Electric field

Remarks

By Coulomb's law, a test charge $q$ at rest at $x$ feels the force $q E(x)$. Example: a single charge $q$ at the origin has the field $E(x) = q x / (4 \pi \varepsilon_0 \abs{x}^3)$, of size $\abs{q} / (4 \pi \varepsilon_0 r^2)$ at distance $r = \abs{x}$. Non-example: the field $q x / (4 \pi \varepsilon_0 \abs{x}^2)$ is not the field of a point charge in $\R^3$. It decays like $1/r$ instead of $1/r^2$. For a bounded density with compact support the integral converges absolutely, since $\abs{x - y}^{-2}$ is integrable near $y = x$ in $\R^3$, and $E$ is continuous. The unit of the electric field is the volt per metre, equal to the newton per coulomb.