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:

$𝐄\mathbf{(}𝐫)=k_e\sum _{n=1}^N\frac{q_n}{|𝐫-{𝐫}_n{|}^2}\frac{𝐫-{𝐫}_n}{|𝐫-{𝐫}_n|}\to k_e\int d^3{r}^{\prime }\frac{\rho ({𝐫}^{\prime })}{|𝐫-{𝐫}^{\prime }{|}^2}\frac{𝐫-{𝐫}^{\prime }}{|𝐫-{𝐫}^{\prime }|},$

where $\rho$ is charge density, and

$\frac{𝐫-{𝐫}^{\prime }}{|𝐫-{𝐫}^{\prime }|}$

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