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 }
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} '}
r
{\displaystyle \mathbf {r} }
