The electric field intensity associated with a single particle bearing charge \(q_1\), located at the origin, is (Section 5.1)
\begin{equation}
{\bf E}({\bf r}) = \hat{\bf r}\frac{q_1}{4\pi\epsilon r^2} \tag{5.7}\end{equation}
If this particle is instead located at some position \({\bf r}_1\), then the above expression may be written as follows:
\begin{equation}
{\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|}~\frac{q_1}{4\pi\epsilon \left|{\bf r}-{\bf r}_1\right|^2} \tag{5.8}\end{equation}
or, combining like terms in the denominator:
\begin{equation}
{\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|^3}~\frac{q_1}{4\pi\epsilon} \tag{5.9}\end{equation}
Now let us consider the field due to multiple such particles. Under the usual assumptions about the permittivity of the medium (reminder: Section 2.8), the property of superposition applies. Using this principle, we conclude:
The electric field resulting from a set of charged particles is equal to the sum of the fields associated with the individual particles.
Stated mathematically:
\begin{equation}
{\bf E}({\bf r}) = \sum_{n=1}^{N}{\bf E}({\bf r};{\bf r}_n) \tag{5.10}\end{equation}
where \(N\) is the number of particles. Thus, we have
\begin{equation}
{\bf E}({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{{\bf r}-{\bf r}_n}{\left|{\bf r}-{\bf r}_n\right|^3}~q_n} \tag{5.11}\end{equation}