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:

$$\mathbf{E(\mathbf r})=k_e\sum_{n=1}^N \frac{q_n}{|\mathbf r-\mathbf r_n|^2} \frac{\mathbf{r}-\mathbf{r}_n}{|\mathbf{r}-\mathbf{r}_n|} \rightarrow k_e\int d^3r'\frac{\rho (\mathbf r')}{|\mathbf r-\mathbf r'|^2} \frac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r}'|}, $$

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 $$\mathbf r$$.