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.