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.