How field lines depend on area or on solid angle made by area ?

Figure shows a set of field lines.

Imagine two equal and small elements of area placed at points $R$ and $S$ normal to the field lines.

The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points. The picture shows that the field at $R$ is stronger than at $S$. The angle subtended by $\Delta l$ at $\mathrm{O}$ can be approximated as $\Delta \theta=\frac{\Delta l}{r}$. Likewise, in threedimensions, the solid angle $\Delta \Omega=\frac{\Delta S}{r^{2}}$.

In a given solid angle the number of radial field lines is the same.

For two points $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ at distances $r_{1}$ and $r_{2}$ from the charge, the element of area subtending the solid angle $\Delta \Omega$ is $r_{1}{ }^{2} \Delta \Omega$ at $\mathrm{P}_{1}$ and an element of area $r_{2}{ }^{2} \Delta \Omega$ at $\mathrm{P}_{2}$.

The number of lines $n$ cutting these area elements are the same. The number of field lines, cutting unit area element is therefore $\frac{n}{r_{1}^{2} \Delta \Omega}$ at $\mathrm{P}_{1}$ and $\frac{n}{r_{2}^{2} \Delta \Omega}$ at $\mathrm{P}_{2}$, respectively. Since $n$ and $\Delta \Omega$ are common, the strength of the field depends on $\frac{1}{r^{2}}$.