Electric Field Intensity due to a System of Charges (Principle of Superposition)
Question: Find an expression for electric field intensity at a point due to a system of point charges.
Principle of Superposition of Electric Fields: According to the principle of superposition, net electric field strength at a point due to a group of point charges is equal to the vector sum of all the electric field strengths produced due to individual point charges at that point.
Suppose we have $n$ point charges $q_1, q_2, \ldots, q_n$ such that their position vectors are $\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_n$ respectively. Let $q_0$ be the positive test charge at point $P$ where the total electric field $\vec{E}$ due to $n$ charges is to be determined. Let the position vector of the point $P$ be $\vec{r}$ .
Electric fields due to point charges $q_1, q_2, \ldots, q_n$ respectively at point $P$ are given by,
$\vec{E}_1= \frac{1}{4\pi\epsilon_0} \frac{q_1(\vec{r} - \vec{r}_1)}{|\vec{r} - \vec{r}_1|^3} $
$\vec{E}_2= \frac{1}{4\pi\epsilon_0} \frac{q_2(\vec{r} - \vec{r}_2)}{|\vec{r} - \vec{r}_2|^3}$
$\vec{E}_n= \frac{1}{4\pi\epsilon_0} \frac{q_n(\vec{r} - \vec{r}_n)}{|\vec{r} - \vec{r}_n|^3} $
and
Therefore, resultant electric field at point P is given by
$\vec{E} = \vec{E}_1 + \vec{E}_2 + \dots + \vec{E}_n$
$\vec{E}= \frac{1}{4\pi\epsilon_0} \frac{q_1(\vec{r} - \vec{r}_1)}{|\vec{r} - \vec{r}_1|^3}+ \frac{1}{4\pi\epsilon_0} \frac{q_2(\vec{r} - \vec{r}_2)}{|\vec{r} - \vec{r}_2|^3}+$
$\dots + \frac{1}{4\pi\epsilon_0} \frac{q_n(\vec{r} - \vec{r}_n)}{|\vec{r} - \vec{r}_n|^3}$
$\vec{E}= \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i(\vec{r} - \vec{r}_i)}{|\vec{r} - \vec{r}_i|^3} \qquad \dots (i)$
or
which is the required expression for the electric field intensity due to a system of $n$ point charge.