Question

Explain MAXWELL first equation.

Answer 1

$\int \vec{E} \cdot d\vec{A}=\frac{q}{\epsilon_0}$

This is Maxwell’s first equation. It represents completely covering the surface with a large number of tiny patches having areas $d\vec{A}.$ We represent these small areas as vectors pointing outwards, because we can then take the dot product with the electric field to select the component of that field pointing perpendicularly outwards (it would count negatively if the field were pointing inwards)—this is the only component of the field that contributes to actual flow across the surface. (Just as a river flowing parallel to its banks has no flow across the banks).