The electric field intensity associated with a single particle bearing charge $$q_1$$, located at the origin, is (Section [m0102_Coulombs_Law]) ${\bf E}({\bf r}) = \hat{\bf r}\frac{q_1}{4\pi\epsilon r^2}$ If this particle is instead located at some position $${\bf r}_1$$, then the above expression may be written as follows: ${\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|}~\frac{q_1}{4\pi\epsilon \left|{\bf r}-{\bf r}_1\right|^2}$ or, combining like terms in the denominator: ${\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|^3}~\frac{q_1}{4\pi\epsilon}$
Stated mathematically: ${\bf E}({\bf r}) = \sum_{n=1}^{N}{\bf E}({\bf r};{\bf r}_n)$ where $$N$$ is the number of particles. Thus, we have ${\bf E}({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{{\bf r}-{\bf r}_n}{\left|{\bf r}-{\bf r}_n\right|^3}~q_n}$