15.2: Maxwell's First Equation
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 \(\textbf{D}\) over a closed surface is equal to the charge enclosed by that surface. That is
\[ \int_{\text{surface}} \textbf{D} \cdot \boldsymbol{d\sigma} = \int_{\text{volume}} \rho \, dv \tag{15.2.1} \label{15.2.1}\]
Here \(\rho\) 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
\[ \int_{\text{surface}} \text{div}\, \textbf{D}\, dv = \int_{\text{volume}} \rho \, dv \tag{15.2.2} \label{15.2.2}\]
Therefore
\[\text{div} \textbf{D} = \rho, \tag{15.2.3} \label{15.2.3}\]
or, in the nabla notation,
\[\nabla \cdot \textbf{D} = \rho. \tag{15.2.3} \label{15.2.4}\]
This is the first of Maxwell's equations.