15.2: Maxwell's First Equation
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Maxwell's first equation, which describes the electrostatic field, is derived immediately from Gauss's theorem, which in turn is a consequence of Coulomb's inverse square law. Gauss's theorem states that the surface integral of the electrostatic field D over a closed surface is equal to the charge enclosed by that surface. That is
∫surfaceD⋅dσ=∫volumeρdv
Here ρ is the charge per unit volume.
But the surface integral of a vector field over a closed surface is equal to the volume integral of its divergence, and therefore
∫surfacedivDdv=∫volumeρdv
Therefore
divD=ρ,
or, in the nabla notation,
This is the first of Maxwell's equations.