Physics with Calculus/Electromagnetism/Continuous Charge Distributions

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:

$\mathbf {E(\mathbf {r} } )=k_{e}\sum _{n=1}^{N}{\frac {q_{n}}{|\mathbf {r} -\mathbf {r} _{n}|^{2}}}{\frac {\mathbf {r} -\mathbf {r} _{n}}{|\mathbf {r} -\mathbf {r} _{n}|}}\rightarrow k_{e}\int d^{3}r'{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|^{2}}}{\frac {\mathbf {r} -\mathbf {r'} }{|\mathbf {r} -\mathbf {r} '|}},$

where $\rho$ is charge density, and

${\frac {\mathbf {r} -\mathbf {r'} }{|\mathbf {r} -\mathbf {r} '|}}$

is a unit vector pointing from the source point at $\mathbf {r} '$ to the field point at $\mathbf {r}$.