Kishan Sinha

The interaction between charge \(Q\) and \(q\) is described by Coulomb's inverse-square law: \[ \vec{F}_{q} = \frac{1}{4 \pi \epsilon_{0}} \frac{Qq}{r^{2}} \hat{r}_{q} \] where \(\vec{F}_{q}\) is the force on charge \(q\) due to \(Q\) and \(\hat{r}\) is the unit vector along the position vector of \(q\) with respect to \(Q\). The electrostatic force on \(Q\) due to \(q\) is equal in magnitude but opposite in direction to \(\vec{F}_{q}\). Everything we need to know to study interactions between static charges is contained in the Coulomb's law. However, in its present form the Coulomb's law does not lend itself to efficient use. For instance, say we want to study the effect of charge \(Q\) on \(q\). We will here employ Coulomb's to figure out the magnitude and direction of the force on \(q\) due to \(Q\). If we replace \(q\) with a different charge, say, \(q'\), we will have to evaluate the Coulomb's law all over again! We will need to repeat ...

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 = \rh...

There is an interesting congruence between the mathematical structure of electric field and the geometrical behavior of lines emanating from a point-source. In this section, we will analyze this congruence and its implications in electrostatics. Let's start by considering a simple case of a point in three-dimensional space with \(N\) number of lines emanating radially outward from the point as shown in the figure below. The lines are distributed uniformly around the point. Now, let's draw consider a spherical shell of radius \(r\) centered at this point. The number of lines per unit surface area of the sphere, or the number density \(\rho\) of lines, is: \[ \begin{equation*} \rho = \frac{N}{4 \pi r^{2}} \end{equation*} \] where \(4 \pi r^{2}\) is the surface area of the sphere. Thus, it follows that as the radius of the spherical shell increases, the density of lines (\(\rho\), lines per unit area of the shell) decreases as \(\frac{1}{r^{2}}\). This is becaus...