Gauss' law of electrostatics

In the previous section, we established mathematical equivalence between electric field of a point charge $Q$ and lines emanating from a point-source. The key condition that needed to be satisfied was

\[ N = \frac{Q}{\epsilon_{0}} \]

We now consider the implications of this condition. It is clear from the outset that this condition establishes a one-to-one relationship between the total number of lines \(N\) and the magnitude of the point change \(Q\). The higher the \(Q\), the more the number of lines \(N\); and \(N\) depends on \(Q\) only. We had used a spherical enclosure in the previously section for its mathematical simplicity, but had we used an enclosure with irregular surface such as shown in figure 3, the total number of lines \(N\) passing through this irregular would have remained unaltered since \(N\) depends solely on \(Q\). Let's try to express this fact mathematically.

In case of a sphere, we can write the total number of lines as \(N = \rho \times 4 \pi r^{2}\), where \(\rho\) is the density of lines on the surface of the sphere and \(4 \pi r^{2}\) is the total surface area. In case of an irregular surface, however, we cannot simply take the product of the total surface area and the density \(-\) the surface being irregular, the density of lines is different in different regions of the surface. In such a case, we evaluate the number of lines passing through infinitesimal area elements \(dA\) on the surface and then add (integrate) all of them to get an expression for the total number of lines:

\[ dN = \rho (\vec{r}) \, dA \]

where \(dN\) is the number of lines passing through the area \(dA\) located at \(\vec{r}\) with respect to the point-source and \(\rho (\vec{r})\) is the density of lines at \(\vec{r}\). We writing the above equation, we have ignored the fact that the area elements \(dA\) in different regions of the surface intersect with lines at different angles. In such a case, the correct equation for \(dN\) is

\[ dN = \vec{\rho}(\vec{r}) \cdot \vec{dA} \]

where \(\vec{\rho} (\vec{r})\) is the density of lines at \(\vec{r}\) in vector form and \(\vec{dA}\) is the area element. The total number lines is

\[ N = \oint dN = \oint \vec{\rho}(\vec{r}) \cdot \vec{dA} \]

where \(\oint \) represents integration over a closed surface such as we have considered in this and the previous section. Using the equivalence between electric field and lines emanating from a point source, we get

\[ \frac{Q}{\epsilon_{0}} = \oint \vec{E}(\vec{r}) \cdot \vec{dA} \]

This expression is known as Gauss' law of electrostatics. This equation holds as long as the surface, however irregular, is closed.