Electric Fields of Charge Distributions

An extended distribution is handled by cutting it into elements small enough to be point charges, writing $dE = k\,dq/r^2$ with $dq = \lambda\,dx$, $\sigma\,dA$ or $\rho\,dV$, resolving into components, and integrating. Sometimes the geometry is generous: on a ring's axis every element shares one distance and one projection angle, so both leave the integral and $E = kQz/(z^2+R^2)^{3/2}$ follows in a line. Usually it is not: beside a finite rod both $r = \sqrt{x^2+y^2}$ and the projection factor vary with the element, and the result $kQ/(y\sqrt{y^2+L^2/4})$ has to be earned. Every answer gets checked against its far-field limit, where it must become $kQ/r^2$.

Four errors run through this topic. Collapsing a nearby distribution to a point at its centroid, which is exact only for spheres and only outside them. Pulling $r$ or the projection angle out of the integral in a geometry where they vary, which turns the integral back into $kQ/r^2$ by force. Ignoring a symmetry that is real, or inventing one at a field point that has no mirror image, such as near one end of a rod. And aiming the resultant along the distribution itself, parallel to the rod or tangent to the ring, when the surviving direction is the symmetry axis, precisely because the other components cancel in pairs.