If N charges is present, the electric field is obtained by summing over the contributions of each charge. This can be converted into an integral:
E ( r ) = k e ∑ n = 1 N q n | r − r n | 2 r − r n | r − r n | → k e ∫ d 3 r ′ ρ ( r ′ ) | r − r ′ | 2 r − r ′ | r − r ′ | , {\displaystyle \mathbf {E(\mathbf {r} } )=k_{e}\sum _{n=1}^{N}{\frac {q_{n}}{|\mathbf {r} -\mathbf {r} _{n}|^{2}}}{\frac {\mathbf {r} -\mathbf {r} _{n}}{|\mathbf {r} -\mathbf {r} _{n}|}}\rightarrow k_{e}\int d^{3}r'{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{2}}}{\frac {\mathbf {r} -\mathbf {r'} }{|\mathbf {r} -\mathbf {r} '|}},}
where ρ {\displaystyle \rho } is charge density, and
r − r ′ | r − r ′ | {\displaystyle {\frac {\mathbf {r} -\mathbf {r'} }{|\mathbf {r} -\mathbf {r} '|}}}
is a unit vector pointing from the source point at r ′ {\displaystyle \mathbf {r} '} to the field point at r {\displaystyle \mathbf {r} } .
is charge density, and
to the field point at 
.
