Let a rod of length L carry a uniform line charge density, λ. What is the electric field at a perpendicular distance d from its midpoint? It is the vector integral of the electric field due to infinitesimal line elements. Choose the origin at the midpoint of the rod. Consider the electric field due to a line element at point x along the rod, which has a mirror element at -x. The x-components of their fields at P exactly cancel. Since this is true for each pair, the resultant field is perpendicular to the rod. The perpendicular components of each pair reinforce. Since they are of the same magnitude, the resultant field is twice the integral over half of the rod, which depends on its total charge q, L, and d. At large distances, the expression reduces to the field of a point charge, indicating that the line element is too small for its distribution to be significant. At small distances, the expression reduces to one that falls inversely with the distance.