# Maxwell’s First Equation in Electrostatics

From the Divergence theorem, we have

$\overrightarrow{\bigtriangledown&space;}.\overrightarrow{D}=&space;div&space;\overrightarrow{D}=(&space;\frac{\partial&space;D&space;_{x}}{\partial&space;x}+&space;\frac{\partial&space;D&space;_{y}}{\partial&space;y}+&space;\frac{\partial&space;D&space;_{z}}{\partial&space;z})$

$\lim_{dv->0}\frac{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}{dv}=\lim_{dv->0}(&space;\frac{\partial&space;D&space;_{x}}{\partial&space;x}+&space;\frac{\partial&space;D&space;_{y}}{\partial&space;y}+&space;\frac{\partial&space;D&space;_{z}}{\partial&space;z})$

${\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}=\(&space;\frac{\partial&space;D&space;_{x}}{\partial&space;x}+&space;\frac{\partial&space;D&space;_{y}}{\partial&space;y}+&space;\frac{\partial&space;D&space;_{z}}{\partial&space;z})dv$

from Gauss’s law $\oint_{s}\overrightarrow{D}.\overrightarrow{ds}&space;=Q_{enclosed}$

${\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}=\(&space;\frac{\partial&space;D&space;_{x}}{\partial&space;x}+&space;\frac{\partial&space;D&space;_{y}}{\partial&space;y}+&space;\frac{\partial&space;D&space;_{z}}{\partial&space;z})dv&space;=&space;Q_{enclosed}$

dividing it by $dv(or)&space;\Delta&space;v$ differential volume on both sides

$\frac{{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}}{dv}&space;=&space;\frac{Q_{enclosed}}{dv}$

by applying limit on both  sides

$\lim_{dv->0}\frac{{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}}{dv}&space;=&space;\lim_{dv->0}&space;\frac{Q_{enclosed}}{dv}$

$div\overrightarrow{D}&space;=&space;\rho&space;_{v}$

$\overrightarrow{\bigtriangledown&space;}.\overrightarrow{D}&space;=&space;\rho&space;_{v}$

This equation is known as Maxwell’s first equation and is also known as point form of Gauss’s law /Differential form of Gauss’s law.

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