Physics with Calculus/Electromagnetism/Continuous Charge Distributions

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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^3r'\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.