and prove Gauss Theorem .

Deriving Gauss's law from Coulomb's law

Gauss's law can be derived from Coulomb's law, which states that the electric field due to a stationary point charge is:

$mathbf{E}(mathbf{r}) = frac{q}{4pi epsilon_0} frac{mathbf{e_r}}{r^2}$

where

e_r is the radial unit vector,

r is the radius, |r|,

$epsilon_0$ is the electric constant,

q is the charge of the particle, which is assumed to be located at the origin.

Using the expression from Coulomb's law, we get the total field at r by using an integral to sum the field at r due to the infinitesimal charge at each other point s in space, to give

$mathbf{E}(mathbf{r}) = frac{1}{4piepsilon_0} int frac{rho(mathbf{s})(mathbf{r}-mathbf{s})}{|mathbf{r}-mathbf{s}|^3} , d^3 mathbf{s}$

where ρ is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem^[5]

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